Geometry
Geometry is a branch of mathematics concerned with questions of shape, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a practical way for dealing with lengths and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.
While geometry has evolved throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, planes, surfaces and curves, as well as the more advanced notions of manifolds and topology or metric. Geometry has applications to many fields, including art, physics, as well as to other branches of mathematics. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense; the mandatory educational curriculum of the majority of nations includes the study of points, planes, triangles, similarity, solid figures and analytic geometry. Euclidean geometry has applications in computer science and various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry, it has applications in physics, including in general relativity. Topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this means dealing with large-scale properties of spaces, such as connectedness and compactness.
Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues using techniques of real analysis. It has close connections to convex analysis and functional analysis and important applications in number theory. Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques, it has applications including cryptography and string theory. Discrete geometry is concerned with questions of relative position of simple geometric objects, such as points and circles, it shares many principles with combinatorics. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc; the earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles and volumes, which were developed to meet some practical need in surveying, construction and various crafts.
The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. Clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space; these geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.
Eudoxus developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom and proof. Although most of the contents of the Elements were known, Euclid arranged them into a single, coherent logical framework; the Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, gave remarkably accurate approximations of Pi.
He studied the sp
Isotoxal figure
In geometry, a polytope, or a tiling, is isotoxal or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation and/or reflection that will move one edge to the other, while leaving the region occupied by the object unchanged; the term isotoxal is derived from the Greek τοξον meaning arc. An isotoxal polygon is an equilateral polygon; the duals of isotoxal polygons are isogonal polygons. In general, an isotoxal 2n-gon will have Dn dihedral symmetry. A rhombus is an isotoxal polygon with D2 symmetry. All regular polygons are isotoxal, having double the minimum symmetry order: a regular n-gon has Dn dihedral symmetry. A regular 2n-gon is an isotoxal polygon and can be marked with alternately colored vertices, removing the line of reflection through the mid-edges. Regular polyhedra are isohedral and isotoxal. Quasiregular polyhedra are not isohedral. Not every polyhedron or 2-dimensional tessellation constructed from regular polygons is isotoxal.
For instance, the truncated icosahedron has two types of edges: hexagon-hexagon and hexagon-pentagon, it is not possible for a symmetry of the solid to move a hexagon-hexagon edge onto a hexagon-pentagon edge. An isotoxal polyhedron has the same dihedral angle for all edges. There are nine convex isotoxal polyhedra formed from the Platonic solids, 8 formed by the Kepler–Poinsot polyhedra, six more as quasiregular star polyhedra and their duals. There are at least 5 polygonal tilings of the Euclidean plane that are isotoxal, infinitely many isotoxal polygonal tilings of the hyperbolic plane, including the Wythoff constructions from the regular hyperbolic tilings, non-right groups. Table of polyhedron dihedral angles Vertex-transitive Face-transitive Cell-transitive Peter R. Cromwell, Cambridge University Press 1997, ISBN 0-521-55432-2, p. 371 Transitivity Grünbaum, Branko. C.. Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. CS1 maint: Multiple names: authors list Coxeter, Harold Scott MacDonald.
"Uniform polyhedra", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 246: 401–450, doi:10.1098/rsta.1954.0003, ISSN 0080-4614, JSTOR 91532, MR 0062446
Octahedron
In geometry, an octahedron is a polyhedron with eight faces, twelve edges, six vertices. The term is most used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. A regular octahedron is the dual polyhedron of a cube, it is a rectified tetrahedron. It is a square bipyramid in any of three orthogonal orientations, it is a triangular antiprism in any of four orientations. An octahedron is the three-dimensional case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan metric. If the edge length of a regular octahedron is a, the radius of a circumscribed sphere is r u = a 2 2 ≈ 0.707 ⋅ a and the radius of an inscribed sphere is r i = a 6 6 ≈ 0.408 ⋅ a while the midradius, which touches the middle of each edge, is r m = a 2 = 0.5 ⋅ a The octahedron has four special orthogonal projections, centered, on an edge, vertex and normal to a face. The second and third correspond to A2 Coxeter planes.
The octahedron can be represented as a spherical tiling, projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not 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. In an x–y–z Cartesian coordinate system, the octahedron with center coordinates and radius r is the set of all points such that | x − a | + | y − b | + | z − c | = r; the surface area A and the volume V of a regular octahedron of edge length a are: A = 2 3 a 2 ≈ 3.464 a 2 V = 1 3 2 a 3 ≈ 0.471 a 3 Thus the volume is four times that of a regular tetrahedron with the same edge length, while the surface area is twice. If an octahedron has been stretched so that it obeys the equation | x x m | + | y y m | + | z z m | = 1, the formulas for the surface area and volume expand to become A = 4 x m y m z m × 1 x m 2 + 1 y m 2 + 1 z m 2, V = 4 3 x m y m z m.
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 2 2. {\displaystyle x_=y_=z_=
Truncated triapeirogonal tiling
In geometry, the truncated triapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of tr. The dual of this tiling represents the fundamental domains of, *∞32 symmetry. There are 3 small index subgroup constructed from by mirror alternation. In these images fundamental domains are alternately colored black and white, mirrors exist on the boundaries between colors. A special index 4 reflective subgroup, is, its direct subgroup +, semidirect subgroup. Given with generating mirrors its index 4 subgroup has generators. An index 6 subgroup constructed as, becomes; this tiling can be considered a member of a sequence of uniform patterns with vertex figure and Coxeter-Dynkin diagram. For p < 6, the members of the sequence are omnitruncated polyhedra, shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling. List of uniform planar tilings Tilings of regular polygons Uniform tilings in hyperbolic plane
Cuboctahedron
In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, 24 identical edges, each separating a triangle from a square; as such, it is a quasiregular polyhedron, i.e. an Archimedean solid, not only vertex-transitive but edge-transitive. It is the only radially equilateral convex polyhedron, its dual polyhedron is the rhombic dodecahedron. The cuboctahedron was known to Plato: Heron's Definitiones quotes Archimedes as saying that Plato knew of a solid made of 8 triangles and 6 squares. Heptaparallelohedron Fuller applied the name "Dymaxion" to this shape, used in an early version of the Dymaxion map, he called it the "Vector Equilibrium" because of its radial equilateral symmetry. He called a cuboctahedron consisting of rigid struts connected by flexible vertices a "jitterbug". With Oh symmetry, order 48, it is a rectified cube or rectified octahedron With Td symmetry, order 24, it is a cantellated tetrahedron or rhombitetratetrahedron.
With D3d symmetry, order 12, it is a triangular gyrobicupola. The area A and the volume V of the cuboctahedron of edge length a are: A = a 2 ≈ 9.464 1016 a 2 V = 5 3 2 a 3 ≈ 2.357 0226 a 3. The cuboctahedron has four special orthogonal projections, centered on a vertex, an edge, the two types of faces and square; the last two correspond to the B2 and A2 Coxeter planes. The skew projections show a hexagon passing through the center of the cuboctahedron; the cuboctahedron can be represented as a spherical tiling, projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not lengths. Straight lines on the sphere are projected as circular arcs on the plane; the Cartesian coordinates for the vertices of a cuboctahedron centered at the origin are: An alternate set of coordinates can be made in 4-space, as 12 permutations of: This construction exists as one of 16 orthant facets of the cantellated 16-cell. The cuboctahedron's 12 vertices can represent the root vectors of the simple Lie group A3.
With the addition of 6 vertices of the octahedron, these vertices represent the 18 root vectors of the simple Lie group B3. The cuboctahedron can be dissected into two triangular cupolas by a common hexagon passing through the center of the cuboctahedron. If these two triangular cupolas are twisted so triangles and squares line up, Johnson solid J27, the triangular orthobicupola, is created; the cuboctahedron can be dissected into 6 square pyramids and 8 tetrahedra meeting at a central point. This dissection is expressed in the alternated cubic honeycomb where pairs of square pyramids are combined into octahedra; the cuboctahedron is the unique convex polyhedron in which the long radius is the same as the edge length. This radial equilateral symmetry is a property of only a few polytopes, including the two-dimensional hexagon, the three-dimensional cuboctahedron, the four-dimensional 24-cell and 8-cell. Radially equilateral polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.
Therefore, all the interior elements which meet at the center of these polytopes have equilateral triangle inward faces, as in the dissection of the cuboctahedron into 6 square pyramids and 8 tetrahedra. Each of these radially equilateral polytopes occurs as cells of a characteristic space-filling tessellation: the tiling of regular hexagons, the rectified cubic honeycomb, the 24-cell honeycomb and the tesseractic honeycomb, respectively; each tessellation has a dual tessellation. The densest known regular sphere-packing in two and four dimensions uses the cell centers of one of these tessellations as sphere centers. A cuboctahedron has octahedral symmetry, its first stellation is the compound of a cube and its dual octahedron, with the vertices of the cuboctahedron located at the midpoints of the edges of either. A cuboctahedron can be obtained by taking an equatorial cross section of a four-dimensional 24-cell or 16-cell. A hexagon can be obtained by taking an equatorial cross section of a cuboctahedron.
The cuboctahedron is a rectified cube and a rectified octahedron. It is a cantellated tetrahedron. With this construction it is given the Wythoff symbol: 3 3 | 2. A skew cantellation of the tetrahedron produces a solid with faces parallel to those of the cuboctahedron, namely eight triangles of two sizes, six rectangles. While its edges are unequal, this solid remains vertex-uniform: the solid has the full tetrahedral symmet
Orbifold notation
In geometry, orbifold notation is a system, invented by William Thurston and popularized by the mathematician John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advantage of the notation is that it describes these groups in a way which indicates many of the groups' properties: in particular, it describes the orbifold obtained by taking the quotient of Euclidean space by the group under consideration. Groups representable in this notation include the point groups on the sphere, the frieze groups and wallpaper groups of the Euclidean plane, their analogues on the hyperbolic plane; the following types of Euclidean transformation can occur in a group described by orbifold notation: reflection through a line translation by a vector rotation of finite order around a point infinite rotation around a line in 3-space glide-reflection, i.e. reflection followed by translation. All translations which occur are assumed to form a discrete subgroup of the group symmetries being described.
Each group is denoted in orbifold notation by a finite string made up from the following symbols: positive integers 1, 2, 3, … the infinity symbol, ∞ the asterisk, * the symbol o, called a wonder and a handle because it topologically represents a torus closed surface. Patterns repeat by two translation; the symbol ×, called a miracle and represents a topological crosscap where a pattern repeats as a mirror image without crossing a mirror line. A string written in boldface represents a group of symmetries of Euclidean 3-space. A string not written in boldface represents a group of symmetries of the Euclidean plane, assumed to contain two independent translations; each symbol corresponds to a distinct transformation: an integer n to the left of an asterisk indicates a rotation of order n around a gyration point an integer n to the right of an asterisk indicates a transformation of order 2n which rotates around a kaleidoscopic point and reflects through a line an × indicates a glide reflection the symbol ∞ indicates infinite rotational symmetry around a line.
By abuse of language, we might say that such a group is a subgroup of symmetries of the Euclidean plane with only one independent translation. The frieze groups occur in this way; the exceptional symbol o indicates that there are two linearly independent translations. An orbifold symbol is called good if it is not one of the following: p, pq, *p, *pq, for p,q>=2, p≠q. An object is chiral; the corresponding orbifold is non-orientable otherwise. The Euler characteristic of an orbifold can be read from its Conway symbol; each feature has a value: n without or before an asterisk counts as n − 1 n n after an asterisk counts as n − 1 2 n asterisk and × count as 1 o counts as 2. Subtracting the sum of these values from 2 gives the Euler characteristic. If the sum of the feature values is 2, the order is infinite, i.e. the notation represents a wallpaper group or a frieze group. Indeed, Conway's "Magic Theorem" indicates that the 17 wallpaper groups are those with the sum of the feature values equal to 2.
Otherwise, the order is 2 divided by the Euler characteristic. The following groups are isomorphic: 1* and *11 22 and 221 *22 and *221 2* and 2*1; this is. The symmetry of a 2D object without translational symmetry can be described by the 3D symmetry type by adding a third dimension to the object which does not add or spoil symmetry. For example, for a 2D image we can consider a piece of carton with that image displayed on one side, thus we have n• and *n•. The bullet is added on one- and two-dimensional groups to imply the existence of a fixed point. A 1D image can be drawn horizontally on a piece of carton, with a provision to avoid additional symmetry with respect to the line of the image, e.g. by drawing a horizontal bar under the image. Thus the discrete symmetry groups in one dimension are *•, *1•, ∞• and *∞•. Another way of constructing a 3D object from a 1D or 2D object for describing the symmetry is taking the Cartesian product of the object and an asymmetric 2D or 1D object, respectively.
*Schönflies's point group notation is extended here as infinite cases of the equivalent dihedral points symmetries §The diagram shows one fundamental domain in yellow, with reflection lines in blue, glide reflection lines in dashed green, translation normals in red, 2-fold gyration points as small green squares. A first few hyperbolic groups, ordered by their Euler characteristic are: Mutation of orbifolds Fibrifold notation - an extension of orbifold notation for 3d space groups John H. Conway, Olaf Delgado Friedrichs, Daniel H. Huson, W
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R. Hyperbolic plane geometry is the geometry of saddle surfaces and pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. A modern use of hyperbolic geometry is in the theory of special relativity Minkowski spacetime and gyrovector space; when geometers first realised they were working with something other than the standard Euclidean geometry they described their geometry under many different names. In the former Soviet Union, it is called Lobachevskian geometry, named after one of its discoverers, the Russian geometer Nikolai Lobachevsky; this page is about the 2-dimensional hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry. Hyperbolic geometry can be extended to three and more dimensions.
Hyperbolic geometry is more related to Euclidean geometry than it seems: the only axiomatic difference is the parallel postulate. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry. There are two kinds of absolute geometry and hyperbolic. All theorems of absolute geometry, including the first 28 propositions of book one of Euclid's Elements, are valid in Euclidean and hyperbolic geometry. Propositions 27 and 28 of Book One of Euclid's Elements prove the existence of parallel/non-intersecting lines; this difference has many consequences: concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry. Further, because of the angle of parallelism, hyperbolic geometry has an absolute scale, a relation between distance and angle measurements. Single lines in hyperbolic geometry have the same properties as single straight lines in Euclidean geometry. For example, two points uniquely define a line, lines can be infinitely extended.
Two intersecting lines have the same properties as two intersecting lines in Euclidean geometry. For example, two lines can intersect in no more than one point, intersecting lines have equal opposite angles, adjacent angles of intersecting lines are supplementary; when we add a third line there are properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, given 2 intersecting lines there are infinitely many lines that do not intersect either of the given lines; these properties all are independent of the model used if the lines may look radically different. Non-intersecting lines in hyperbolic geometry have properties that differ from non-intersecting lines in Euclidean geometry: For any line R and any point P which does not lie on R, in the plane containing line R and point P there are at least two distinct lines through P that do not intersect R; this implies that there are through P an infinite number of coplanar lines that do not intersect R.
These non-intersecting lines are divided into two classes: Two of the lines are limiting parallels: there is one in the direction of each of the ideal points at the "ends" of R, asymptotically approaching R, always getting closer to R, but never meeting it. All other non-intersecting lines have a point of minimum distance and diverge from both sides of that point, are called ultraparallel, diverging parallel or sometimes non-intersecting; some geometers use parallel lines instead of limiting parallel lines, with ultraparallel lines being just non-intersecting. These limiting parallels make an angle θ with PB. For ultraparallel lines, the ultraparallel theorem states that there is a unique line in the hyperbolic plane, perpendicular to each pair of ultraparallel lines. In hyperbolic geometry, the circumference of a circle of radius r is greater than 2 π r. Let R = 1 − K, where K is the Gaussian curvature of the plane. In hyperbolic geometry, K is negative, so the square root is of a positive number.
The circumference of a circle of radius r is equal to: 2 π R sinh r R. And the area of the enclosed disk is: 4 π R 2 sinh 2 r 2 R = 2 π R 2. Therefore, in hyperbolic geometry the ratio of a circle's circumference to its radius is always greater than 2 π, though