Rhombic triacontahedron
In geometry, the rhombic triacontahedron, sometimes called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 32 vertices of two types, it is a Catalan solid, the dual polyhedron of the icosidodecahedron. It is a zonohedron; the ratio of the long diagonal to the short diagonal of each face is equal to the golden ratio, φ, so that the acute angles on each face measure 2 tan−1 = tan−1, or 63.43°. A rhombus so obtained. Being the dual of an Archimedean solid, the rhombic triacontahedron is face-transitive, meaning the symmetry group of the solid acts transitively on the set of faces; 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 triacontahedron is somewhat special in being one of the nine edge-transitive convex polyhedra, the others being the five Platonic solids, the cuboctahedron, the icosidodecahedron, the rhombic dodecahedron.
The rhombic triacontahedron is interesting in that its vertices include the arrangement of four Platonic solids. It contains five cubes, an icosahedron and a dodecahedron; the centers of the faces contain five octahedra. It can be made from a truncated octahedron by dividing the hexagonal faces into 3 rhombi: If the edge length of a rhombic triacontahedron is a, surface area, the radius of an inscribed sphere and midradius, which touches the middle of each edge are: S = 12 5 a 2 ≈ 26.8328 a 2 V = 4 5 + 2 5 a 3 ≈ 12.3107 a 3 r i = φ 2 1 + φ 2 a = 1 + 2 5 a ≈ 1.37638 a r m = a ≈ 1.44721 a where φ is the golden ratio. The insphere is tangent to the faces at their face centroids. Short diagonals belong only to the edges of the inscribed regular dodecahedron, while long diagonals are included only in edges of the inscribed icosahedron; the rhombic triacontahedron can be dissected into 20 golden rhombohedra: 10 acute ones and 10 obtuse ones. The rhombic triacontahedron has four symmetry positions, two centered on vertices, one mid-face, one mid-edge.
Embedded in projection "10" are the "fat" rhombus and "skinny" rhombus which tile together to produce the non-periodic tessellation referred to as Penrose tiling. The rhombic triacontahedron has 227 supported stellations; the total number of stellations of the rhombic triacontahedron is 358,833,097. This polyhedron is a part of a sequence of rhombic polyhedra and tilings with Coxeter group symmetry; the cube can be seen as a rhombic hexahedron where the rhombi are rectangles. The rhombic triacontahedron forms a 32 vertex convex hull of one projection of a 6-cube to three dimensions. Danish designer Holger Strøm used the rhombic triacontahedron as a basis for the design of his buildable lamp IQ-light. Woodworker Jane Kostick builds boxes in the shape of a rhombic triacontahedron; the simple construction is based on the less than obvious relationship between the rhombic triacontahedron and the cube. Roger von Oech's "Ball of Whacks" comes in the shape of a rhombic triacontahedron; the rhombic triacontahedron is used as the "d30" thirty-sided die, sometimes useful in some roleplaying games or other places.
Golden rhombus Rhombille tiling 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, Rhombic triacontahedron at MathWorld. Rhombic Triacontrahedron – Interactive Polyhedron Model Virtual Reality Polyhedra – The Encyclopedia of Polyhedra Stellations of Rhombic Triacontahedron EarthStar globe – Rhombic Triacontahedral map projection IQ-light—Danish designer Holger Strøm's lamp Make your own a wooden construction of a rhombic triacontahedron box – by woodworker Jane Kostick 120 Rhombic Triaconta
Tetrahedron
In geometry, a tetrahedron known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces; the tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, may thus be called a 3-simplex. The tetrahedron is one kind of pyramid, a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle, so a tetrahedron is known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper, it has two such nets. For any tetrahedron there exists a sphere on which all four vertices lie, another sphere tangent to the tetrahedron's faces. A regular tetrahedron is one, it is one of the five regular Platonic solids. In a regular tetrahedron, all faces are the same size and shape and all edges are the same length.
Regular tetrahedra alone do not tessellate, but if alternated with regular octahedra in the ratio of two tetrahedra to one octahedron, they form the alternated cubic honeycomb, a tessellation. The regular tetrahedron is self-dual; the compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. The following Cartesian coordinates define the four vertices of a tetrahedron with edge length 2, centered at the origin, two level edges: and Expressed symmetrically as 4 points on the unit sphere, centroid at the origin, with lower face level, the vertices are: v1 = v2 = v3 = v4 = with the edge length of sqrt. Still another set of coordinates are based on an alternated cube or demicube with edge length 2; 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, the pair together form the stellated octahedron, whose vertices are those of the original cube. Tetrahedron:, Dual tetrahedron:, For a regular tetrahedron of edge length a: With respect to the base plane the slope of a face is twice that of an edge, corresponding to the fact that the horizontal distance covered from the base to the apex along an edge is twice that along the median of a face.
In other words, if C is the centroid of the base, the distance from C to a vertex of the base is twice that from C to the midpoint of an edge of the base. This follows from the fact that the medians of a triangle intersect at its centroid, this point divides each of them in two segments, one of, twice as long as the other. For a regular tetrahedron with side length a, radius R of its circumscribing sphere, distances di from an arbitrary point in 3-space to its four vertices, we have d 1 4 + d 2 4 + d 3 4 + d 4 4 4 + 16 R 4 9 = 2.
Kepler–Poinsot polyhedron
In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra. They may be obtained by stellating the regular convex dodecahedron and icosahedron, differ from these in having regular pentagrammic faces or vertex figures, they can all be seen as three-dimensional analogues of the pentagram in another. These figures have pentagrams as faces or vertex figures; the small and great stellated dodecahedron have nonconvex regular pentagram faces. The great dodecahedron and great icosahedron have convex polygonal faces, but pentagrammic vertex figures. In all cases, two faces can intersect along a line, not an edge of either face, so that part of each face passes through the interior of the figure; such lines of intersection are not part of the polyhedral structure and are sometimes called false edges. Where three such lines intersect at a point, not a corner of any face, these points are false vertices; the images below show spheres at the true vertices, blue rods along the true edges.
For example, the small stellated dodecahedron has 12 pentagram faces with the central pentagonal part hidden inside the solid. The visible parts of each face comprise five isosceles triangles which touch at five points around the pentagon. We could treat these triangles as 60 separate faces to obtain a new, irregular polyhedron which looks outwardly identical; each edge would now be divided into three shorter edges, the 20 false vertices would become true ones, so that we have a total of 32 vertices. The hidden inner pentagons are no longer part of the polyhedral surface, can disappear. Now Euler's formula holds: 60 − 90 + 32 = 2; however this polyhedron is no longer the one described by the Schläfli symbol, so can not be a Kepler–Poinsot solid though it still looks like one from outside. A Kepler–Poinsot polyhedron covers its circumscribed sphere more than once, with the centers of faces acting as winding points in the figures which have pentagrammic faces, the vertices in the others; because of this, they are not topologically equivalent to the sphere as Platonic solids are, in particular the Euler relation χ = V − E + F = 2 does not always hold.
Schläfli held that all polyhedra must have χ = 2, he rejected the small stellated dodecahedron and great dodecahedron as proper polyhedra. This view was never held. A modified form of Euler's formula, using density of the vertex figures and faces was given by Arthur Cayley, holds both for convex polyhedra, the Kepler–Poinsot polyhedra: d v V − E + d f F = 2 D; the Kepler–Poinsot polyhedra exist in dual pairs. Duals have the same Petrie polygon, or more Petrie polygons with the same two dimensional projection; the following images show the two dual compounds with the same edge radius. They show that the Petrie polygons are skew. Two relationships described in the article below are easily seen in the images: That the violet edges are the same, that the green faces lie in the same planes. John Conway defines the Kepler–Poinsot polyhedra as greatenings and stellations of the convex solids. In his naming convention the small stellated dodecahedron is just the stellated dodecahedron. Stellation changes pentagonal faces into pentagrams.
Greatening maintains the type of faces and resizing them into parallel planes. The great icosahedron is one of the stellations of the icosahedron; the three others are all the stellations of the dodecahedron. The great stellated dodecahedron is a faceting of the dodecahedron; the three others are facetings of the icosahedron. If the intersections are treated as new edges and vertices, the figures obtained will not be regular, but they can still be considered stellations; the great stellated dodecahedron shares its vertices with the dodecahedron. The other three Kepler–Poinsot polyhedra share theirs with the icosahedron; the skeletons of the solids sharing vertices are topologically equivalent. The small and great stellated dodecahedron can be seen as a regular and a great dodecahedron with their edges and faces extended until they intersect; the pentagon faces of these cores are the invisible parts of the star polyhedra's pentagram faces. For the small stellated dodecahedron the hull is φ times bigger than the core, for the great it is φ + 1 = φ 2 times bigger.
Traditionally the two star polyhedra have been defined as augmentations, i.e. as dodecahedron and icosahedron with pyramids added to their faces. Kepler calls the small stellation an augmented dodecahedron. In his view the great stellation is related to the icosahedron as the small one is to the dodecahedron; these naïve definitions are still used. E.g. MathWorld states that the two star polyhedra can be constructed by adding pyramids to the faces of the Platonic solids; this is just a help to visualize the shape of these solids, not a claim that the edg
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
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube is one of the five Platonic solids, it has 6 faces, 12 edges, 8 vertices. The cube is a square parallelepiped, an equilateral cuboid and a right rhombohedron, it is a regular square prism in three orientations, a trigonal trapezohedron in four orientations. The cube is dual to the octahedron, it has octahedral symmetry. The cube is the only convex polyhedron; the cube has four special orthogonal projections, centered, on a vertex, edges and normal to its vertex figure. The first and third correspond to the B2 Coxeter planes; the cube 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. For a cube centered at the origin, with edges parallel to the axes and with an edge length of 2, the Cartesian coordinates of the vertices are while the interior consists of all points with −1 < xi < 1 for all i.
In analytic geometry, a cube's surface with center and edge length of 2a is the locus of all points such that max = a. For a cube of edge length a: As the volume of a cube is the third power of its sides a × a × a, third powers are called cubes, by analogy with squares and second powers. A cube has the largest volume among cuboids with a given surface area. A cube has the largest volume among cuboids with the same total linear size. For a cube whose circumscribing sphere has radius R, for a given point in its 3-dimensional space with distances di from the cube's eight vertices, we have: ∑ i = 1 8 d i 4 8 + 16 R 4 9 = 2. Doubling the cube, or the Delian problem, was the problem posed by ancient Greek mathematicians of using only a compass and straightedge to start with the length of the edge of a given cube and to construct the length of the edge of a cube with twice the volume of the original cube, they were unable to solve this problem, in 1837 Pierre Wantzel proved it to be impossible because the cube root of 2 is not a constructible number.
The cube has three uniform colorings, named by the colors of the square faces around each vertex: 111, 112, 123. The cube has three classes of symmetry, which can be represented by vertex-transitive coloring the faces; the highest octahedral symmetry Oh has all the faces the same color. The dihedral symmetry D4h comes from the cube being a prism, with all four sides being the same color; the lowest symmetry D2h is a prismatic symmetry, with sides alternating colors, so there are three colors, paired by opposite sides. Each symmetry form has a different Wythoff symbol. A cube has eleven nets: that is, there are eleven ways to flatten a hollow cube by cutting seven edges. To color the cube so that no two adjacent faces have the same color, one would need at least three colors; the cube is the cell of the only regular tiling of three-dimensional Euclidean space. It is unique among the Platonic solids in having faces with an number of sides and it is the only member of that group, a zonohedron; the cube can be cut into six identical square pyramids.
If these square pyramids are attached to the faces of a second cube, a rhombic dodecahedron is obtained. The analogue of a cube in four-dimensional Euclidean space has a special name—a tesseract or hypercube. More properly, a hypercube is the analogue of the cube in n-dimensional Euclidean space and a tesseract is the order-4 hypercube. A hypercube is called a measure polytope. There are analogues of the cube in lower dimensions too: a point in dimension 0, a line segment in one dimension and a square in two dimensions; the quotient of the cube by the antipodal map yields the hemicube. If the original cube has edge length 1, its dual polyhedron has edge length 2 / 2; the cube is a special case in various classes of general polyhedra: The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron. These two together form the stella octangula; the int
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
Wythoff symbol
In geometry, the Wythoff symbol represents a Wythoff construction of a uniform polyhedron or plane tiling, from a Schwarz triangle. It was first used by Coxeter, Longuet-Higgins and Miller in their enumeration of the uniform polyhedra. A Wythoff symbol consists of a vertical bar, it represents one uniform polyhedron or tiling, although the same tiling/polyhedron can have different Wythoff symbols from different symmetry generators. For example, the regular cube can be represented by 3 | 4 2 with Oh symmetry, 2 4 | 2 as a square prism with 2 colors and D4h symmetry, as well as 2 2 2 | with 3 colors and D 2 h symmetry. With a slight extension, Wythoff's symbol can be applied to all uniform polyhedra. However, the construction methods do not lead to all uniform tilings in Euclidean or hyperbolic space. In three dimensions, Wythoff's construction begins by choosing a generator point on the triangle. If the distance of this point from each of the sides is non-zero, the point must be chosen to be an equal distance from each edge.
A perpendicular line is dropped between the generator point and every face that it does not lie on. The three numbers in Wythoff's symbol, p, q and r, represent the corners of the Schwarz triangle used in the construction, which are π / p, π / q and π / r radians respectively; the triangle is represented with the same numbers, written. The vertical bar in the symbol specifies a categorical position of the generator point within the fundamental triangle according to the following: p | q r indicates that the generator lies on the corner p, p q | r indicates that the generator lies on the edge between p and q, p q r | indicates that the generator lies in the interior of the triangle. In this notation the mirrors are labeled by the reflection-order of the opposite vertex; the p, q, r values are listed before the bar. The one impossible symbol | p q r implies the generator point is on all mirrors, only possible if the triangle is degenerate, reduced to a point; this unused symbol is therefore arbitrarily reassigned to represent the case where all mirrors are active, but odd-numbered reflected images are ignored.
The resulting figure has rotational symmetry only. The generator point can either be off each mirror, activated or not; this distinction creates 8 possible forms, neglecting one where the generator point is on all the mirrors. The Wythoff symbol is functionally similar to the more general Coxeter-Dynkin diagram, in which each node represents a mirror and the arcs between them – marked with numbers – the angles between the mirrors. A node is circled. There are seven generator points with each set of p, q, r: There are three special cases: p q | – This is a mixture of p q r | and p q s |, containing only the faces shared by both. | p q r – Snub forms are given by this otherwise unused symbol. | p q r s – A unique snub form for U75 that isn't Wythoff-constructible. There are 4 symmetry classes of reflection on the sphere, three in the Euclidean plane. A few of the infinitely many such patterns in the hyperbolic plane are listed. Point groups: dihedral symmetry, p = 2, 3, 4 … tetrahedral symmetry octahedral symmetry icosahedral symmetry Euclidean groups: *442 symmetry: 45°-45°-90° triangle *632 symmetry: 30°-60°-90° triangle *333 symmetry: 60°-60°-60° triangleHyperbolic groups: *732 symmetry *832 symmetry *433 symmetry *443 symmetry *444 symmetry *542 symmetry *642 symmetry...
The above symmetry groups only include the integer solutions on the sphere. The list of Schwarz triangles includes rational numbers, determine the full set of solutions of nonconvex uniform polyhedra. In the tilings above, each triangle is a fundamental domain, colored by and odd reflections. Selected tilings created by the Wythoff con