Pentagon

In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be self-intersecting. A self-intersecting regular pentagon is called a pentagram. A regular pentagon has Schläfli symbol and interior angles are 108°. A regular pentagon has five lines of reflectional symmetry, rotational symmetry of order 5; the diagonals of a convex regular pentagon are in the golden ratio to its sides. Its height and width are given by Height = 5 + 2 5 2 ⋅ Side ≈ 1.539 ⋅ Side, Width = Diagonal = 1 + 5 2 ⋅ Side ≈ 1.618 ⋅ Side, Diagonal = R 5 + 5 2 = 2 R cos 18 ∘ = 2 R cos π 10 ≈ 1.902 R, where R is the radius of the circumcircle. The area of a convex regular pentagon with side length t is given by A = t 2 25 + 10 5 4 = 5 t 2 tan 4 ≈ 1.720 t 2. A pentagram or pentangle is a regular star pentagon, its Schläfli symbol is. Its sides form the diagonals of a regular convex pentagon – in this arrangement the sides of the two pentagons are in the golden ratio.

When a regular pentagon is circumscribed by a circle with radius R, its edge length t is given by the expression t = R 5 − 5 2 = 2 R sin 36 ∘ = 2 R sin π 5 ≈ 1.176 R, its area is A = 5 R 2 4 5 + 5 2. The area of any regular polygon is: A = 1 2 P r where P is the perimeter of the polygon, r is the inradius. Substituting the regular pentagon's values for P and r gives the formula A = 1 2 ⋅ 5 t ⋅ t tan 2 = 5 t 2 tan 4 with side length “f” Like every regular convex polygon, the regular convex pentagon has an inscribed circle; the apothem, the radius r of the inscribed circle, of a regular pentagon is related to the side length t by r = t 2 tan = t 2 5 − 20 ≈ 0.6882 ⋅ t. Like every regular convex polygon, the regular convex pentagon has a circumscribed circle. For a regular pentagon with successive vertices A, B, C, D, E, if P is any point on the circumcircle between points B and C PA + PD = PB + PC + PE; the regular pentagon is constructible with compass and straightedge. A variety of methods are known for constructing a regular pentagon.

Some are discussed below. One method to construct a regular pentagon in a given circle is described by Richmond and further discussed in Cromwell's "Polyhedra."The top panel shows the construction used in Richmond's method to create the side of the inscribed pentagon. The circle defining the pentagon has unit radius, its center is located at point

Cupola (geometry)

In geometry, a cupola is a solid formed by joining two polygons, one with twice as many edges as the other, by an alternating band of isosceles triangles and rectangles. If the triangles are equilateral and the rectangles are squares, while the base and its opposite face are regular polygons, the triangular and pentagonal cupolae all count among the Johnson solids, can be formed by taking sections of the cuboctahedron, rhombicuboctahedron, rhombicosidodecahedron, respectively. A cupola can be seen as a prism where one of the polygons has been collapsed in half by merging alternate vertices. A cupola can be given an extended Schläfli symbol || t, representing a regular polygon joined by a parallel of its truncation, t or. Cupolae are a subclass of the prismatoids, its dual contains a shape, sort of a weld between half of an n-sided trapezohedron and a 2n-sided pyramid. The above-mentioned three polyhedra are the only non-trivial convex cupolae with regular faces: The "hexagonal cupola" is a plane figure, the triangular prism might be considered a "cupola" of degree 2.

However, cupolae of higher-degree polygons may be constructed with irregular triangular and rectangular faces. The definition of the cupola does not require the base to be a regular polygon, but it is convenient to consider the case where the cupola has its maximal symmetry, Cnv. In that case, the top is a regular n-gon, while the base is either a regular 2n-gon or a 2n-gon which has two different side lengths alternating and the same angles as a regular 2n-gon, it is convenient to fix the coordinate system so that the base lies in the xy-plane, with the top in a plane parallel to the xy-plane. The z-axis is the n-fold axis, the mirror planes pass through the z-axis and bisect the sides of the base, they either bisect the sides or the angles of the top polygon, or both. The vertices of the base can be designated V1 through V2n, while the vertices of the top polygon can be designated V2n+1 through V3n. With these conventions, the coordinates of the vertices can be written as: V2j−1: V2j: V2n+j: where j = 1, 2... n.

Since the polygons V1V2V2n+2V2n+1, etc. are rectangles, this puts a constraint on the values of rb, rt, α. The distance V1V2 is equal to rb1⁄2 = rb1⁄2 = rb1⁄2 = rb1⁄2while the distance V2n+1V2n+2 is equal to rt1⁄2 = rt1⁄2 = rt1⁄2; these are to be equal, if this common edge is denoted by s, rb = s / 1⁄2 rt = s / 1⁄2These values are to be inserted into the expressions for the coordinates of the vertices given earlier. Star cupolae exist for all bases where 6/5 < n/d < d is odd. At the limits the cupolae collapse into plane figures: beyond the limits the triangles and squares can no longer span the distance between the two polygons; when d is the bottom base becomes degenerate: we can form a cuploid or semicupola by withdrawing this degenerate face and instead letting the triangles and squares connect to each other here. In particular, the tetrahemihexahedron may be seen as a -cuploid; the cupolae are all orientable. When n/d > 2 in a cuploid, the triangles and squares do not cover the entire base, a small membrane is left in the base that covers empty space.

Hence the and cuploids pictured above have membranes. The height h of an -cupola or cuploid is given by the formula h = 1 − 1 4 sin 2 . In particular, h = 0 at the limits of n/d = 6 and n/d = 6/5, h is maximized at n/d = 2. In the images above, the star cupolae have been given a consistent colour scheme to aid identifying their faces: the base n/d-gon is red, the base 2n/d-gon is yellow, the squares are blue, the triangles are green; the cuploids have the base n/d-gon red, the squares yellow, the triangles blue, as the other base has been withdrawn. An n-gonal anticupola is constructed from a regular 2n-gonal base, 3n triangles as two types, a regular n-gonal top. For n=2, the top digon face is reduced to a single edge; the vertices of the top polygon are aligned with vertices in the lower polygon. The symmetry is order 2n. An anticupola can't be constructed with all regular faces. If the top n-gon and triangles are regular, the base 2n-gon can not be regular. In such a case, n=6 generates a regular hexagon and surrounding equilateral triangles of a snub hexagonal tiling, which can be closed into a zero volume polygon with the base a symmetric 12-gon shaped like a larger hexagon, having adjacent pairs of colinear edges.

Two anticupola can be augmented toget

Rhombic dodecahedron

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, 14 vertices of two types, it is a Catalan solid, the dual polyhedron of the cuboctahedron. The rhombic dodecahedron is a zonohedron, its polyhedral dual is the cuboctahedron. The long diagonal of each face is √2 times the length of the short diagonal, so that the acute angles on each face measure arccos, or 70.53°. Being the dual of an Archimedean polyhedron, the rhombic dodecahedron is face-transitive, meaning the symmetry group of the solid acts transitively on the set of faces. In elementary terms, this means that for any two faces A and B there is a rotation or reflection of the solid that leaves it occupying the same region of space while moving face A to face B; the rhombic dodecahedron is one of the nine edge-transitive convex polyhedra, the others being the five Platonic solids, the cuboctahedron, the icosidodecahedron and the rhombic triacontahedron. The rhombic dodecahedron can be used to tessellate three-dimensional space.

It can be stacked to fill a space. This polyhedron in a space-filling tessellation can be seen as the Voronoi tessellation of the face-centered cubic lattice, it is the Brillouin zone of body centered cubic crystals. Some minerals such as garnet form a rhombic dodecahedral crystal habit. Honey bees use the geometry of rhombic dodecahedra to form honeycombs from a tessellation of cells each of, a hexagonal prism capped with half a rhombic dodecahedron; the rhombic dodecahedron appears in the unit cells of diamond and diamondoids. In these cases, four vertices are absent; the graph of the rhombic dodecahedron is nonhamiltonian. A rhombic dodecahedron can be dissected with its center into 4 trigonal trapezohedra; these rhombohedra are the cells of a trigonal trapezohedral honeycomb. This is analogous to the dissection of a regular hexagon dissected into rhombi, tiled in the plane as a rhombille. If the edge length of a rhombic dodecahedron is a, the radius of an inscribed sphere is r i = 6 3 a ≈ 0.816 496 5809 a, OEIS: A157697the radius of the midsphere is r m = 2 2 3 a ≈ 0.942 809 041 58 a, OEIS: A179587.and the radius of the circumscribed sphere is r o = 2 3 3 a ≈ 1.154 700 538 a, OEIS: A020832.

The area A and the volume V of the rhombic dodecahedron of edge length a are: A = 8 2 a 2 ≈ 11.313 7085 a 2 V = 16 3 9 a 3 ≈ 3.079 201 44 a 3 The rhombic dodecahedron has four special orthogonal projections along its axes of symmetry, centered on a face, an edge, the two types of vertex and fourfold. The last two correspond to the B2 and A2 Coxeter planes; the eight vertices where three faces meet at their obtuse angles have Cartesian coordinates: The coordinates of the six vertices where four faces meet at their acute angles are:, The rhombic dodecahedron can be seen as a degenerate limiting case of a pyritohedron, with permutation of coordinates and with parameter h = 1. The rhombic dodecahedron is a parallelohedron, a space-filling polyhedron, being the dual to the tetroctahedrille or half cubic honeycomb, described by two Coxeter diagrams: and. With D3d symmetry, it can be seen as an elongated trigonal trapezohedron. Other symmetry constructions of the rhombic dodecahedron are space-filling, as parallelotopes they are similar to variations of space-filling truncated octahedra.

For example, with 4 square faces, 60-degree rhombic faces, D4h dihedral symmetry, order 16. It be seen as a cuboctahedron with square pyramids augmented on the bottom. In 1960 Stanko Bilinski discovered a second rhombic dodecahedron with 12 congruent rhombus faces, the Bilinski dodecahedron, it has the same different geometry. The rhombic faces in this form have the golden ratio. Another topologically equivalent variation, sometimes called a deltoidal dodecahedron or trapezoidal dodecahedron, is isohedral with tetrahedral symmetry order 24, distorting rhombic faces into kites, it has 8 vertices adjusted in or out in alternate sets of 4, with the limiting case a tetrahedral envelope. Variations can be parametrized by. Is the rhombic solution; as approaches 1/2, approaches infinity. (

Triakis tetrahedron

In geometry, a triakis tetrahedron is a Catalan solid with 12 faces. Each Catalan solid is the dual of an Archimedean solid; the dual of the triakis tetrahedron is the truncated tetrahedron. The triakis tetrahedron can be seen, it is similar to the net for the 5-cell, as the net for a tetrahedron is a triangle with other triangles added to each edge, the net for the 5-cell a tetrahedron with pyramids attached to each face. This interpretation is expressed in the name; the length of the shorter edges is 3/5 that of the longer edges. If the triakis tetrahedron has shorter edge length 1, it has area 5/3√11 and volume 25/36√2; the triakis tetrahedron can be made as a degenerate limit of a tetaroid: A triakis tetrahedron with equilateral triangle faces represents a net of the four-dimensional regular polytope known as the 5-cell. If the triangles are right-angled isosceles, the faces will form a cubic volume; this can be seen by adding the 6 edges of tetrahedron inside of a cube. This chiral figure is one of thirteen stellations allowed by Miller's rules.

The triakis tetrahedron is a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have reflectional symmetry. Truncated triakis tetrahedron Williams, Robert; the Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. Wenninger, Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208 The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 Eric W. Weisstein, Triakis tetrahedron at MathWorld

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_=

Solid geometry

In mathematics, solid geometry is the traditional name for the geometry of three-dimensional Euclidean space. Stereometry deals with the measurements of volumes of various solid figures including pyramids and other polyhedrons; the Pythagoreans dealt with the regular solids, but the pyramid, prism and cylinder were not studied until the Platonists. Eudoxus established their measurement, proving the pyramid and cone to have one-third the volume of a prism and cylinder on the same base and of the same height, he was also the discoverer of a proof that the volume enclosed by a sphere is proportional to the cube of its radius. Basic topics in solid geometry and stereometry include Advanced topics include projective geometry of three dimensions further polyhedra descriptive geometry. Various techniques and tools are used in solid geometry. Among them, analytic geometry and vector techniques have a major impact by allowing the systematic use of linear equations and matrix algebra, which are important for higher dimensions.

A major application of solid geometry and stereometry is in computer graphics. Euclidean geometry Dimension Point Planimetry Shape Surface Surface area Archimedes Kiselev, A. P.. Geometry. Book II. Stereometry. Translated by Givental, Alexander. Sumizdat

Dodecahedron

In geometry, a dodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron, a Platonic solid. There are three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120; the pyritohedron, a common crystal form in pyrite, is an irregular pentagonal dodecahedron, having the same topology as the regular one but pyritohedral symmetry while the tetartoid has tetrahedral symmetry. The rhombic dodecahedron, seen as a limiting case of the pyritohedron, has octahedral symmetry; the elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra, are space-filling. There are a large number of other dodecahedra; the convex regular dodecahedron is one of the five regular Platonic solids and can be represented by its Schläfli symbol. The dual polyhedron is the regular icosahedron, having five equilateral triangles around each vertex; the convex regular dodecahedron has three stellations, all of which are regular star dodecahedra.

They form three of the four Kepler–Poinsot polyhedra. They are the small stellated dodecahedron, the great dodecahedron, the great stellated dodecahedron; the small stellated dodecahedron and great dodecahedron are dual to each other. All of these regular star dodecahedra have regular pentagrammic faces; the convex regular dodecahedron and great stellated dodecahedron are different realisations of the same abstract regular polyhedron. In crystallography, two important dodecahedra can occur as crystal forms in some symmetry classes of the cubic crystal system that are topologically equivalent to the regular dodecahedron but less symmetrical: the pyritohedron with pyritohedral symmetry, the tetartoid with tetrahedral symmetry: A pyritohedron is a dodecahedron with pyritohedral symmetry. Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not constrained to be regular, the underlying atomic arrangement has no true fivefold symmetry axes.

Its 30 edges are divided into two sets -- containing 6 edges of the same length. The only axes of rotational symmetry are three mutually perpendicular twofold axes and four threefold axes. Although regular dodecahedra do not exist in crystals, the pyritohedron form occurs in the crystals of the mineral pyrite, it may be an inspiration for the discovery of the regular Platonic solid form; the true regular dodecahedron can occur as a shape for quasicrystals with icosahedral symmetry, which includes true fivefold rotation axes. Its name comes from one of the two common crystal habits shown by pyrite, the other one being the cube; the coordinates of the eight vertices of the original cube are: The coordinates of the 12 vertices of the cross-edges are: where h is the height of the wedge-shaped "roof" above the faces of the cube. When h = 1, the six cross-edges degenerate to points and a rhombic dodecahedron is formed; when h = 0, the cross-edges are absorbed in the facets of the cube, the pyritohedron reduces to a cube.

When h = −1 + √5/2, the multiplicative inverse of the golden ratio, the result is a regular dodecahedron. When h = −1 − √5/2, the conjugate of this value, the result is a regular great stellated dodecahedron. A reflected pyritohedron is made by swapping; the two pyritohedra can be superimposed to give the compound of two dodecahedra. The image to the left shows the case; the pyritohedron has a geometric degree of freedom with limiting cases of a cubic convex hull at one limit of colinear edges, a rhombic dodecahedron as the other limit as 6 edges are degenerated to length zero. The regular dodecahedron represents a special intermediate case where all edges and angles are equal. A tetartoid is a dodecahedron with chiral tetrahedral symmetry. Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes. Although regular dodecahedra do not exist in crystals, the tetartoid form does.

The name tetartoid comes from the Greek root for one-fourth because it has one fourth of full octahedral symmetry, half of pyritohedral symmetry. The mineral cobaltite can have this symmetry form, its topology can be as a cube with square faces bisected into 2 rectangles like the pyritohedron, the bisection lines are slanted retaining 3-fold rotation at the 8 corners. The following points are vertices of a tetartoid pentagon under tetrahedral symmetry:, it can be seen as a tetrahedron, with edges divided into 3 segments, along with a center point of each triangular face. In Conway polyhedron notation it can be seen as a gyro tetrahedron. A lower symmetry form of the regular dodecahedron can be constructed as the dual of a polyhedra constructed from two triangular anticupola connected base-to-base, called a triangular gyrobianticupo