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.
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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; the vertex of an angle is the point where two rays begin or meet, where two line segments join or meet, where two lines intersect, or any appropriate combination of rays and lines that result in two straight "sides" meeting at one place. A vertex is a corner point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection of edges, faces or facets of the object. In a polygon, a vertex is called "convex" if the internal angle of the polygon, that is, the angle formed by the two edges at the vertex, with the polygon inside the angle, is less than π radians. More a vertex of a polyhedron or polytope is convex if the intersection of the polyhedron or polytope with a sufficiently small sphere centered at the vertex is convex, concave otherwise. Polytope vertices are related to vertices of graphs, in that the 1-skeleton of a polytope is a graph, the vertices of which correspond to the vertices of the polytope, in that a graph can be viewed as a 1-dimensional simplicial complex the vertices of which are the graph's vertices.
However, in graph theory, vertices may have fewer than two incident edges, not allowed for geometric vertices. There is a connection between geometric vertices and the vertices of a curve, its points of extreme curvature: in some sense the vertices of a polygon are points of infinite curvature, if a polygon is approximated by a smooth curve there will be a point of extreme curvature near each polygon vertex. However, a smooth curve approximation to a polygon will have additional vertices, at the points where its curvature is minimal. A vertex of a plane tiling or tessellation is a point. More a tessellation can be viewed as a kind of topological cell complex, as can the faces of a polyhedron or polytope. 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: mouths. A principal vertex xi of a simple polygon P is called an ear if the diagonal that bridges xi lies 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 polyhedron's surface has Euler characteristic V − E + F = 2, where V is the number of vertices, E is the number of edges, F is the number of faces; this equation is known as Euler's 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, hence 8 vertices. In computer graphics, objects are represented as triangulated polyhedra in which the object vertices are associated not only with three spatial coordinates but with other graphical information necessary to render the object such as colors, reflectance properties and surface normal. Weisstein, Eric W. "Polygon Vertex". MathWorld. Weisstein, Eric W. "Polyhedron Vertex". MathWorld. Weisstein, Eric W. "Principal Vertex". MathWorld
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
Octagon
In geometry, an octagon is an eight-sided polygon or 8-gon. A regular octagon has Schläfli symbol and can be constructed as a quasiregular truncated square, t, which alternates two types of edges. A truncated octagon, t; the sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°. If squares are constructed all internally or all externally on the sides of an octagon the midpoints of the segments connecting the centers of opposite squares form a quadrilateral, both equidiagonal and orthodiagonal; the midpoint octagon of a reference octagon has its eight vertices at the midpoints of the sides of the reference octagon. If squares are constructed all internally or all externally on the sides of the midpoint octagon the midpoints of the segments connecting the centers of opposite squares themselves form the vertices of a square. A regular octagon is a closed figure with sides of the same length and internal angles of the same size, it has eight lines of reflective symmetry and rotational symmetry of order 8.
A regular octagon is represented by the Schläfli symbol. The internal angle at each vertex of a regular octagon is 135°; the central angle is 45°. The area of a regular octagon of side length a is given by A = 2 cot π 8 a 2 = 2 a 2 ≃ 4.828 a 2. In terms of the circumradius R, the area is A = 4 sin π 4 R 2 = 2 2 R 2 ≃ 2.828 R 2. In terms of the apothem r, the area is A = 8 tan π 8 r 2 = 8 r 2 ≃ 3.314 r 2. These last two coefficients bracket the value of the area of the unit circle; the area can be expressed as A = S 2 − a 2, where S is the span of the octagon, or the second-shortest diagonal. This is proven if one takes an octagon, draws a square around the outside and takes the corner triangles and places them with right angles pointed inward, forming a square; the edges of this square are each the length of the base. Given the length of a side a, the span S is S = a 2 + a + a 2 = a ≈ 2.414 a. The area is as above: A = 2 − a 2 = 2 a 2 ≈ 4.828 a 2. Expressed in terms of the span, the area is A = 2 S 2 ≈ 0.828 S 2.
Another simple formula for the area is A = 2 a S. More the span S is known, the length of the sides, a, is to be determined, as when cutting a square piece of material into a regular octagon. From the above, a ≈ S / 2.414. The two end lengths e on each side, as well as being e = a / 2, may be calculated as e = / 2; the circumradius of the regular octagon in terms of the side length a is R = a, the inradius is r = a. The regular octagon, in ter
Dual polyhedron
In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a symmetric class. Thus, the regular polyhedra – the Platonic solids and Kepler–Poinsot polyhedra – form dual pairs, where the regular tetrahedron is self-dual; the dual of an isogonal polyhedron, having equivalent vertices, is one, isohedral, having equivalent faces. The dual of an isotoxal polyhedron is isotoxal. Duality is related to reciprocity or polarity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.
There are many kinds of duality. The kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality; the duality of polyhedra is defined in terms of polar reciprocation about a concentric sphere. Here, each vertex is associated with a face plane so that the ray from the center to the vertex is perpendicular to the plane, the product of the distances from the center to each is equal to the square of the radius. In coordinates, for reciprocation about the sphere x 2 + y 2 + z 2 = r 2, the vertex is associated with the plane x 0 x + y 0 y + z 0 z = r 2; the vertices of the dual are the poles reciprocal to the face planes of the original, the faces of the dual lie in the polars reciprocal to the vertices of the original. Any two adjacent vertices define an edge, these will reciprocate to two adjacent faces which intersect to define an edge of the dual; this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, r 1 and r 2 the distances from its centre to the pole and its polar, then: r 1.
R 2 = r 0 2 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, as in the Dorman Luke construction described below. However, it is possible to reciprocate a polyhedron about any sphere, the resulting form of the dual will depend on the size and position of the sphere; the choice of center for the sphere is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will intersect at a single point, this is taken to be the centroid. Failing that, a circumscribed sphere, inscribed sphere, or midsphere is used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, the corresponding element of its dual will go to infinity. Since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required'plane at infinity'; some theorists prefer to say that there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, in a manner suitable for making models.
The concept of duality here is related to the duality in projective geometry, where lines and edges are interchanged. Projective polarity works well enough for convex polyhedra, but for non-convex figures such as star polyhedra, when we seek to rigorously define this form of polyhedral duality in terms of projective polarity, various problems appear. Because of the definitional issues for geometric duality of non-convex polyhedra, Grünbaum argues that any proper definition of a non-convex polyhedron should include a notion of a dual polyhedron. Any convex polyhedron can be distorted into a canonical form, in which a unit midsphere exists tangent to every edge, such that the average position of the points of tangency is the center of the sphere; this form is unique up to congruences. If we reciprocate such a canonical polyhedron about its midsphere, the dual polyhedron will share the same edge-tangency points and so must be canonical, it is the canonical dual, the two together form a canonical dual pair.
When a pair of polyhedra cannot be obtained by reciprocation from each other, they may be called duals of each other as long as the vertices of one correspond to the faces of the other, the edges of one correspond to the edges of the other, in an incidence-preserving way. Such pairs of polyhedra are abstractly dual; the vertices and edges of a convex polyhedron form a graph, embedded on a topological sphere, the surface of the polyhedron. The same graph can be projected to form
Dodecagon
In geometry, a dodecagon or 12-gon is any twelve-sided polygon. A regular dodecagon is a figure with internal angles of the same size, it has twelve lines of reflective symmetry and rotational symmetry of order 12. A regular dodecagon is represented by the Schläfli symbol and can be constructed as a truncated hexagon, t, or a twice-truncated triangle, tt; the internal angle at each vertex of a regular dodecagon is 150°. The area of a regular dodecagon of side length a is given by: A = 3 cot a 2 = 3 a 2 ≃ 11.19615242 a 2 And in terms of the apothem r, the area is: A = 12 tan r 2 = 12 r 2 ≃ 3.2153903 r 2 In terms of the circumradius R, the area is: A = 6 sin R 2 = 3 R 2 The span S of the dodecagon is the distance between two parallel sides and is equal to twice the apothem. A simple formula for area is: A = 3 a S This can be verified with the trigonometric relationship: S = a The perimeter of a regular dodecagon in terms of circumradius is: p = 24 R tan = 12 R 2 − 3 ≃ 6.21165708246 R The perimeter in terms of apothem is: p = 24 r tan = 24 r ≃ 6.43078061835 r This coefficient is double the coefficient found in the apothem equation for area.
As 12 = 22 × 3, regular dodecagon is constructible using compass-and-straightedge construction: Coxeter states that every zonogon can be dissected into m/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular dodecagon, m=6, it can be divided into 15: 3 squares, 6 wide 30° rhombs and 6 narrow 15° rhombs; this decomposition is based with 15 of 240 faces. The sequence OEIS sequence A006245 defines the number of solutions as 908, including up to 12-fold rotations and chiral forms in reflection. One of the ways the mathematical manipulative pattern blocks are used is in creating a number of different dodecagons, they are related to the rhombic dissections, with 3 60° rhombi merged into hexagons, half-hexagon trapezoids, or divided into 2 equilateral triangles. The regular dodecagon has Dih12 symmetry, order 24. There are 15 distinct subgroup cyclic symmetries; each subgroup symmetry allows one or more degrees of freedom for irregular forms.
Only the g12 subgroup has no degrees of freedom. A regular dodecagon can fill a plane vertex with other regular polygons in 4 ways: Here are 3 example periodic plane tilings that use regular dodecagons, defined by their vertex configuration: A skew dodecagon is a skew polygon with 12 vertices and edges but not existing on the same plane; the interior of such an dodecagon is not defined. A skew zig-zag. A regular skew dodecagon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew dodecagon and can be seen in the vertices and side edges of a hexagonal antiprism with the same D5d, order
Truncated octahedron
In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces, 36 edges, 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron, it is the Goldberg polyhedron GIV, containing square and hexagonal faces. Like the cube, it can tessellate 3-dimensional space, as a permutohedron, its dual polyhedron is the tetrakis hexahedron. If the original truncated octahedron has unit edge length, its dual tetrakis cube has edge lengths 9/8√2 and 3/2√2. A truncated octahedron is constructed from a regular octahedron with side length 3a by the removal of six right square pyramids, one from each point; these pyramids have both base side length and lateral side length of a, to form equilateral triangles. The base area is a2. Note that this shape is similar to half an octahedron or Johnson solid J1. From the properties of square pyramids, we can now find the slant height, s, the height, h, of the pyramid: h = e 2 − 1 2 a 2 = 2 2 a s = h 2 + 1 4 a 2 = 1 2 a 2 + 1 4 a 2 = 3 2 a The volume, V, of the pyramid is given by: V = 1 3 a 2 h = 2 6 a 3 Because six pyramids are removed by truncation, there is a total lost volume of √2a3.
The truncated octahedron has five special orthogonal projections, centered, on a vertex, on two types of edges, two types of faces: Hexagon, square. The last two correspond to the B2 and A2 Coxeter planes; the truncated 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. All permutations of are Cartesian coordinates of the vertices of a truncated octahedron of edge length a = √ 2 centered at the origin; the vertices are thus the corners of 12 rectangles whose long edges are parallel to the coordinate axes. The edge vectors have Cartesian permutations of these; the face normals of the 6 square faces are, and. The face normals of the 8 hexagonal faces are; the dot product between pairs of two face normals is the cosine of the dihedral angle between adjacent faces, either −1/3 or −1/√3. The dihedral angle is 1.910633 radians at edges shared by two hexagons or 2.186276 radians at edges shared by a hexagon and a square.
The truncated octahedron can be dissected into a central octahedron, surrounded by 8 triangular cupola on each face, 6 square pyramids above the vertices. Removing the central octahedron and 2 or 4 triangular cupola creates two Stewart toroids, with dihedral and tetrahedral symmetry: The truncated octahedron can be represented by more symmetric coordinates in four dimensions: all permutations of form the vertices of a truncated octahedron in the three-dimensional subspace x + y + z + w = 10. Therefore, the truncated octahedron is the permutohedron of order 4: each vertex corresponds to a permutation of and each edge represents a single pairwise swap of two elements; the area A and the volume V of a truncated octahedron of edge length a are: A = a 2 ≈ 26.784 6097 a 2 V = 8 2 a 3 ≈ 11.313 7085 a 3. There are two uniform colorings, with tetrahedral symmetry and octahedral symmetry, two 2-uniform coloring with dihedral symmetry as a truncated triangular antiprism; the construcational names are given for each.
Their Conway polyhedron notation is given in parentheses. The truncated octahedron exists in the structure of the faujasite crystals; the truncated octahedron is one of a family of uniform polyhedra related to the cube and regular octahedron. It exists as the omnitruncate of the tetrahedron family: This polyhedron is a member of a sequence of uniform patterns with vertex figure a