1.
Three-dimensional space
–
Three-dimensional space is a geometric setting in which three values are required to determine the position of an element. This is the meaning of the term dimension. In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space, when n =3, the set of all such locations is called three-dimensional Euclidean space. It is commonly represented by the symbol ℝ3 and this serves as a three-parameter model of the physical universe in which all known matter exists. However, this space is one example of a large variety of spaces in three dimensions called 3-manifolds. Furthermore, in case, these three values can be labeled by any combination of three chosen from the terms width, height, depth, and breadth. In mathematics, analytic geometry describes every point in space by means of three coordinates. Three coordinate axes are given, each perpendicular to the two at the origin, the point at which they cross. They are usually labeled x, y, and z, below are images of the above-mentioned systems. Two distinct points determine a line. Three distinct points are either collinear or determine a unique plane, four distinct points can either be collinear, coplanar or determine the entire space. Two distinct lines can intersect, be parallel or be skew. Two parallel lines, or two intersecting lines, lie in a plane, so skew lines are lines that do not meet. Two distinct planes can either meet in a line or are parallel. Three distinct planes, no pair of which are parallel, can meet in a common line. In the last case, the three lines of intersection of each pair of planes are mutually parallel, a line can lie in a given plane, intersect that plane in a unique point or be parallel to the plane. In the last case, there will be lines in the plane that are parallel to the given line, a hyperplane is a subspace of one dimension less than the dimension of the full space. The hyperplanes of a space are the two-dimensional subspaces, that is
2.
Regular polyhedron
–
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive, in classical contexts, many different equivalent definitions are used, a common one is that faces are congruent regular polygons which are assembled in the same way around each vertex. A regular polyhedron is identified by its Schläfli symbol of the form, there are 5 finite convex regular polyhedra, known as the Platonic solids. These are the, tetrahedron, cube, octahedron, dodecahedron and icosahedron, there are also four regular star polyhedra, making nine regular polyhedra in all. All the dihedral angles of the polyhedron are equal All the vertex figures of the polyhedron are regular polygons, All the solid angles of the polyhedron are congruent. A regular polyhedron has all of three related spheres which share its centre, An insphere, tangent to all faces, an intersphere or midsphere, tangent to all edges. A circumsphere, tangent to all vertices, the regular polyhedra are the most symmetrical of all the polyhedra. They lie in just three symmetry groups, which are named after them, Tetrahedral Octahedral Icosahedral Any shapes with icosahedral or octahedral symmetry will also contain tetrahedral symmetry, the five Platonic solids have an Euler characteristic of 2. Some of the stars have a different value. The sum of the distances from any point in the interior of a polyhedron to the sides is independent of the location of the point. However, the converse does not hold, not even for tetrahedra, in a dual pair of polyhedra, the vertices of one polyhedron correspond to the faces of the other, and vice versa. The regular polyhedra show this duality as follows, The tetrahedron is self-dual, the cube and octahedron are dual to each other. The icosahedron and dodecahedron are dual to each other, the small stellated dodecahedron and great dodecahedron are dual to each other. The great stellated dodecahedron and great icosahedron are dual to each other, the Schläfli symbol of the dual is just the original written backwards, for example the dual of is. See also Regular polytope, History of discovery, stones carved in shapes resembling clusters of spheres or knobs have been found in Scotland and may be as much as 4,000 years old. Some of these stones show not only the symmetries of the five Platonic solids, examples of these stones are on display in the John Evans room of the Ashmolean Museum at Oxford University. Why these objects were made, or how their creators gained the inspiration for them, is a mystery, the earliest known written records of the regular convex solids originated from Classical Greece. When these solids were all discovered and by whom is not known, euclids reference to Plato led to their common description as the Platonic solids
3.
Convex set
–
In convex geometry, a convex set is a subset of an affine space that is closed under convex combinations. For example, a cube is a convex set, but anything that is hollow or has an indent, for example. The boundary of a set is always a convex curve. The intersection of all convex sets containing a given subset A of Euclidean space is called the hull of A. It is the smallest convex set containing A, a convex function is a real-valued function defined on an interval with the property that its epigraph is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets, the branch of mathematics devoted to the study of properties of convex sets and convex functions is called convex analysis. The notion of a set can be generalized as described below. Let S be a space over the real numbers, or, more generally. A set C in S is said to be if, for all x and y in C and all t in the interval. In other words, every point on the segment connecting x and y is in C. This implies that a set in a real or complex topological vector space is path-connected. Furthermore, C is strictly convex if every point on the segment connecting x and y other than the endpoints is inside the interior of C. A set C is called convex if it is convex. The convex subsets of R are simply the intervals of R, some examples of convex subsets of the Euclidean plane are solid regular polygons, solid triangles, and intersections of solid triangles. Some examples of convex subsets of a Euclidean 3-dimensional space are the Archimedean solids, the Kepler-Poinsot polyhedra are examples of non-convex sets. A set that is not convex is called a non-convex set, the complement of a convex set, such as the epigraph of a concave function, is sometimes called a reverse convex set, especially in the context of mathematical optimization. If S is a set in n-dimensional space, then for any collection of r, r >1. Ur in S, and for any nonnegative numbers λ1, + λr =1, then one has, ∑ k =1 r λ k u k ∈ S
4.
Polyhedron
–
In geometry, a polyhedron is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- + -hedron, a convex polyhedron is the convex hull of finitely many points, not all on the same plane. Cubes and pyramids are examples of convex polyhedra, a polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions. Convex polyhedra are well-defined, with several equivalent standard definitions, however, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of polyhedron have been given within particular contexts, some more rigorous than others, some of these definitions exclude shapes that have often been counted as polyhedra or include shapes that are often not considered as valid polyhedra. As Branko Grünbaum observed, The Original Sin in the theory of polyhedra goes back to Euclid, the writers failed to define what are the polyhedra. Nevertheless, there is agreement that a polyhedron is a solid or surface that can be described by its vertices, edges, faces. Natural refinements of this definition require the solid to be bounded, to have a connected interior, and possibly also to have a connected boundary. However, the polyhedra defined in this way do not include the self-crossing star polyhedra, their faces may not form simple polygons, definitions based on the idea of a bounding surface rather than a solid are also common. If a planar part of such a surface is not itself a convex polygon, ORourke requires it to be subdivided into smaller convex polygons, cromwell gives a similar definition but without the restriction of three edges per vertex. Again, this type of definition does not encompass the self-crossing polyhedra, however, there exist topological polyhedra that cannot be realized as acoptic polyhedra. One modern approach is based on the theory of abstract polyhedra and these can be defined as partially ordered sets whose elements are the vertices, edges, and faces of a polyhedron. A vertex or edge element is less than an edge or face element when the vertex or edge is part of the edge or face, additionally, one may include a special bottom element of this partial order and a top element representing the whole polyhedron. However, these requirements are relaxed, to instead require only that the sections between elements two levels apart from line segments. Geometric polyhedra, defined in other ways, can be described abstractly in this way, a realization of an abstract polyhedron is generally taken to be a mapping from the vertices of the abstract polyhedron to geometric points, such that the points of each face are coplanar. A geometric polyhedron can then be defined as a realization of an abstract polyhedron, realizations that forgo the requirement of planarity, that impose additional requirements of symmetry, or that map the vertices to higher dimensional spaces have also been considered. Unlike the solid-based and surface-based definitions, this perfectly well for star polyhedra. However, without restrictions, this definition allows degenerate or unfaithful polyhedra
5.
Congruence (geometry)
–
In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. This means that either object can be repositioned and reflected so as to coincide precisely with the other object, so two distinct plane figures on a piece of paper are congruent if we can cut them out and then match them up completely. Turning the paper over is permitted, in elementary geometry the word congruent is often used as follows. The word equal is often used in place of congruent for these objects, two line segments are congruent if they have the same length. Two angles are congruent if they have the same measure, two circles are congruent if they have the same diameter. The related concept of similarity applies if the objects differ in size, for two polygons to be congruent, they must have an equal number of sides. Two polygons with n sides are congruent if and only if they each have identical sequences side-angle-side-angle-. for n sides. Congruence of polygons can be established graphically as follows, First, match, second, draw a vector from one of the vertices of the one of the figures to the corresponding vertex of the other figure. Translate the first figure by this vector so that two vertices match. Third, rotate the translated figure about the matched vertex until one pair of corresponding sides matches, fourth, reflect the rotated figure about this matched side until the figures match. If at any time the step cannot be completed, the polygons are not congruent, two triangles are congruent if their corresponding sides are equal in length, in which case their corresponding angles are equal in measure. SSS, If three pairs of sides of two triangles are equal in length, then the triangles are congruent, ASA, If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. The ASA Postulate was contributed by Thales of Miletus, in most systems of axioms, the three criteria—SAS, SSS and ASA—are established as theorems. In the School Mathematics Study Group system SAS is taken as one of 22 postulates, AAS, If two pairs of angles of two triangles are equal in measurement, and a pair of corresponding non-included sides are equal in length, then the triangles are congruent. For American usage, AAS is equivalent to an ASA condition, RHS, also known as HL, If two right-angled triangles have their hypotenuses equal in length, and a pair of shorter sides are equal in length, then the triangles are congruent. The SSA condition which specifies two sides and a non-included angle does not by itself prove congruence, in order to show congruence, additional information is required such as the measure of the corresponding angles and in some cases the lengths of the two pairs of corresponding sides. The opposite side is longer when the corresponding angles are acute. This is the case and two different triangles can be formed from the given information, but further information distinguishing them can lead to a proof of congruence
6.
Regular polygon
–
In Euclidean geometry, a regular polygon is a polygon that is equiangular and equilateral. Regular polygons may be convex or star, in the limit, a sequence of regular polygons with an increasing number of sides becomes a circle, if the perimeter is fixed, or a regular apeirogon, if the edge length is fixed. These properties apply to all regular polygons, whether convex or star, a regular n-sided polygon has rotational symmetry of order n. All vertices of a regular polygon lie on a common circle and that is, a regular polygon is a cyclic polygon. Together with the property of equal-length sides, this implies that every regular polygon also has a circle or incircle that is tangent to every side at the midpoint. Thus a regular polygon is a tangential polygon, a regular n-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes. The symmetry group of a regular polygon is dihedral group Dn, D2, D3. It consists of the rotations in Cn, together with reflection symmetry in n axes that pass through the center, if n is even then half of these axes pass through two opposite vertices, and the other half through the midpoint of opposite sides. If n is odd then all pass through a vertex. All regular simple polygons are convex and those having the same number of sides are also similar. An n-sided convex regular polygon is denoted by its Schläfli symbol, for n <3 we have two degenerate cases, Monogon, degenerate in ordinary space. Digon, a line segment, degenerate in ordinary space. In certain contexts all the polygons considered will be regular, in such circumstances it is customary to drop the prefix regular. For instance, all the faces of uniform polyhedra must be regular, for n >2 the number of diagonals is n 2, i. e.0,2,5,9. for a triangle, square, pentagon, hexagon. The diagonals divide the polygon into 1,4,11,24, for a regular n-gon inscribed in a unit-radius circle, the product of the distances from a given vertex to all other vertices equals n. For a regular simple n-gon with circumradius R and distances di from a point in the plane to the vertices. For a regular n-gon, the sum of the distances from any interior point to the n sides is n times the apothem. This is a generalization of Vivianis theorem for the n=3 case, the sum of the perpendiculars from a regular n-gons vertices to any line tangent to the circumcircle equals n times the circumradius
7.
Polygon
–
In elementary geometry, a polygon /ˈpɒlɪɡɒn/ is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit. These segments are called its edges or sides, and the points where two edges meet are the vertices or corners. The interior of the polygon is called its body. An n-gon is a polygon with n sides, for example, a polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The basic geometrical notion of a polygon has been adapted in various ways to suit particular purposes, mathematicians are often concerned only with the bounding closed polygonal chain and with simple polygons which do not self-intersect, and they often define a polygon accordingly. A polygonal boundary may be allowed to intersect itself, creating star polygons and these and other generalizations of polygons are described below. The word polygon derives from the Greek adjective πολύς much, many and it has been suggested that γόνυ knee may be the origin of “gon”. Polygons are primarily classified by the number of sides, Polygons may be characterized by their convexity or type of non-convexity, Convex, any line drawn through the polygon meets its boundary exactly twice. As a consequence, all its interior angles are less than 180°, equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. Non-convex, a line may be found which meets its boundary more than twice, equivalently, there exists a line segment between two boundary points that passes outside the polygon. Simple, the boundary of the polygon does not cross itself, there is at least one interior angle greater than 180°. Star-shaped, the interior is visible from at least one point. The polygon must be simple, and may be convex or concave, self-intersecting, the boundary of the polygon crosses itself. Branko Grünbaum calls these coptic, though this term does not seem to be widely used, star polygon, a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped, equiangular, all corner angles are equal. Cyclic, all lie on a single circle, called the circumcircle. Isogonal or vertex-transitive, all lie within the same symmetry orbit. The polygon is cyclic and equiangular
8.
Face (geometry)
–
In solid geometry, a face is a flat surface that forms part of the boundary of a solid object, a three-dimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, in elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include side of a polyhedron, for example, any of the six squares that bound a cube is a face of the cube. Sometimes face is used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, some other polygons, which are not faces, are also important for polyhedra and tessellations. These include Petrie polygons, vertex figures and facets, any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Eulers polyhedron formula, thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, in higher-dimensional geometry the faces of a polytope are features of all dimensions. A face of dimension k is called a k-face, for example, the polygonal faces of an ordinary polyhedron are 2-faces. In set theory, the set of faces of a polytope includes the polytope itself, for any n-polytope, −1 ≤ k ≤ n. For example, with meaning, the faces of a cube include the empty set, its vertices, edges and squares. Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P, from this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, abstract theory still requires that the set of faces include the polytope itself and the empty set. A cell is an element of a 4-dimensional polytope or 3-dimensional tessellation. Cells are facets for 4-polytopes and 3-honeycombs, examples, In higher-dimensional geometry, the facets of a n-polytope are the -faces of dimension one less than the polytope itself. A polytope is bounded by its facets, for example, The facets of a line segment are its 0-faces or vertices. The facets of a polygon are its 1-faces or edges, the facets of a polyhedron or plane tiling are its 2-faces. The facets of a 4D polytope or 3-honeycomb are its 3-faces, the facets of a 5D polytope or 4-honeycomb are its 4-faces
9.
Vertex (geometry)
–
In geometry, a vertex is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. A vertex is a point of a polygon, polyhedron, or other higher-dimensional polytope. However, in theory, vertices may have fewer than two incident edges, which is usually not allowed for geometric vertices. However, a smooth approximation to a polygon will also have additional vertices. A polygon vertex xi of a simple polygon P is a principal polygon vertex if the diagonal intersects the boundary of P only at x and x, there are two types of principal vertices, ears and mouths. A principal vertex xi of a simple polygon P is called an ear if the diagonal that bridges xi lies entirely in P, according to the two ears theorem, every simple polygon has at least two ears. A principal vertex xi of a simple polygon P is called a mouth if the diagonal lies outside the boundary of P. Any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges and this equation is known as Eulers polyhedron formula. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces, for example, a cube has 12 edges and 6 faces, and hence 8 vertices
10.
Tetrahedron
–
In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra, the tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a polygon base. In the case of a tetrahedron the base is a triangle, like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. For any tetrahedron there exists a sphere on which all four vertices lie, a regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have known since antiquity. In a regular tetrahedron, not only are all its faces the same size and shape, regular tetrahedra alone do not tessellate, but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron, the compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. This form has Coxeter diagram and Schläfli symbol h, the tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, and the together form the stellated octahedron. In other words, if C is the centroid of the base and this follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other. The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron, the symmetries of a regular tetrahedron correspond to half of those of a cube, those that map the tetrahedra to themselves, and not to each other. The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion, the regular tetrahedron has 24 isometries, forming the symmetry group Td, isomorphic to the symmetric group, S4. The first corresponds to the A2 Coxeter plane, the two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these intersects the tetrahedron the resulting cross section is a rectangle. When the intersecting plane is one of the edges the rectangle is long. When halfway between the two edges the intersection is a square, the aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly, if the tetrahedron is bisected on this plane, both halves become wedges
11.
Cube
–
Beryllium copper, also known as copper beryllium, beryllium bronze and spring copper, is a copper alloy with 0. 5—3% beryllium and sometimes other elements. Beryllium copper combines high strength with non-magnetic and non-sparking qualities and it has excellent metalworking, forming and machining properties. It has many specialized applications in tools for hazardous environments, musical instruments, precision measurement devices, bullets, beryllium alloys present a toxic inhalation hazard during manufacture. Beryllium copper is a ductile, weldable, and machinable alloy and it is resistant to non-oxidizing acids, to plastic decomposition products, to abrasive wear, and to galling. It can be heat-treated for increased strength, durability, and electrical conductivity, beryllium copper attains the greatest strength of any copper-based alloy. In solid form and as finished objects, beryllium copper presents no known health hazard, however, inhalation of dust, mist, or fume containing beryllium can cause the serious lung condition, chronic beryllium disease. That disease affects primarily the lungs, restricting the exchange of oxygen between the lungs and the bloodstream, the International Agency for Research on Cancer lists beryllium as a Group 1 Human Carcinogen. The National Toxicology Program also lists beryllium as a carcinogen, beryllium copper is a non-ferrous alloy used in springs, spring wire, load cells, and other parts that must retain their shape under repeated stress and strain. It has high electrical conductivity, and is used in low-current contacts for batteries, beryllium copper is non-sparking but physically tough and nonmagnetic, fulfilling the requirements of ATEX directive for Zones 0,1, and 2. Beryllium copper screwdrivers, pliers, wrenches, cold chisels, knives, and hammers are available for environments with explosive hazards, such oil rigs, coal mines, an alternative metal sometimes used for non-sparking tools is aluminium bronze. Compared to steel tools, beryllium copper tools are more expensive, not as strong, and less durable, beryllium copper is frequently used for percussion instruments for its consistent tone and resonance, especially tambourines and triangles. Beryllium copper has been used for armour piercing bullets, though usage is unusual because bullets made from steel alloys are much less expensive and have similar properties. Beryllium copper is used for measurement-while-drilling tools in the drilling industry. A non-magnetic alloy is required, as magnetometers are used for field-strength data received from the tool, beryllium copper gaskets are used to create an RF-tight, electronic seal on doors used with EMC testing and anechoic chambers. For a time, beryllium copper was used in the manufacture of clubs, particularly wedges. Though some golfers prefer the feel of BeCu club heads, regulatory issues, kiefer Plating of Elkhart, Indiana built some beryllium-copper trumpet bells for the Schilke Music Co. of Chicago. These light-weight bells produce a sound preferred by some musicians, beryllium copper wire is produced in many forms, round, square, flat and shaped, in coils, on spools and in straight lengths. Beryllium copper valve seats and guides are used in high performance engines with coated titanium valves
12.
Octahedron
–
In geometry, an octahedron is a polyhedron with eight faces, twelve edges, and six vertices. A regular octahedron is a Platonic solid composed of eight equilateral triangles, a regular octahedron is the dual polyhedron of a cube. It is a square bipyramid in any of three orthogonal orientations and it is also a triangular antiprism in any of four orientations. An octahedron is the case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan metric, the second and third correspond to the B2 and A2 Coxeter planes. The octahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. An octahedron with edge length √2 can be placed with its center at the origin and its vertices on the coordinate axes, the Cartesian coordinates of the vertices are then. In an x–y–z Cartesian coordinate system, the octahedron with center coordinates, additionally the inertia tensor of the stretched octahedron is I =. These reduce to the equations for the regular octahedron when x m = y m = z m = a 22, the interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. Correspondingly, an octahedron is the result of cutting off from a regular tetrahedron. One can also divide the edges of an octahedron in the ratio of the mean to define the vertices of an icosahedron. There are five octahedra that define any given icosahedron in this fashion, octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space, called the octet truss by Buckminster Fuller. This is the only such tiling save the regular tessellation of cubes, another is a tessellation of octahedra and cuboctahedra. The octahedron is unique among the Platonic solids in having a number of faces meeting at each vertex. Consequently, it is the member of that group to possess mirror planes that do not pass through any of the faces. Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid, truncation of two opposite vertices results in a square bifrustum. The octahedron is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices and it is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size
13.
Regular dodecahedron
–
It is one of the five Platonic solids. It has 12 faces,20 vertices,30 edges, and 160 diagonals. Note that, given a regular dodecahedron of edge length one, ru is the radius of a sphere about a cube of edge length ϕ. In perspective projection, viewed above a face, the regular dodecahedron can be seen as a linear-edged schlegel diagram. These projections are used in showing the four-dimensional 120-cell, a regular 4-dimensional polytope, constructed from 120 dodecahedra. The regular dodecahedron can also be represented as a spherical tiling, the following Cartesian coordinates define the 20 vertices of a regular dodecahedron centered at the origin and suitably scaled and oriented, where ϕ =1 + √5/2 is the golden ratio ≈1.618. The edge length is 2/ϕ = √5 −1, the containing sphere has a radius of √3. 5650512°. A137218 If the original regular dodecahedron has edge length 1, its dual icosahedron has edge length ϕ, If the five Platonic solids are built with same volume, the regular dodecahedron has the shortest edges. The map-coloring number of a regular dodecahedrons faces is 4, the distance between the vertices on the same face not connected by an edge is ϕ times the edge length. If two edges share a vertex, then the midpoints of those edges form an equilateral triangle with the body center. The regular dodecahedron is the third in a set of truncated trapezohedra which can be constructed by truncating the two axial vertices of a pentagonal trapezohedron. The stellations of the regular dodecahedron make up three of the four Kepler–Poinsot polyhedra, a rectified regular dodecahedron forms an icosidodecahedron. The regular dodecahedron has icosahedral symmetry Ih, Coxeter group, order 120, when a regular dodecahedron is inscribed in a sphere, it occupies more of the spheres volume than an icosahedron inscribed in the same sphere. A regular dodecahedron has 12 faces and 20 vertices, whereas a regular icosahedron has 20 faces and 12 vertices, a cube can be embedded within a regular dodecahedron, affixed to eight of its equidistant vertices, in five different positions. In fact, five cubes may overlap and interlock inside the regular dodecahedron to result in the compound of five cubes, the ratio of the edge of a regular dodecahedron to the edge of a cube embedded inside such a regular dodecahedron is 1, ϕ, or,1. The ratio of a regular dodecahedrons volume to the volume of a cube embedded inside such a regular dodecahedron is 1, 2/2 + ϕ, or 1 + ϕ/2,1, or,4. For example, a cube with a volume of 64. Thus, the difference in volume between the regular dodecahedron and the enclosed cube is always one half the volume of the cube times ϕ
14.
Regular icosahedron
–
In geometry, a regular icosahedron is a convex polyhedron with 20 faces,30 edges and 12 vertices. It is one of the five Platonic solids, and also the one with the most sides and it has five equilateral triangular faces meeting at each vertex. It is represented by its Schläfli symbol, or sometimes by its vertex figure as 3.3.3.3.3 or 35 and it is the dual of the dodecahedron, which is represented by, having three pentagonal faces around each vertex. A regular icosahedron is a pentagonal bipyramid and a biaugmented pentagonal antiprism in any of six orientations. The name comes from Greek εἴκοσι, meaning twenty, and ἕδρα, the plural can be either icosahedrons or icosahedra. The surface area A and the volume V of a regular icosahedron of edge length a are, note that these vertices form five sets of three concentric, mutually orthogonal golden rectangles, whose edges form Borromean rings. If the original icosahedron has edge length 1, its dual dodecahedron has edge length √5 − 1/2 = 1/ϕ = ϕ −1, the 12 edges of a regular octahedron can be subdivided in the golden ratio so that the resulting vertices define a regular icosahedron. The locations of the vertices of a regular icosahedron can be described using spherical coordinates, if two vertices are taken to be at the north and south poles, then the other ten vertices are at latitude ±arctan ≈ ±26. 57°. These ten vertices are at evenly spaced longitudes, alternating between north and south latitudes and this projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane, an icosahedron has 43,380 distinct nets. To color the icosahedron, such that no two adjacent faces have the color, requires at least 3 colors. A problem dating back to the ancient Greeks is to determine which of two shapes has larger volume, an icosahedron inscribed in a sphere, or a dodecahedron inscribed in the same sphere, the problem was solved by Hero, Pappus, and Fibonacci, among others. Apollonius of Perga discovered the result that the ratio of volumes of these two shapes is the same as the ratio of their surface areas. Both volumes have formulas involving the golden ratio, but taken to different powers, as it turns out, the icosahedron occupies less of the spheres volume than the dodecahedron. The following construction of the icosahedron avoids tedious computations in the number field ℚ necessary in more elementary approaches, the existence of the icosahedron amounts to the existence of six equiangular lines in ℝ3. Indeed, intersecting such a system of lines with a Euclidean sphere centered at their common intersection yields the twelve vertices of a regular icosahedron as can easily be checked. Conversely, supposing the existence of an icosahedron, lines defined by its six pairs of opposite vertices form an equiangular system. In order to such an equiangular system, we start with this 6 ×6 square matrix
15.
Mathematical beauty
–
Mathematical beauty describes the notion that some mathematicians may derive aesthetic pleasure from their work, and from mathematics in general. They express this pleasure by describing mathematics as beautiful, Mathematicians describe mathematics as an art form or, at a minimum, as a creative activity. Comparisons are often made music and poetry. The true spirit of delight, the exaltation, the sense of being more than Man, Paul Erdős expressed his views on the ineffability of mathematics when he said, Why are numbers beautiful. Its like asking why is Beethovens Ninth Symphony beautiful, if you dont see why, someone cant tell you. If they arent beautiful, nothing is, Mathematicians describe an especially pleasing method of proof as elegant. Depending on context, this may mean, A proof that uses a minimum of additional assumptions or previous results, a proof that is unusually succinct. A proof that derives a result in a surprising way A proof that is based on new, a method of proof that can be easily generalized to solve a family of similar problems. In the search for an elegant proof, mathematicians often look for different independent ways to prove a result—the first proof that is found may not be the best. The theorem for which the greatest number of different proofs have been discovered is possibly the Pythagorean theorem, another theorem that has been proved in many different ways is the theorem of quadratic reciprocity—Carl Friedrich Gauss alone published eight different proofs of this theorem. Some mathematicians see beauty in mathematical results that establish connections between two areas of mathematics that at first sight appear to be unrelated and these results are often described as deep. While it is difficult to find agreement on whether a result is deep. One is Eulers identity, e i π +1 =0 and this is a special case of Eulers formula, which the physicist Richard Feynman called our jewel and the most remarkable formula in mathematics. Other examples of deep results include unexpected insights into mathematical structures, for example, Gausss Theorema Egregium is a deep theorem which relates a local phenomenon to a global phenomenon in a surprising way. In particular, the area of a triangle on a surface is proportional to the excess of the triangle. Another example is the theorem of calculus. The opposite of deep is trivial, sometimes, however, a statement of a theorem can be original enough to be considered deep, even though its proof is fairly obvious. In his A Mathematicians Apology, Hardy suggests that a proof or result possesses inevitability, unexpectedness
16.
Symmetry
–
Symmetry in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, symmetry has a precise definition, that an object is invariant to any of various transformations. Although these two meanings of symmetry can sometimes be told apart, they are related, so they are discussed together. The opposite of symmetry is asymmetry, a geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object, an object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape. An object has symmetry if it can be translated without changing its overall shape. An object has symmetry if it can be simultaneously translated and rotated in three-dimensional space along a line known as a screw axis. An object has symmetry if it does not change shape when it is expanded or contracted. Fractals also exhibit a form of symmetry, where small portions of the fractal are similar in shape to large portions. Other symmetries include glide reflection symmetry and rotoreflection symmetry, a dyadic relation R is symmetric if and only if, whenever its true that Rab, its true that Rba. Thus, is the age as is symmetrical, for if Paul is the same age as Mary. Symmetric binary logical connectives are and, or, biconditional, nand, xor, the set of operations that preserve a given property of the object form a group. In general, every kind of structure in mathematics will have its own kind of symmetry, examples include even and odd functions in calculus, the symmetric group in abstract algebra, symmetric matrices in linear algebra, and the Galois group in Galois theory. In statistics, it appears as symmetric probability distributions, and as skewness, symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations. This concept has one of the most powerful tools of theoretical physics. See Noethers theorem, and also, Wigners classification, which says that the symmetries of the laws of physics determine the properties of the found in nature. Important symmetries in physics include continuous symmetries and discrete symmetries of spacetime, internal symmetries of particles, in biology, the notion of symmetry is mostly used explicitly to describe body shapes. Bilateral animals, including humans, are more or less symmetric with respect to the plane which divides the body into left
17.
Ancient Greek philosophy
–
Ancient Greek philosophy arose in the 6th century BC and continued throughout the Hellenistic period and the period in which Ancient Greece was part of the Roman Empire. Philosophy was used to sense out of the world in a non-religious way. It dealt with a variety of subjects, including political philosophy, ethics, metaphysics, ontology, logic, biology, rhetoric. Many philosophers around the world agree that Greek philosophy has influenced much of Western culture since its inception, alfred North Whitehead once noted, The safest general characterization of the European philosophical tradition is that it consists of a series of footnotes to Plato. Clear, unbroken lines of lead from ancient Greek and Hellenistic philosophers to Early Islamic philosophy, the European Renaissance. Some claim that Greek philosophy, in turn, was influenced by the wisdom literature. But they taught themselves to reason, Philosophy as we understand it is a Greek creation. Subsequent philosophic tradition was so influenced by Socrates as presented by Plato that it is conventional to refer to philosophy developed prior to Socrates as pre-Socratic philosophy. The periods following this, up to and after the wars of Alexander the Great, are those of classical Greek, the pre-Socratics were primarily concerned with cosmology, ontology and mathematics. They were distinguished from non-philosophers insofar as they rejected mythological explanations in favor of reasoned discourse, Thales of Miletus, regarded by Aristotle as the first philosopher, held that all things arise from water. It is not because he gave a cosmogony that John Burnet calls him the first man of science, according to tradition, Thales was able to predict an eclipse and taught the Egyptians how to measure the height of the pyramids. He began from the observation that the world seems to consist of opposites, therefore, they cannot truly be opposites but rather must both be manifestations of some underlying unity that is neither. This underlying unity could not be any of the classical elements, for example, water is wet, the opposite of dry, while fire is dry, the opposite of wet. Anaximenes in turn held that the arche was air, although John Burnet argues that by this he meant that it was a transparent mist, the aether. Xenophanes was born in Ionia, where the Milesian school was at its most powerful, Burnet says that Xenophanes was not, however, a scientific man, with many of his naturalistic explanations having no further support than that they render the Homeric gods superfluous or foolish. He has been claimed as an influence on Eleatic philosophy, although that is disputed, and a precursor to Epicurus, a representative of a total break between science and religion. Pythagoras lived at roughly the time that Xenophanes did and, in contrast to the latter. Parmenides of Elea cast his philosophy against those who held it is and is not the same, and all travel in opposite directions, —presumably referring to Heraclitus
18.
Plato
–
Plato was a philosopher in Classical Greece and the founder of the Academy in Athens, the first institution of higher learning in the Western world. He is widely considered the most pivotal figure in the development of philosophy, unlike nearly all of his philosophical contemporaries, Platos entire work is believed to have survived intact for over 2,400 years. Along with his teacher, Socrates, and his most famous student, Aristotle, Plato laid the foundations of Western philosophy. Alfred North Whitehead once noted, the safest general characterization of the European philosophical tradition is that it consists of a series of footnotes to Plato. In addition to being a figure for Western science, philosophy. Friedrich Nietzsche, amongst other scholars, called Christianity, Platonism for the people, Plato was the innovator of the written dialogue and dialectic forms in philosophy, which originate with him. He was not the first thinker or writer to whom the word “philosopher” should be applied, few other authors in the history of Western philosophy approximate him in depth and range, perhaps only Aristotle, Aquinas and Kant would be generally agreed to be of the same rank. Due to a lack of surviving accounts, little is known about Platos early life, the philosopher came from one of the wealthiest and most politically active families in Athens. Ancient sources describe him as a bright though modest boy who excelled in his studies, the exact time and place of Platos birth are unknown, but it is certain that he belonged to an aristocratic and influential family. Based on ancient sources, most modern scholars believe that he was born in Athens or Aegina between 429 and 423 BCE. According to a tradition, reported by Diogenes Laertius, Ariston traced his descent from the king of Athens, Codrus. Platos mother was Perictione, whose family boasted of a relationship with the famous Athenian lawmaker, besides Plato himself, Ariston and Perictione had three other children, these were two sons, Adeimantus and Glaucon, and a daughter Potone, the mother of Speusippus. The brothers Adeimantus and Glaucon are mentioned in the Republic as sons of Ariston, and presumably brothers of Plato, but in a scenario in the Memorabilia, Xenophon confused the issue by presenting a Glaucon much younger than Plato. Then, at twenty-eight, Hermodorus says, went to Euclides in Megara, as Debra Nails argues, The text itself gives no reason to infer that Plato left immediately for Megara and implies the very opposite. Thus, Nails dates Platos birth to 424/423, another legend related that, when Plato was an infant, bees settled on his lips while he was sleeping, an augury of the sweetness of style in which he would discourse about philosophy. Ariston appears to have died in Platos childhood, although the dating of his death is difficult. Perictione then married Pyrilampes, her mothers brother, who had served many times as an ambassador to the Persian court and was a friend of Pericles, Pyrilampes had a son from a previous marriage, Demus, who was famous for his beauty. Perictione gave birth to Pyrilampes second son, Antiphon, the half-brother of Plato and these and other references suggest a considerable amount of family pride and enable us to reconstruct Platos family tree
19.
Timaeus (dialogue)
–
Timaeus is one of Platos dialogues, mostly in the form of a long monologue given by the title character Timaeus of Locri, written c.360 BC. The work puts forward speculation on the nature of the physical world, participants in the dialogue include Socrates, Timaeus, Hermocrates, and Critias. Some scholars believe that it is not the Critias of the Thirty Tyrants who is appearing in this dialogue, but his grandfather and it has been suggested that Timaeus influenced a book about Pythagoras, written by Philolaus. The dialogue takes place the day after Socrates described his ideal state, in Platos works such a discussion occurs in the Republic. Hermocrates wishes to oblige Socrates and mentions that Critias knows just the account to do so, Critias proceeds to tell the story of Solons journey to Egypt where he hears the story of Atlantis, and how Athens used to be an ideal state that subsequently waged war against Atlantis. Critias believes that he is getting ahead of himself, and mentions that Timaeus will tell part of the account from the origin of the universe to man, the history of Atlantis is postponed to Critias. The main content of the dialogue, the exposition by Timaeus, Timaeus begins with a distinction between the physical world, and the eternal world. The physical one is the world changes and perishes, therefore it is the object of opinion. The eternal one never changes, therefore it is apprehended by reason, the speeches about the two worlds are conditioned by the different nature of their objects. Indeed, a description of what is changeless, fixed and clearly intelligible will be changeless and fixed, while a description of changes and is likely, will also change. As being is to becoming, so is truth to belief, therefore, in a description of the physical world, one should not look for anything more than a likely story. Timaeus suggests that since nothing becomes or changes without cause, then the cause of the universe must be a demiurge or a god, and since the universe is fair, the demiurge must have looked to the eternal model to make it, and not to the perishable one. Hence, using the eternal and perfect world of forms or ideals as a template, he set about creating our world, Timaeus continues with an explanation of the creation of the universe, which he ascribes to the handiwork of a divine craftsman. The demiurge, being good, wanted there to be as good as was the world. The demiurge is said to bring out of substance by imitating an unchanging. The ananke, often translated as necessity, was the only other co-existent element or presence in Platos cosmogony, later Platonists clarified that the eternal model existed in the mind of the Demiurge. Timaeus describes the substance as a lack of homogeneity or balance, in which the four elements were shapeless, mixed, considering that order is favourable over disorder, the essential act of the creator was to bring order and clarity to this substance. Therefore, all the properties of the world are to be explained by the choice of what is fair and good, or
20.
Classical element
–
Ancient cultures in Egypt, Babylonia, Japan, Tibet, and India had similar lists, sometimes referring in local languages to air as wind and the fifth element as void. The Chinese Wu Xing system lists Wood, Fire, Earth, Metal and these different cultures and even individual philosophers had widely varying explanations concerning their attributes and how they related to observable phenomena as well as cosmology. Sometimes these theories overlapped with mythology and were personified in deities, some of these interpretations included atomism but other interpretations considered the elements to be divisible into infinitely small pieces without changing their nature. Centuries of empirical investigation have proven that all the ancient systems were incorrect explanations of the physical world. It is now known that atomic theory is an explanation, and that atoms can be classified into more than a hundred chemical elements such as oxygen, iron. These elements form chemical compounds and mixtures, and under different temperatures and pressures, the concept of the five elements formed a basis of analysis in both Hinduism and Buddhism. In Hinduism, particularly in a context, the four states-of-matter describe matter. Similar lists existed in ancient China and Japan, in Buddhism the four great elements, to which two others are sometimes added, are not viewed as substances, but as categories of sensory experience. A Greek text called the Kore Kosmou ascribed to Hermes Trismegistus, names the four fire, water, air. And, on the contrary, again some are made enemies of fire, and some of water, some of earth, and some of air, and some of two of them, and some of three, and some of all. For instance, son, the locust and all flies flee fire, the eagle and the hawk and all high-flying birds flee water, fish, air and earth, the snake avoids the open air. Not that some of the animals as well do not love fire, for instance salamanders and it is because one or another of the elements doth form their bodies outer envelope. Each soul, accordingly, while it is in its body is weighted and constricted by these four. According to Galen, these elements were used by Hippocrates in describing the body with an association with the four humours, yellow bile, black bile, blood. Medical care was flexible and primarily about helping the patient stay in or return to his/her own personal natural balanced state. In other Babylonian texts these phenomena are considered independent of their association with deities, though they are not treated as the component elements of the universe, the five elements are associated with the five senses, and act as the gross medium for the experience of sensations. The basest element, earth, created using all the elements, can be perceived by all five senses – hearing, touch, sight, taste. The next higher element, water, has no odor but can be heard, felt, seen, next comes fire, which can be heard, felt and seen
21.
Johannes Kepler
–
Johannes Kepler was a German mathematician, astronomer, and astrologer. A key figure in the 17th-century scientific revolution, he is best known for his laws of motion, based on his works Astronomia nova, Harmonices Mundi. These works also provided one of the foundations for Isaac Newtons theory of universal gravitation, Kepler was a mathematics teacher at a seminary school in Graz, where he became an associate of Prince Hans Ulrich von Eggenberg. Later he became an assistant to the astronomer Tycho Brahe in Prague and he was also a mathematics teacher in Linz, and an adviser to General Wallenstein. Kepler lived in an era when there was no distinction between astronomy and astrology, but there was a strong division between astronomy and physics. Kepler was born on December 27, the feast day of St John the Evangelist,1571 and his grandfather, Sebald Kepler, had been Lord Mayor of the city. By the time Johannes was born, he had two brothers and one sister and the Kepler family fortune was in decline and his father, Heinrich Kepler, earned a precarious living as a mercenary, and he left the family when Johannes was five years old. He was believed to have died in the Eighty Years War in the Netherlands and his mother Katharina Guldenmann, an innkeepers daughter, was a healer and herbalist. Born prematurely, Johannes claimed to have weak and sickly as a child. Nevertheless, he often impressed travelers at his grandfathers inn with his phenomenal mathematical faculty and he was introduced to astronomy at an early age, and developed a love for it that would span his entire life. At age six, he observed the Great Comet of 1577, in 1580, at age nine, he observed another astronomical event, a lunar eclipse, recording that he remembered being called outdoors to see it and that the moon appeared quite red. However, childhood smallpox left him with vision and crippled hands. In 1589, after moving through grammar school, Latin school, there, he studied philosophy under Vitus Müller and theology under Jacob Heerbrand, who also taught Michael Maestlin while he was a student, until he became Chancellor at Tübingen in 1590. He proved himself to be a mathematician and earned a reputation as a skilful astrologer. Under the instruction of Michael Maestlin, Tübingens professor of mathematics from 1583 to 1631 and he became a Copernican at that time. In a student disputation, he defended heliocentrism from both a theoretical and theological perspective, maintaining that the Sun was the source of motive power in the universe. Despite his desire to become a minister, near the end of his studies, Kepler was recommended for a position as teacher of mathematics and he accepted the position in April 1594, at the age of 23. Keplers first major work, Mysterium Cosmographicum, was the first published defense of the Copernican system
22.
Solar System
–
The Solar System is the gravitationally bound system comprising the Sun and the objects that orbit it, either directly or indirectly. Of those objects that orbit the Sun directly, the largest eight are the planets, with the remainder being significantly smaller objects, such as dwarf planets, of the objects that orbit the Sun indirectly, the moons, two are larger than the smallest planet, Mercury. The Solar System formed 4.6 billion years ago from the collapse of a giant interstellar molecular cloud. The vast majority of the mass is in the Sun. The four smaller inner planets, Mercury, Venus, Earth and Mars, are terrestrial planets, being composed of rock. The four outer planets are giant planets, being more massive than the terrestrials. All planets have almost circular orbits that lie within a flat disc called the ecliptic. The Solar System also contains smaller objects, the asteroid belt, which lies between the orbits of Mars and Jupiter, mostly contains objects composed, like the terrestrial planets, of rock and metal. Beyond Neptunes orbit lie the Kuiper belt and scattered disc, which are populations of trans-Neptunian objects composed mostly of ices, within these populations are several dozen to possibly tens of thousands of objects large enough that they have been rounded by their own gravity. Such objects are categorized as dwarf planets, identified dwarf planets include the asteroid Ceres and the trans-Neptunian objects Pluto and Eris. In addition to two regions, various other small-body populations, including comets, centaurs and interplanetary dust clouds. Six of the planets, at least four of the dwarf planets, each of the outer planets is encircled by planetary rings of dust and other small objects. The solar wind, a stream of charged particles flowing outwards from the Sun, the heliopause is the point at which pressure from the solar wind is equal to the opposing pressure of the interstellar medium, it extends out to the edge of the scattered disc. The Oort cloud, which is thought to be the source for long-period comets, the Solar System is located in the Orion Arm,26,000 light-years from the center of the Milky Way. For most of history, humanity did not recognize or understand the concept of the Solar System, the invention of the telescope led to the discovery of further planets and moons. The principal component of the Solar System is the Sun, a G2 main-sequence star that contains 99. 86% of the known mass. The Suns four largest orbiting bodies, the giant planets, account for 99% of the mass, with Jupiter. The remaining objects of the Solar System together comprise less than 0. 002% of the Solar Systems total mass, most large objects in orbit around the Sun lie near the plane of Earths orbit, known as the ecliptic
23.
Mysterium Cosmographicum
–
Mysterium Cosmographicum is an astronomy book by the German astronomer Johannes Kepler, published at Tübingen in 1596 and in a second edition in 1621. This was the first attempt since Copernicus to say that the theory of heliocentrism is physically true, from this he realized that he had stumbled on the same ratio between the orbits of Saturn and Jupiter. He wrote, By a certain mere accident I chanced to come closer to the state of affairs. I thought it was by divine intervention that I gained fortuitously what I was never able to obtain by any amount of toil. But after doing further calculations he realized he could not use the two-dimensional polygons to represent all the planets, Johannes Keplers first major astronomical work, Mysterium Cosmographicum, was the first published defense of the Copernican system. After failing to find an arrangement of polygons that fit known astronomical observations. However, Kepler later rejected this formula, because it was not precise enough, as he indicated in the title, Kepler thought he had revealed God’s geometrical plan for the universe. His first manuscript of Mysterium contained an extensive chapter reconciling heliocentrism with biblical passages that seemed to support geocentrism, the effusive dedication, to powerful patrons as well as to the men who controlled his position in Graz, also provided a crucial doorway into the patronage system. Though the details would be modified in light of his later work, jardine has pointed out that it would be sounder to read Keplers CU more as a work against skepticism than in the context of the modern realism/instrumentalism debate. On the one hand, “causality” is a notion implying the most general idea of “actual scientific knowledge” which guides and stimulates each investigation. In this sense, Kepler already embarked in his MC on an investigation by asking for the cause of the number, the sizes. In terms of the impact of Mysterium, it can be seen as an important first step in modernizing the theory proposed by Nicolaus Copernicus in his De Revolutionibus. Kepler regarded Copernicus arguments as being merely a posteriori and set out to provide the a priori demonstrations of the astronomical phenomena, especially when dealing with the geometry of the universe, Kepler consistently utilizes Platonic and Neo-Platonic frameworks of thought. The entirety of the idea is based on the same formal cause postulated by Plato for the structure of the universe. In addition, an efficient cause is need to answer the reason for the structure. In an argument from design, Kepler postulates the existence and necessity of God the Creator as this efficient cause, because he was promised use of these observations by Brahe, Kepler sought him out in the beginning of 1600. Brahe, however, only gave him the data on Mars, the Mysterium Cosmographicum was featured on the Austrian 10 euro Johannes Kepler silver commemorative coin minted in 2002. George W. Hart, Johannes Keplers polyhedra Johannes Kepler Caspar, Kepler, pp. 60–65, see also, Barker and Goldstein, Theological Foundations of Keplers Astronomy
24.
Carved Stone Balls
–
Carved Stone Balls are petrospheres, usually round and rarely oval. They have from 3 to 160 protruding knobs on the surface and they range from having no ornamentation to extensive and highly varied engraved patterns. A wide range of theories have been produced to explain their use or significance and they are not to be confused with the much larger smooth round stone spheres of Costa Rica. Carved Stone Balls are up to 5200 years old, coming from the late Neolithic to at least the Bronze Age, nearly all have been found in north-east Scotland, the majority in Aberdeenshire, the fertile land lying to the east of the Grampian Mountains. A similar distribution to that of Pictish symbols led to the suggestion that Carved Stone Balls are Pictish artefacts. The core distribution also reflects that of the Recumbent stone circles, as objects they are very easy to transport and a few have been found on Iona, Skye, Harris, Uist, Lewis, Arran, Hawick, Wigtownshire and fifteen from Orkney. Outside Scotland examples have found in Ireland at Ballymena, and in England at Durham, Cumbria, Lowick. The larger balls are all from Aberdeenshire, bar one from Newburgh in Fife, by the late 1970s a total of 387 had been recorded. Of these, by far the greatest concentration was found in Aberdeenshire, by 1983 the number had risen to 411. By 2015 a total of over 425 balls were recorded, a collection of over 30 carved balls from Scotland, Ireland and northern England is in the British Museums collection. Many of the balls have not had their discovery site recorded, five were found at Skara Brae village and one at the Dunadd hillfort. The distribution of the balls is similar to that of mace-heads, the lack of context is likely to distort the interpretation. Random finds are likely to have been picked up and entered a collection if they were aesthetically appealing. Damaged and plain balls were likely to find a market than decorated examples so some more decorated examples might be fraudulent. In 2013, archaeologists discovered a stone ball at Ness of Brodgar. Many are said to be made of greenstone, but this is a term for all varieties of dark, greenish igneous rocks, including diorites, serpentinite. Forty-three are sandstone, including Old Red Sandstone,26 greenstone and 12 quartzite, nine were serpentinite and these had been carved. Some were made of gabbro, and a material to carve
25.
Late Neolithic
–
It ended when metal tools became widespread. The Neolithic is a progression of behavioral and cultural characteristics and changes, including the use of wild and domestic crops, the beginning of the Neolithic culture is considered to be in the Levant about 10, 200–8800 BC. It developed directly from the Epipaleolithic Natufian culture in the region, whose people pioneered the use of wild cereals, which then evolved into true farming. The Natufian period was between 12,000 and 10,200 BC, and the so-called proto-Neolithic is now included in the Pre-Pottery Neolithic between 10,200 and 8800 BC. By 10, 200–8800 BC, farming communities arose in the Levant and spread to Asia Minor, North Africa, Mesopotamia is the site of the earliest developments of the Neolithic Revolution from around 10,000 BC. Early Neolithic farming was limited to a range of plants, both wild and domesticated, which included einkorn wheat, millet and spelt, and the keeping of dogs, sheep. By about 6900–6400 BC, it included domesticated cattle and pigs, the establishment of permanently or seasonally inhabited settlements, not all of these cultural elements characteristic of the Neolithic appeared everywhere in the same order, the earliest farming societies in the Near East did not use pottery. Early Japanese societies and other East Asian cultures used pottery before developing agriculture, unlike the Paleolithic, when more than one human species existed, only one human species reached the Neolithic. The term Neolithic derives from the Greek νέος néos, new and λίθος líthos, stone, the term was invented by Sir John Lubbock in 1865 as a refinement of the three-age system. In the Middle East, cultures identified as Neolithic began appearing in the 10th millennium BC, early development occurred in the Levant and from there spread eastwards and westwards. Neolithic cultures are attested in southeastern Anatolia and northern Mesopotamia by around 8000 BC. The total excavated area is more than 1,200 square yards, the Neolithic 1 period began roughly 10,000 years ago in the Levant. A temple area in southeastern Turkey at Göbekli Tepe dated around 9500 BC may be regarded as the beginning of the period. This site was developed by nomadic tribes, evidenced by the lack of permanent housing in the vicinity. At least seven stone circles, covering 25 acres, contain limestone pillars carved with animals, insects, Stone tools were used by perhaps as many as hundreds of people to create the pillars, which might have supported roofs. Other early PPNA sites dating to around 9500–9000 BC have been found in Jericho, Israel, Gilgal in the Jordan Valley, the start of Neolithic 1 overlaps the Tahunian and Heavy Neolithic periods to some degree. The major advance of Neolithic 1 was true farming, in the proto-Neolithic Natufian cultures, wild cereals were harvested, and perhaps early seed selection and re-seeding occurred. The grain was ground into flour, emmer wheat was domesticated, and animals were herded and domesticated
26.
Scotland
–
Scotland is a country that is part of the United Kingdom and covers the northern third of the island of Great Britain. It shares a border with England to the south, and is surrounded by the Atlantic Ocean, with the North Sea to the east. In addition to the mainland, the country is made up of more than 790 islands, including the Northern Isles, the Kingdom of Scotland emerged as an independent sovereign state in the Early Middle Ages and continued to exist until 1707. By inheritance in 1603, James VI, King of Scots, became King of England and King of Ireland, Scotland subsequently entered into a political union with the Kingdom of England on 1 May 1707 to create the new Kingdom of Great Britain. The union also created a new Parliament of Great Britain, which succeeded both the Parliament of Scotland and the Parliament of England. Within Scotland, the monarchy of the United Kingdom has continued to use a variety of styles, titles, the legal system within Scotland has also remained separate from those of England and Wales and Northern Ireland, Scotland constitutes a distinct jurisdiction in both public and private law. Glasgow, Scotlands largest city, was one of the worlds leading industrial cities. Other major urban areas are Aberdeen and Dundee, Scottish waters consist of a large sector of the North Atlantic and the North Sea, containing the largest oil reserves in the European Union. This has given Aberdeen, the third-largest city in Scotland, the title of Europes oil capital, following a referendum in 1997, a Scottish Parliament was re-established, in the form of a devolved unicameral legislature comprising 129 members, having authority over many areas of domestic policy. Scotland is represented in the UK Parliament by 59 MPs and in the European Parliament by 6 MEPs, Scotland is also a member nation of the British–Irish Council, and the British–Irish Parliamentary Assembly. Scotland comes from Scoti, the Latin name for the Gaels, the Late Latin word Scotia was initially used to refer to Ireland. By the 11th century at the latest, Scotia was being used to refer to Scotland north of the River Forth, alongside Albania or Albany, the use of the words Scots and Scotland to encompass all of what is now Scotland became common in the Late Middle Ages. Repeated glaciations, which covered the land mass of modern Scotland. It is believed the first post-glacial groups of hunter-gatherers arrived in Scotland around 12,800 years ago, the groups of settlers began building the first known permanent houses on Scottish soil around 9,500 years ago, and the first villages around 6,000 years ago. The well-preserved village of Skara Brae on the mainland of Orkney dates from this period and it contains the remains of an early Bronze Age ruler laid out on white quartz pebbles and birch bark. It was also discovered for the first time that early Bronze Age people placed flowers in their graves, in the winter of 1850, a severe storm hit Scotland, causing widespread damage and over 200 deaths. In the Bay of Skaill, the storm stripped the earth from a large irregular knoll, when the storm cleared, local villagers found the outline of a village, consisting of a number of small houses without roofs. William Watt of Skaill, the laird, began an amateur excavation of the site, but after uncovering four houses
27.
Ancient Greeks
–
Ancient Greece was a civilization belonging to a period of Greek history from the Greek Dark Ages of the 12th-9th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and this was followed by the period of Classical Greece, an era that began with the Greco-Persian Wars, lasting from the 5th to 4th centuries BC. Due to the conquests by Alexander the Great of Macedonia, Hellenistic civilization flourished from Central Asia to the end of the Mediterranean Sea. Classical Greek culture, especially philosophy, had a influence on ancient Rome. For this reason Classical Greece is generally considered to be the culture which provided the foundation of modern Western culture and is considered the cradle of Western civilization. Classical Antiquity in the Mediterranean region is considered to have begun in the 8th century BC. Classical Antiquity in Greece is preceded by the Greek Dark Ages and this period is succeeded, around the 8th century BC, by the Orientalizing Period during which a strong influence of Syro-Hittite, Jewish, Assyrian, Phoenician and Egyptian cultures becomes apparent. The end of the Dark Ages is also dated to 776 BC. The Archaic period gives way to the Classical period around 500 BC, Ancient Periods Astronomical year numbering Dates are approximate, consult particular article for details The history of Greece during Classical Antiquity may be subdivided into five major periods. The earliest of these is the Archaic period, in which artists made larger free-standing sculptures in stiff, the Archaic period is often taken to end with the overthrow of the last tyrant of Athens and the start of Athenian Democracy in 508 BC. It was followed by the Classical period, characterized by a style which was considered by observers to be exemplary, i. e. classical, as shown in the Parthenon. This period saw the Greco-Persian Wars and the Rise of Macedon, following the Classical period was the Hellenistic period, during which Greek culture and power expanded into the Near and Middle East. This period begins with the death of Alexander and ends with the Roman conquest, Herodotus is widely known as the father of history, his Histories are eponymous of the entire field. Herodotus was succeeded by authors such as Thucydides, Xenophon, Demosthenes, Plato, most of these authors were either Athenian or pro-Athenian, which is why far more is known about the history and politics of Athens than those of many other cities. Their scope is limited by a focus on political, military and diplomatic history, ignoring economic. In the 8th century BC, Greece began to emerge from the Dark Ages which followed the fall of the Mycenaean civilization, literacy had been lost and Mycenaean script forgotten, but the Greeks adopted the Phoenician alphabet, modifying it to create the Greek alphabet. The Lelantine War is the earliest documented war of the ancient Greek period and it was fought between the important poleis of Chalcis and Eretria over the fertile Lelantine plain of Euboea. Both cities seem to have suffered a decline as result of the long war, a mercantile class arose in the first half of the 7th century BC, shown by the introduction of coinage in about 680 BC
28.
Pythagoras
–
Pythagoras of Samos was an Ionian Greek philosopher, mathematician, and the putative founder of the movement called Pythagoreanism. Most of the information about Pythagoras was written centuries after he lived. He was born on the island of Samos, and travelled, visiting Egypt and Greece, around 530 BC, he moved to Croton, in Magna Graecia, and there established some kind of school or guild. In 520 BC, he returned to Samos, Pythagoras made influential contributions to philosophy and religion in the late 6th century BC. He is often revered as a mathematician and scientist and is best known for the Pythagorean theorem which bears his name. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his colleagues, some accounts mention that the philosophy associated with Pythagoras was related to mathematics and that numbers were important. It was said that he was the first man to himself a philosopher, or lover of wisdom, and Pythagorean ideas exercised a marked influence on Plato. Burkert states that Aristoxenus and Dicaearchus are the most important accounts, Aristotle had written a separate work On the Pythagoreans, which is no longer extant. However, the Protrepticus possibly contains parts of On the Pythagoreans and his disciples Dicaearchus, Aristoxenus, and Heraclides Ponticus had written on the same subject. These writers, late as they are, were among the best sources from whom Porphyry and Iamblichus drew, while adding some legendary accounts. Herodotus, Isocrates, and other writers agree that Pythagoras was the son of Mnesarchus and born on the Greek island of Samos. His father is said to have been a gem-engraver or a wealthy merchant, a late source gives his mothers name as Pythais. As to the date of his birth, Aristoxenus stated that Pythagoras left Samos in the reign of Polycrates, at the age of 40, around 530 BC he arrived in the Greek colony of Croton in what was then Magna Graecia. There he founded his own school the members of which he engaged to a disciplined. He furthermore aquired some political influence, on Greeks and non-Greeks of the region, following a conflict with the neighbouring colony of Sybaris, internal discord drove most of the Pythagoreans out of Croton. Pythagoras left the city before the outbreak of civil unrest and moved to Metapontum, after his death, his house was transformed into a sanctuary of Demeter, out of veneration for the philosopher, by the local population. In ancient sources there was disagreement and inconsistency about the late life of Pythagoras. His tomb was shown at Metapontum in the time of Cicero, according to Walter Burkert, Most obvious is the contradiction between Aristoxenus and Dicaearchus, regarding the catastrophe that overwhelmed the Pythagorean society
29.
Earth (classical element)
–
Earth is one of the classical elements, in some systems numbering four along with air, fire, and water. Earth is one of the four elements in ancient Greek philosophy. It was commonly associated with qualities of heaviness, matter and the terrestrial world, due to the hero cults, and chthonic underworld deities, the element of earth is also associated with the sensual aspects of both life and death in later occultism. Empedocles of Acragas proposed four archai by which to understand the cosmos, fire, air, water, plato believed the elements were geometric forms and he assigned the cube to the element of earth in his dialogue Timaeus. Aristotle, believed earth was the heaviest element, and his theory of natural place suggested that any earth–laden substances, would fall quickly, straight down, in ancient Greek medicine, each of the four humours became associated with an element. Black bile was the humor identified with earth, since both were cold and dry, in alchemy, earth was believed to be primarily dry, and secondarily cold. Beyond those classical attributes, the chemical substance salt, was associated with earth and its symbol was a downward-pointing triangle. Prithvi is the Hindu earth and mother goddess, according to one such tradition, she is the personification of the Earth itself, according to another, its actual mother, being Prithvi Tattwa, the essence of the element earth. As Prithvi Mata, or Mother Earth, she contrasts with Dyaus Pita, in the Rigveda, earth and sky are frequently addressed as a duality, often indicated by the idea of two complementary half-shells. In addition, the element Earth is associated with Budha or Mercury who represents communication, business, mathematics, people born under the astrological signs of Taurus, Virgo and Capricorn are thought to have dominant earth personalities. Earth and the other Greek classical elements were incorporated into the Golden Dawn system, zelator is the elemental grade attributed to earth, this grade is also attributed to the Qabbalistic sphere Malkuth. The elemental weapon of earth is the Pentacle, each of the elements has several associated spiritual beings. The archangel of earth is Uriel, the angel is Phorlakh, the ruler is Kerub, the king is Ghob, and the earth elementals are called gnomes. Earth is considered to be passive, it is represented by the symbol for Taurus, many of these associations have since spread throughout the occult community. It is sometimes represented by its Tattva or by a downward pointing triangle with a line through it. Earth is one of the five elements appear in most Wiccan and Pagan traditions. Wicca in particular was influenced by the Golden Dawn system of magic, and Aleister Crowleys mysticism which was in turn inspired by the Golden Dawn. In East Asia, metal is seen as the equivalent of earth and is represented by the White Tiger, known as 白虎 in Chinese, Byakko in Japanese, Bạch Hổ in Vietnamese
30.
Air (classical element)
–
Air is often seen as a universal power or pure substance. Its fundamental importance to life can be seen in such as aspire, inspire, perspire and spirit. Air is one of the four elements in ancient Greek philosophy. According to Plato, it is associated with the octahedron, air is considered to be hot and wet. The ancient Greeks used two words for air, aer meant the dim lower atmosphere, and aether meant the bright upper atmosphere above the clouds. Plato, for instance writes that So it is air, there is the brightest variety which we call aether, the muddiest which we call mist and darkness. Among the early Greek Pre-Socratic philosophers, Anaximenes named air as the arche, aristophanes parodied such teachings in his play The Clouds by putting a prayer to air in the mouth of Socrates. Air was one of many proposed by the Pre-socratics, most of whom tried to reduce all things to a single substance. However, Empedocles of Acragas selected four archai for his four roots, Air, fire, water, Ancient and modern opinions differ as to whether he identified air by the divine name Hera, Aidoneus or even Zeus. Empedocles’ roots became the four elements of Greek philosophy. Plato took over the four elements of Empedocles, in the Timaeus, his major cosmological dialogue, the Platonic solid associated with air is the octahedron which is formed from eight equilateral triangles. This places air between fire and water which Plato regarded as appropriate because it is intermediate in its mobility, sharpness and he also said of air that its minuscule components are so smooth that one can barely feel them. Platos student Aristotle developed a different explanation for the elements based on pairs of qualities, the four elements were arranged concentrically around the center of the universe to form the sublunary sphere. According to Aristotle, air is hot and wet and occupies a place between fire and water among the elemental spheres. Aristotle definitively separated air from aether, for him, aether was an unchanging, almost divine substance that was found only in the heavens, where it formed celestial spheres. In ancient Greek medicine, each of the four humours became associated with an element, blood was the humor identified with air, since both were hot and wet. The alchemical symbol for air is a triangle, bisected by a horizontal line. In Hinduism, Vayu, also known as Vāta वात, Pavana पवन, or Prāna, is a deity, who is the father of Bhima
31.
Water (classical element)
–
Water is one of the elements in ancient Greek philosophy, in the Asian Indian system Panchamahabhuta, and in the Chinese cosmological and physiological system Wu Xing. In contemporary esoteric traditions, it is associated with the qualities of emotion and intuition. Water was one of many proposed by the Pre-socratics, most of whom tried to reduce all things to a single substance. However, Empedocles of Acragas selected four archai for his four roots, air, fire, water, Empedocles roots became the four classical elements of Greek philosophy. Plato took over the four elements of Empedocles, in the Timaeus, his major cosmological dialogue, the Platonic solid associated with water is the icosahedron which is formed from twenty equilateral triangles. This makes water the element with the greatest number of sides, plato’s student Aristotle developed a different explanation for the elements based on pairs of qualities. The four elements were arranged concentrically around the center of the Universe to form the sublunary sphere, according to Aristotle, water is both cold and wet and occupies a place between air and earth among the elemental spheres. In ancient Greek medicine, each of the four humours became associated with an element, phlegm was the humor identified with water, since both were cold and wet. In alchemy, the element of mercury was often associated with water. Ap is the Vedic Sanskrit term for water, in Classical Sanskrit occurring only in the plural is not an element. v, āpas, the term is from PIE hxap water. In Hindu philosophy, the term refers to water as an element, one of the Panchamahabhuta, in Hinduism, it is also the name of the deva, a personification of water. The element water is associated with Chandra or the moon and Shukra. People born under the signs of Cancer, Scorpio and Pisces are thought to have dominant water personalities. Water personalities tend to be emotional, deep, nurturing, sympathetic, empathetic, water and the other Greek classical elements were incorporated into the Golden Dawn system. The elemental weapon of water is the cup, each of the elements has several associated spiritual beings. The archangel of water is Gabriel, the angel is Taliahad, the ruler is Tharsis, the king is Nichsa and it is referred to the upper right point of the pentagram in the Supreme Invoking Ritual of the Pentagram. Many of these associations have since spread throughout the occult community, water is one of the five elements that appear in most Wiccan traditions. Wicca in particular was influenced by the Golden Dawn system of magic and Aleister Crowleys mysticism, water Sea and river deity Different versions of the classical elements
32.
Fire (classical element)
–
Fire has been an important part of all cultures and religions from pre-history to modern day and was vital to the development of civilization. It has been regarded in many different contexts throughout history, Fire is one of the four classical elements in ancient Greek philosophy and science. It was commonly associated with the qualities of energy, assertiveness, in one Greek myth, Prometheus stole fire from the gods to protect the otherwise helpless humans, but was punished for this charity. Fire was one of many proposed by the Pre-socratics, most of whom sought to reduce the cosmos, or its creation. Heraclitus considered fire to be the most fundamental of all elements and he believed fire gave rise to the other three elements, All things are an interchange for fire, and fire for all things, just like goods for gold and gold for goods. He had a reputation for obscure philosophical principles and for speaking in riddles and this is a concept that anticipates both the four classical elements of Empedocles and Aristotles transmutation of the four elements into one another. This world, which is the same for all, no one of gods or men has made, but it always was and will be, an ever-living fire, with measures of it kindling, and measures going out. Heraclitus regarded the soul as being a mixture of fire and water, with fire being the noble part. He believed the goal of the soul is to be rid of water and become pure fire and he was known as the weeping philosopher and died of hydropsy, a swelling due to abnormal accumulation of fluid beneath the skin. However, Empedocles of Acragas, is best known for having selected all elements as his archai and by the time of Plato, the four Empedoclian elements were well established. This also makes fire the element with the smallest number of sides, and Plato regarded it as appropriate for the heat of fire, which he felt is sharp and stabbing. Fire the hot and dry element, like the elements, was an abstract principle and not identical with the normal solids, liquids and combustion phenomena we experience. According to Aristotle, the four elements rise or fall toward their natural place in concentric layers surrounding the center of the earth, in ancient Greek medicine, each of the four humours became associated with an element. Yellow bile was the humor identified with fire, since both were hot and dry, in alchemy the chemical element of sulfur was often associated with fire and its alchemical symbol and its symbol was an upward-pointing triangle. In alchemic tradition, metals are incubated by fire in the womb of the Earth, Agni is a Hindu and Vedic deity. The word agni is Sanskrit for fire, cognate with Latin ignis, Russian огонь, Agni has three forms, fire, lightning and the sun. Agni is one of the most important of the Vedic gods and he is the god of fire and the acceptor of sacrifices. The sacrifices made to Agni go to the deities because Agni is a messenger from and he is ever-young, because the fire is re-lit every day, yet he is also immortal
33.
Tessellation
–
A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries, a periodic tiling has a repeating pattern. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups, a tiling that lacks a repeating pattern is called non-periodic. An aperiodic tiling uses a set of tile shapes that cannot form a repeating pattern. In the geometry of higher dimensions, a space-filling or honeycomb is called a tessellation of space. A real physical tessellation is a made of materials such as cemented ceramic squares or hexagons. Such tilings may be decorative patterns, or may have such as providing durable and water-resistant pavement. Historically, tessellations were used in Ancient Rome and in Islamic art such as in the decorative geometric tiling of the Alhambra palace, in the twentieth century, the work of M. C. Escher often made use of tessellations, both in ordinary Euclidean geometry and in geometry, for artistic effect. Tessellations are sometimes employed for decorative effect in quilting, Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs. Tessellations were used by the Sumerians in building wall decorations formed by patterns of clay tiles, decorative mosaic tilings made of small squared blocks called tesserae were widely employed in classical antiquity, sometimes displaying geometric patterns. In 1619 Johannes Kepler made a documented study of tessellations. He wrote about regular and semiregular tessellations in his Harmonices Mundi, he was possibly the first to explore and to explain the structures of honeycomb. Some two hundred years later in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the features one of seventeen different groups of isometries. Fyodorovs work marked the beginning of the mathematical study of tessellations. Other prominent contributors include Shubnikov and Belov, and Heinrich Heesch, in Latin, tessella is a small cubical piece of clay, stone or glass used to make mosaics. The word tessella means small square and it corresponds to the everyday term tiling, which refers to applications of tessellations, often made of glazed clay. Tessellation or tiling in two dimensions is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules
34.
Euclidean space
–
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria, the term Euclidean distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions, classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are used to define rational numbers. It means that points of the space are specified with collections of real numbers and this approach brings the tools of algebra and calculus to bear on questions of geometry and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions. From the modern viewpoint, there is only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the coordinate space of the same dimension. In one dimension, this is the line, in two dimensions, it is the Cartesian plane, and in higher dimensions it is a coordinate space with three or more real number coordinates. One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance, for example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that point is shifted in the same direction. The other is rotation about a point in the plane. In order to all of this mathematically precise, the theory must clearly define the notions of distance, angle, translation. Even when used in theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments. The standard way to such space, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product. The reason for working with vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner. Once the Euclidean plane has been described in language, it is actually a simple matter to extend its concept to arbitrary dimensions. For the most part, the vocabulary, formulae, and calculations are not made any more difficult by the presence of more dimensions. Intuitively, the distinction says merely that there is no choice of where the origin should go in the space
35.
Aristotle
–
Aristotle was an ancient Greek philosopher and scientist born in the city of Stagira, Chalkidice, on the northern periphery of Classical Greece. His father, Nicomachus, died when Aristotle was a child, at seventeen or eighteen years of age, he joined Platos Academy in Athens and remained there until the age of thirty-seven. Shortly after Plato died, Aristotle left Athens and, at the request of Philip II of Macedon, teaching Alexander the Great gave Aristotle many opportunities and an abundance of supplies. He established a library in the Lyceum which aided in the production of many of his hundreds of books and he believed all peoples concepts and all of their knowledge was ultimately based on perception. Aristotles views on natural sciences represent the groundwork underlying many of his works, Aristotles views on physical science profoundly shaped medieval scholarship. Their influence extended from Late Antiquity and the Early Middle Ages into the Renaissance, some of Aristotles zoological observations, such as on the hectocotyl arm of the octopus, were not confirmed or refuted until the 19th century. His works contain the earliest known study of logic, which was incorporated in the late 19th century into modern formal logic. Aristotle was well known among medieval Muslim intellectuals and revered as The First Teacher and his ethics, though always influential, gained renewed interest with the modern advent of virtue ethics. All aspects of Aristotles philosophy continue to be the object of academic study today. Though Aristotle wrote many elegant treatises and dialogues – Cicero described his style as a river of gold – it is thought that only around a third of his original output has survived. Aristotle, whose means the best purpose, was born in 384 BC in Stagira, Chalcidice. His father Nicomachus was the physician to King Amyntas of Macedon. Aristotle was orphaned at a young age, although there is little information on Aristotles childhood, he probably spent some time within the Macedonian palace, making his first connections with the Macedonian monarchy. At the age of seventeen or eighteen, Aristotle moved to Athens to continue his education at Platos Academy and he remained there for nearly twenty years before leaving Athens in 348/47 BC. Aristotle then accompanied Xenocrates to the court of his friend Hermias of Atarneus in Asia Minor, there, he traveled with Theophrastus to the island of Lesbos, where together they researched the botany and zoology of the island. Aristotle married Pythias, either Hermiass adoptive daughter or niece and she bore him a daughter, whom they also named Pythias. Soon after Hermias death, Aristotle was invited by Philip II of Macedon to become the tutor to his son Alexander in 343 BC, Aristotle was appointed as the head of the royal academy of Macedon. During that time he gave not only to Alexander
36.
Aether (classical element)
–
According to ancient and medieval science, aether, also spelled æther or ether, also called quintessence, is the material that fills the region of the universe above the terrestrial sphere. The concept of aether was used in several theories to explain natural phenomena, such as the traveling of light. The word αἰθήρ in Homeric Greek means pure, fresh air or clear sky, in Greek mythology, it was thought to be the pure essence that the gods breathed, filling the space where they lived, analogous to the air breathed by mortals. It is also personified as a deity, Aether, the son of Erebus, Aether is related to αἴθω to incinerate, and intransitive to burn, to shine. In Platos Timaeus speaking about air, Plato mentions that there is the most translucent kind which is called by the name of aether, but otherwise he adopted the classical system of four elements. Aristotle, who had been Platos student at the Akademia, agreed on this point with his former mentor, however, in his Book On the Heavens he introduced a new first element to the system of the classical elements of Ionian philosophy. He noted that the four classical elements were subject to change. The first element however, located in the regions and heavenly bodies. It was neither hot nor cold, neither wet nor dry, Aether did not follow Aristotelian physics either. Aether was also incapable of motion of quality or motion of quantity, Aether was only capable of local motion. Aether naturally moved in circles, and had no contrary, or unnatural, aristotle also noted that crystalline spheres made of aether held the celestial bodies. The idea of spheres and natural circular motion of aether led to Aristotles explanation of the observed orbits of stars. Medieval scholastic philosophers granted aether changes of density, in which the bodies of the planets were considered to be more dense than the medium which filled the rest of the universe, Robert Fludd stated that the aether was of the character that it was subtler than light. Fludd cites the 3rd-century view of Plotinus, concerning the aether as penetrative, Quintessence is the Latinate name of the fifth element used by medieval alchemists for a medium similar or identical to that thought to make up the heavenly bodies. It was noted there was very little presence of quintessence within the terrestrial sphere. Due to the low presence of quintessence, earth could be affected by what takes place within the heavenly bodies and this theory was developed in the 14th century text The testament of Lullius, attributed to Ramon Llull. The use of quintessence became popular within medieval alchemy and this elemental system spread rapidly throughout all of Europe and became popular with alchemists, especially in medicinal alchemy. Medicinal alchemy then sought to isolate quintessence and incorporate it within medicine, due to quintessences pure and heavenly quality, it was thought that through consumption one may rid oneself of any impurities or illnesses
37.
Euclid
–
Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the father of geometry. He was active in Alexandria during the reign of Ptolemy I, in the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, Euclid is the anglicized version of the Greek name Εὐκλείδης, which means renowned, glorious. Very few original references to Euclid survive, so little is known about his life, the date, place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other people mentioned with him. He is rarely mentioned by name by other Greek mathematicians from Archimedes onward, the few historical references to Euclid were written centuries after he lived by Proclus c.450 AD and Pappus of Alexandria c.320 AD. Proclus introduces Euclid only briefly in his Commentary on the Elements, Proclus later retells a story that, when Ptolemy I asked if there was a shorter path to learning geometry than Euclids Elements, Euclid replied there is no royal road to geometry. This anecdote is questionable since it is similar to a story told about Menaechmus, a detailed biography of Euclid is given by Arabian authors, mentioning, for example, a birth town of Tyre. This biography is generally believed to be completely fictitious, however, this hypothesis is not well accepted by scholars and there is little evidence in its favor. The only reference that historians rely on of Euclid having written the Elements was from Proclus, although best known for its geometric results, the Elements also includes number theory. The geometrical system described in the Elements was long known simply as geometry, today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean geometries that mathematicians discovered in the 19th century. In addition to the Elements, at least five works of Euclid have survived to the present day and they follow the same logical structure as Elements, with definitions and proved propositions. Data deals with the nature and implications of information in geometrical problems. On Divisions of Figures, which only partially in Arabic translation. It is similar to a first-century AD work by Heron of Alexandria, catoptrics, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors. The attribution is held to be anachronistic however by J J OConnor, phaenomena, a treatise on spherical astronomy, survives in Greek, it is quite similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC. Optics is the earliest surviving Greek treatise on perspective, in its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which emanate from the eye. One important definition is the fourth, Things seen under a greater angle appear greater, proposition 45 is interesting, proving that for any two unequal magnitudes, there is a point from which the two appear equal. Other works are attributed to Euclid, but have been lost
38.
Euclid's Elements
–
Euclids Elements is a mathematical and geometric treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt circa 300 BC. It is a collection of definitions, postulates, propositions, the books cover Euclidean geometry and the ancient Greek version of elementary number theory. Elements is the second-oldest extant Greek mathematical treatise after Autolycus On the Moving Sphere and it has proven instrumental in the development of logic and modern science. According to Proclus, the element was used to describe a theorem that is all-pervading. The word element in the Greek language is the same as letter and this suggests that theorems in the Elements should be seen as standing in the same relation to geometry as letters to language. Euclids Elements has been referred to as the most successful and influential textbook ever written, for centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclids Elements was required of all students. Not until the 20th century, by which time its content was taught through other school textbooks. Scholars believe that the Elements is largely a collection of theorems proven by other mathematicians, the Elements may have been based on an earlier textbook by Hippocrates of Chios, who also may have originated the use of letters to refer to figures. This manuscript, the Heiberg manuscript, is from a Byzantine workshop around 900 and is the basis of modern editions, papyrus Oxyrhynchus 29 is a tiny fragment of an even older manuscript, but only contains the statement of one proposition. Although known to, for instance, Cicero, no record exists of the text having been translated into Latin prior to Boethius in the fifth or sixth century. The Arabs received the Elements from the Byzantines around 760, this version was translated into Arabic under Harun al Rashid circa 800, the Byzantine scholar Arethas commissioned the copying of one of the extant Greek manuscripts of Euclid in the late ninth century. Although known in Byzantium, the Elements was lost to Western Europe until about 1120, the first printed edition appeared in 1482, and since then it has been translated into many languages and published in about a thousand different editions. Theons Greek edition was recovered in 1533, in 1570, John Dee provided a widely respected Mathematical Preface, along with copious notes and supplementary material, to the first English edition by Henry Billingsley. Copies of the Greek text still exist, some of which can be found in the Vatican Library, the manuscripts available are of variable quality, and invariably incomplete. By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text, ancient texts which refer to the Elements itself, and to other mathematical theories that were current at the time it was written, are also important in this process. Such analyses are conducted by J. L. Heiberg and Sir Thomas Little Heath in their editions of the text, also of importance are the scholia, or annotations to the text. These additions, which distinguished themselves from the main text. The Elements is still considered a masterpiece in the application of logic to mathematics, in historical context, it has proven enormously influential in many areas of science
39.
Andreas Speiser
–
Andreas Speiser was a Swiss Mathematician and Philosopher of Science. Speiser studied since 1904 in Göttingen notably with David Hilbert, Felix Klein, in 1917 he became full-time professor at the University of Zurich but later relocated in Basel. During 1924/25 he was president of the Swiss Mathematical Association, Speiser worked on number theory, group theory, and the theory of Riemann surfaces. Speiser also added an appendix on ideal theory to Dicksons book, speisers book Theorie der Gruppen endlicher Ordnung is a classic, richly illustrated work on group theory. Speiser also worked on the history of mathematics and was the editor for the Euler Commissions edition of Leonhard Eulers Opera Omnia. As a philosopher Speiser was chiefly concerned with Plato and wrote a commentary on the Parmenides Dialogue, die Theorie der Gruppen von endlicher Ordnung – mit Anwendungen auf algebraische Zahlen und Gleichungen sowie auf die Kristallographie. Leonhard Euler und die Deutsche Philosophie, atlantis Verlag, Zürich 1939,1940, S. 1-6. Vortrag gehalten an der Generalversammlung des S. I. A. in Basel am 11, ein Parmenideskommentar – Studien zur Platonischen Dialektik. Naturphilosophische Untersuchungen von Euler und Riemann, einteilung der sämtlichen Werke Leonhard Eulers
40.
Astronomer
–
An astronomer is a scientist in the field of astronomy who concentrates their studies on a specific question or field outside of the scope of Earth. They look at stars, planets, moons, comets and galaxies, as well as other celestial objects — either in observational astronomy. Examples of topics or fields astronomers work on include, planetary science, solar astronomy, there are also related but distinct subjects like physical cosmology which studies the Universe as a whole. Astronomers usually fit into two types, Observational astronomers make direct observations of planets, stars and galaxies, and analyze the data, theoretical astronomers create and investigate models of things that cannot be observed. They use this data to create models or simulations to theorize how different celestial bodies work, there are further subcategories inside these two main branches of astronomy such as planetary astronomy, galactic astronomy or physical cosmology. Today, that distinction has disappeared and the terms astronomer. Professional astronomers are highly educated individuals who typically have a Ph. D. in physics or astronomy and are employed by research institutions or universities. They spend the majority of their time working on research, although quite often have other duties such as teaching, building instruments. The number of astronomers in the United States is actually quite small. The American Astronomical Society, which is the organization of professional astronomers in North America, has approximately 7,000 members. This number includes scientists from other such as physics, geology. The International Astronomical Union comprises almost 10,145 members from 70 different countries who are involved in research at the Ph. D. level. Before CCDs, photographic plates were a method of observation. Modern astronomers spend relatively little time at telescopes usually just a few weeks per year, analysis of observed phenomena, along with making predictions as to the causes of what they observe, takes the majority of observational astronomers time. Astronomers who serve as faculty spend much of their time teaching undergraduate and graduate classes, most universities also have outreach programs including public telescope time and sometimes planetariums as a public service to encourage interest in the field. Those who become astronomers usually have a background in maths, sciences. Taking courses that teach how to research, write and present papers are also invaluable, in college/university most astronomers get a Ph. D. in astronomy or physics. Keeping in mind how few there are it is understood that graduate schools in this field are very competitive
41.
Planet
–
The term planet is ancient, with ties to history, astrology, science, mythology, and religion. Several planets in the Solar System can be seen with the naked eye and these were regarded by many early cultures as divine, or as emissaries of deities. As scientific knowledge advanced, human perception of the planets changed, in 2006, the International Astronomical Union officially adopted a resolution defining planets within the Solar System. This definition is controversial because it excludes many objects of mass based on where or what they orbit. The planets were thought by Ptolemy to orbit Earth in deferent, at about the same time, by careful analysis of pre-telescopic observation data collected by Tycho Brahe, Johannes Kepler found the planets orbits were not circular but elliptical. As observational tools improved, astronomers saw that, like Earth, the planets rotated around tilted axes, and some shared such features as ice caps and seasons. Since the dawn of the Space Age, close observation by space probes has found that Earth and the planets share characteristics such as volcanism, hurricanes, tectonics. Planets are generally divided into two types, large low-density giant planets, and smaller rocky terrestrials. Under IAU definitions, there are eight planets in the Solar System, in order of increasing distance from the Sun, they are the four terrestrials, Mercury, Venus, Earth, and Mars, then the four giant planets, Jupiter, Saturn, Uranus, and Neptune. Six of the planets are orbited by one or more natural satellites, several thousands of planets around other stars have been discovered in the Milky Way. e. in the habitable zone. On December 20,2011, the Kepler Space Telescope team reported the discovery of the first Earth-sized extrasolar planets, Kepler-20e and Kepler-20f, orbiting a Sun-like star, Kepler-20. A2012 study, analyzing gravitational microlensing data, estimates an average of at least 1.6 bound planets for every star in the Milky Way, around one in five Sun-like stars is thought to have an Earth-sized planet in its habitable zone. The idea of planets has evolved over its history, from the lights of antiquity to the earthly objects of the scientific age. The concept has expanded to include not only in the Solar System. The ambiguities inherent in defining planets have led to much scientific controversy, the five classical planets, being visible to the naked eye, have been known since ancient times and have had a significant impact on mythology, religious cosmology, and ancient astronomy. In ancient times, astronomers noted how certain lights moved across the sky, as opposed to the fixed stars, ancient Greeks called these lights πλάνητες ἀστέρες or simply πλανῆται, from which todays word planet was derived. In ancient Greece, China, Babylon, and indeed all pre-modern civilizations, it was almost universally believed that Earth was the center of the Universe and that all the planets circled Earth. The first civilization known to have a theory of the planets were the Babylonians
42.
Saturn
–
Saturn is the sixth planet from the Sun and the second-largest in the Solar System, after Jupiter. It is a gas giant with a radius about nine times that of Earth. Although it has only one-eighth the average density of Earth, with its larger volume Saturn is just over 95 times more massive, Saturn is named after the Roman god of agriculture, its astronomical symbol represents the gods sickle. Saturns interior is composed of a core of iron–nickel and rock. This core is surrounded by a layer of metallic hydrogen, an intermediate layer of liquid hydrogen and liquid helium. Saturn has a yellow hue due to ammonia crystals in its upper atmosphere. Saturns magnetic field strength is around one-twentieth of Jupiters, the outer atmosphere is generally bland and lacking in contrast, although long-lived features can appear. Wind speeds on Saturn can reach 1,800 km/h, higher than on Jupiter, sixty-two moons are known to orbit Saturn, of which fifty-three are officially named. This does not include the hundreds of moonlets comprising the rings, Saturn is a gas giant because it is predominantly composed of hydrogen and helium. It lacks a definite surface, though it may have a solid core, Saturns rotation causes it to have the shape of an oblate spheroid, that is, it is flattened at the poles and bulges at its equator. Its equatorial and polar radii differ by almost 10%,60,268 km versus 54,364 km, Jupiter, Uranus, and Neptune, the other giant planets in the Solar System, are also oblate but to a lesser extent. Saturn is the planet of the Solar System that is less dense than water—about 30% less. Although Saturns core is considerably denser than water, the specific density of the planet is 0.69 g/cm3 due to the atmosphere. Jupiter has 318 times the Earths mass, while Saturn is 95 times the mass of the Earth, together, Jupiter and Saturn hold 92% of the total planetary mass in the Solar System. On 8 January 2015, NASA reported determining the center of the planet Saturn, the temperature, pressure, and density inside Saturn all rise steadily toward the core, which causes hydrogen to transition into a metal in the deeper layers. Standard planetary models suggest that the interior of Saturn is similar to that of Jupiter, having a rocky core surrounded by hydrogen. This core is similar in composition to the Earth, but more dense, in 2004, they estimated that the core must be 9–22 times the mass of the Earth, which corresponds to a diameter of about 25,000 km. This is surrounded by a liquid metallic hydrogen layer, followed by a liquid layer of helium-saturated molecular hydrogen that gradually transitions to a gas with increasing altitude
43.
Mercury (planet)
–
Mercury is the smallest and innermost planet in the Solar System. Its orbital period around the Sun of 88 days is the shortest of all the planets in the Solar System and it is named after the Roman deity Mercury, the messenger to the gods. Like Venus, Mercury orbits the Sun within Earths orbit as a planet, so it can only be seen visually in the morning or the evening sky. Also, like Venus and the Moon, the displays the complete range of phases as it moves around its orbit relative to Earth. Seen from Earth, this cycle of phases reoccurs approximately every 116 days, although Mercury can appear as a bright star-like object when viewed from Earth, its proximity to the Sun often makes it more difficult to see than Venus. Mercury is tidally or gravitationally locked with the Sun in a 3,2 resonance, as seen relative to the fixed stars, it rotates on its axis exactly three times for every two revolutions it makes around the Sun. As seen from the Sun, in a frame of reference that rotates with the orbital motion, an observer on Mercury would therefore see only one day every two years. Mercurys axis has the smallest tilt of any of the Solar Systems planets, at aphelion, Mercury is about 1.5 times as far from the Sun as it is at perihelion. Mercurys surface appears heavily cratered and is similar in appearance to the Moons, the polar regions are constantly below 180 K. The planet has no natural satellites. Mercury is one of four planets in the Solar System. It is the smallest planet in the Solar System, with a radius of 2,439.7 kilometres. Mercury is also smaller—albeit more massive—than the largest natural satellites in the Solar System, Ganymede, Mercury consists of approximately 70% metallic and 30% silicate material. Mercurys density is the second highest in the Solar System at 5.427 g/cm3, Mercurys density can be used to infer details of its inner structure. Although Earths high density results appreciably from gravitational compression, particularly at the core, Mercury is much smaller, therefore, for it to have such a high density, its core must be large and rich in iron. Geologists estimate that Mercurys core occupies about 55% of its volume, Research published in 2007 suggests that Mercury has a molten core. Surrounding the core is a 500–700 km mantle consisting of silicates, based on data from the Mariner 10 mission and Earth-based observation, Mercurys crust is estimated to be 35 km thick. One distinctive feature of Mercurys surface is the presence of narrow ridges
44.
Earth
–
Earth, otherwise known as the World, or the Globe, is the third planet from the Sun and the only object in the Universe known to harbor life. It is the densest planet in the Solar System and the largest of the four terrestrial planets, according to radiometric dating and other sources of evidence, Earth formed about 4.54 billion years ago. Earths gravity interacts with objects in space, especially the Sun. During one orbit around the Sun, Earth rotates about its axis over 365 times, thus, Earths axis of rotation is tilted, producing seasonal variations on the planets surface. The gravitational interaction between the Earth and Moon causes ocean tides, stabilizes the Earths orientation on its axis, Earths lithosphere is divided into several rigid tectonic plates that migrate across the surface over periods of many millions of years. About 71% of Earths surface is covered with water, mostly by its oceans, the remaining 29% is land consisting of continents and islands that together have many lakes, rivers and other sources of water that contribute to the hydrosphere. The majority of Earths polar regions are covered in ice, including the Antarctic ice sheet, Earths interior remains active with a solid iron inner core, a liquid outer core that generates the Earths magnetic field, and a convecting mantle that drives plate tectonics. Within the first billion years of Earths history, life appeared in the oceans and began to affect the Earths atmosphere and surface, some geological evidence indicates that life may have arisen as much as 4.1 billion years ago. Since then, the combination of Earths distance from the Sun, physical properties, in the history of the Earth, biodiversity has gone through long periods of expansion, occasionally punctuated by mass extinction events. Over 99% of all species that lived on Earth are extinct. Estimates of the number of species on Earth today vary widely, over 7.4 billion humans live on Earth and depend on its biosphere and minerals for their survival. Humans have developed diverse societies and cultures, politically, the world has about 200 sovereign states, the modern English word Earth developed from a wide variety of Middle English forms, which derived from an Old English noun most often spelled eorðe. It has cognates in every Germanic language, and their proto-Germanic root has been reconstructed as *erþō, originally, earth was written in lowercase, and from early Middle English, its definite sense as the globe was expressed as the earth. By early Modern English, many nouns were capitalized, and the became the Earth. More recently, the name is simply given as Earth. House styles now vary, Oxford spelling recognizes the lowercase form as the most common, another convention capitalizes Earth when appearing as a name but writes it in lowercase when preceded by the. It almost always appears in lowercase in colloquial expressions such as what on earth are you doing, the oldest material found in the Solar System is dated to 4. 5672±0.0006 billion years ago. By 4. 54±0.04 Gya the primordial Earth had formed, the formation and evolution of Solar System bodies occurred along with the Sun
45.
Jupiter
–
Jupiter is the fifth planet from the Sun and the largest in the Solar System. It is a giant planet with a mass one-thousandth that of the Sun, Jupiter and Saturn are gas giants, the other two giant planets, Uranus and Neptune are ice giants. Jupiter has been known to astronomers since antiquity, the Romans named it after their god Jupiter. Jupiter is primarily composed of hydrogen with a quarter of its mass being helium and it may also have a rocky core of heavier elements, but like the other giant planets, Jupiter lacks a well-defined solid surface. Because of its rotation, the planets shape is that of an oblate spheroid. The outer atmosphere is visibly segregated into several bands at different latitudes, resulting in turbulence, a prominent result is the Great Red Spot, a giant storm that is known to have existed since at least the 17th century when it was first seen by telescope. Surrounding Jupiter is a faint planetary ring system and a powerful magnetosphere, Jupiter has at least 67 moons, including the four large Galilean moons discovered by Galileo Galilei in 1610. Ganymede, the largest of these, has a greater than that of the planet Mercury. Jupiter has been explored on several occasions by robotic spacecraft, most notably during the early Pioneer and Voyager flyby missions and later by the Galileo orbiter. In late February 2007, Jupiter was visited by the New Horizons probe, the latest probe to visit the planet is Juno, which entered into orbit around Jupiter on July 4,2016. Future targets for exploration in the Jupiter system include the probable ice-covered liquid ocean of its moon Europa, Earth and its neighbor planets may have formed from fragments of planets after collisions with Jupiter destroyed those super-Earths near the Sun. Astronomers have discovered nearly 500 planetary systems with multiple planets, Jupiter moving out of the inner Solar System would have allowed the formation of inner planets, including Earth. Jupiter is composed primarily of gaseous and liquid matter and it is the largest of the four giant planets in the Solar System and hence its largest planet. It has a diameter of 142,984 km at its equator, the average density of Jupiter,1.326 g/cm3, is the second highest of the giant planets, but lower than those of the four terrestrial planets. Jupiters upper atmosphere is about 88–92% hydrogen and 8–12% helium by percent volume of gas molecules, a helium atom has about four times as much mass as a hydrogen atom, so the composition changes when described as the proportion of mass contributed by different atoms. Thus, Jupiters atmosphere is approximately 75% hydrogen and 24% helium by mass, the atmosphere contains trace amounts of methane, water vapor, ammonia, and silicon-based compounds. There are also traces of carbon, ethane, hydrogen sulfide, neon, oxygen, phosphine, the outermost layer of the atmosphere contains crystals of frozen ammonia. The interior contains denser materials - by mass it is roughly 71% hydrogen, 24% helium, through infrared and ultraviolet measurements, trace amounts of benzene and other hydrocarbons have also been found