Stone carving is an activity where pieces of rough natural stone are shaped by the controlled removal of stone. Owing to the permanence of the material, stone work has survived, created during our prehistory. Work carried out by paleolithic societies to create flint tools is more referred to as knapping. Stone carving, done to produce lettering is more referred to as lettering; the process of removing stone from the earth is called quarrying. Stone carving is one of the processes; the term refers to the activity of masons in dressing stone blocks for use in architecture, building or civil engineering. It is a phrase used by archaeologists and anthropologists to describe the activity involved in making some types of petroglyphs; the earliest known works of representational art are stone carvings. Marks carved into rock or petroglyphs will survive where painted work will not. Prehistoric Venus figurines such as the Venus of Berekhat Ram may be as old as 800,000 years, are carved in stones such as tuff and limestone.
These earliest examples of the stone carving are the result of hitting or scratching a softer stone with a harder one, although sometimes more resilient materials such as antlers are known to have been used for soft stone. Another early technique was to use an abrasive, rubbed on the stone to remove the unwanted area. Prior to the discovery of steel by any culture, all stone carving was carried out by using an abrasion technique, following rough hewing of the stone block using hammers; the reason for this is that bronze, the hardest available metal until steel, is not hard enough to work any but the softest stone. The Ancient Greeks used the ductility of bronze to trap small granules of carborundum, that are occurring on the island of Milos, thus making a efficient file for abrading the stone; the development of iron made possible stone carving tools, such as chisels and saws made from steel, that were capable of being hardened and tempered to a state hard enough to cut stone without deforming, while not being so brittle as to shatter.
Carving tools have changed little since then. Modern, large quantity techniques still rely on abrasion to cut and remove stone, although at a faster rate with processes such as water erosion and diamond saw cutting. One modern stone carving technique uses a new process: The technique of applying sudden high temperature to the surface; the expansion of the top surface due to the sudden increase in temperature causes it to break away. On a small scale, Oxy-acetylene torches are used. On an industrial scale, lasers are used. On a massive scale, carvings such as the Crazy Horse Memorial carved from the Harney Peak granite of Mount Rushmore and the Confederate Memorial Park in Albany, Georgia are produced using jet heat torches. Carving stone into sculpture is an activity older than civilization itself. Prehistoric sculptures were human forms, such as the Venus of Willendorf and the faceless statues of the Cycladic cultures. Cultures devised animal, human-animal and abstract forms in stone; the earliest cultures used abrasive techniques, modern technology employs pneumatic hammers and other devices.
But for most of human history, sculptors used hammer and chisel as the basic tools for carving stone. The process begins with the selection of a stone for carving; some artists use the stone itself as inspiration. Other artists begin with a form in mind and find a stone to complement their vision; the sculptor may begin by forming a model in clay or wax, sketching the form of the statue on paper or drawing a general outline of the statue on the stone itself. When ready to carve, the artist begins by knocking off large portions of unwanted stone; this is the "roughing out" stage of the sculpting process. For this task they may select a point chisel, a long, hefty piece of steel with a point at one end and a broad striking surface at the other. A pitching tool may be used at this early stage; the pitching tool is useful for removing large, unwanted chunks. Those two chisels are used in combination with a masons driving hammer. Once the general shape of the statue has been determined, the sculptor uses other tools to refine the figure.
A toothed chisel or claw chisel has multiple gouging surfaces which create parallel lines in the stone. These tools are used to add texture to the figure. An artist might mark out specific lines by using calipers to measure an area of stone to be addressed, marking the removal area with pencil, charcoal or chalk; the stone carver uses a shallower stroke at this point in the process in combination with a wooden mallet. The sculptor has changed the stone from a rough block into the general shape of the finished statue. Tools called rasps and rifflers are used to enhance the shape into its final form. A rasp is a flat, steel tool with a coarse surface; the sculptor uses sweeping strokes to remove excess stone as small chips or dust. A riffler is a smaller variation of the rasp, which can be used to create details such as folds of clothing or locks of hair; the final stage of the carving process is polishing. Sandpaper can be used as a first step in sand cloth. Emery, a stone, harder and rougher than the sculpture media, is used in the finishing process.
This abrading, or wearing away, brings out the color of the stone, reveals patterns in the surface and adds a sheen. Tin and iron oxides are
Ball (association football)
A football, soccer ball, or association football ball is the ball used in the sport of association football. The name of the ball varies according to whether the sport is called "football", "soccer", or "association football"; the ball's spherical shape, as well as its size and material composition, are specified by Law 2 of the Laws of the Game maintained by the International Football Association Board. Additional, more stringent, standards are specified by FIFA and subordinate governing bodies for the balls used in the competitions they sanction. Early footballs began as animal bladders or stomachs that would fall apart if kicked too much. Improvements became possible in the 19th century with the introduction of rubber and discoveries of vulcanization by Charles Goodyear; the modern 32-panel ball design was developed in 1962 by Eigil Nielsen, technological research continues today to develop footballs with improved performance. The 32-panel ball design was soon overcome by 24-panel balls as well as 42-panel balls, both of which improved performance compared to before, in 2007.
A black-and-white patterned truncated icosahedron design, brought to prominence by the Adidas Telstar, has become an icon of the sport. Many different designs of balls exist, varying both in physical characteristics. In the year 1863, the first specifications for footballs were laid down by the Football Association. Previous to this, footballs were made out of inflated leather, with leather coverings to help footballs maintain their shapes. In 1872 the specifications were revised, these rules have been left unchanged as defined by the International Football Association Board. Differences in footballs created since this rule came into effect have been to do with the material used in their creation. Footballs have gone through a dramatic change over time. During medieval times balls were made from an outer shell of leather filled with cork shavings. Another method of creating a ball was using animal bladders for the inside of the ball making it inflatable. However, these two styles of creating footballs made it easy for the ball to puncture and were inadequate for kicking.
It was not until the 19th century. In 1838, Charles Goodyear introduced vulcanized rubber, which improved the football. Vulcanisation is the treatment of rubber to give it certain qualities such as strength and resistance to solvents. Vulcanisation of rubber helps the football resist moderate heat and cold. Vulcanisation helped create inflatable bladders that pressurize the outer panel arrangement of the football. Charles Goodyear's innovation made it easier to kick. Most balls of this time had tanned leather with eighteen sections stitched together; these were arranged in six panels of three strips each. During the 1900s, footballs were made out of leather with a lace of the same material used to stitch the panels. Although leather was perfect for bouncing and kicking the ball, when heading the football it was painful; this problem was most due to water absorption of the leather from rain, which caused a considerable increase in weight, causing head or neck injury. By around 2017, this had been associated with dementia in former players.
Another problem of early footballs was that they deteriorated as the leather used in manufacturing the footballs varied in thickness and in quality. The ball without the leather lace was developed and patented by Romano Polo, Antonio Tossolini and Juan Valbonesi in 1931 in Argentina; this innovative ball would be adopted by the Argentine Football Association as the official ball for its competitions since 1932. Elements of the football that today are tested are the deformation of the football when it is kicked or when the ball hits a surface. Two styles of footballs have been tested by the Sports Technology Research Group of Wolfson School of Mechanical and Manufacturing Engineering in Loughborough University; the basic model considered the ball as being a spherical shell with isotropic material properties. The developed model utilised isotropic material properties but included an additional stiffer stitching seam region. Companies such as Umbro, Adidas, Nike and Puma are releasing footballs made out of new materials which are intended to provide more accurate flight and more power to be transferred to the football.
Today's footballs are more complex than past footballs. Most modern footballs consist of twelve regular pentagonal and twenty regular hexagonal panels positioned in a truncated icosahedron spherical geometry; some premium-grade 32-panel balls use non-regular polygons to give a closer approximation to sphericality. The inside of the football is made up of a latex bladder which enables the football to be pressurised; the ball's panel pairs are stitched along the edge. The size of a football is 22 cm in diameter for a regulation size 5 ball. Rules state. Averaging that to 69 cm and dividing by π gives about 22 cm for a diameter; the ball's weight must be in the range of 410 to 450 grams and inflated to a pressure of between 0.6 and 1.1 standard atmospheres at sea level. There are a number of different types of football balls depending on the match and turf including training footballs, match footballs, professional match footballs, beach footballs, street footballs, indoor footballs, turf balls, futsa
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
A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive. It follows. Uniform polyhedra may be quasi-regular or semi-regular; the faces and vertices need not be convex, so many of the uniform polyhedra are star polyhedra. There are two infinite classes of uniform polyhedra together with 75 others. Infinite classes prisms antiprisms Convex exceptional 5 Platonic solids – regular convex polyhedra 13 Archimedean solids – 2 quasiregular and 11 semiregular convex polyhedra Star exceptional 4 Kepler–Poinsot polyhedra – regular nonconvex polyhedra 53 uniform star polyhedra – 5 quasiregular and 48 semiregularhence 5 + 13 + 4 + 53 = 75. There are many degenerate uniform polyhedra with pairs of edges that coincide, including one found by John Skilling called the great disnub dirhombidodecahedron. Dual polyhedra to uniform polyhedra are face-transitive and have regular vertex figures, are classified in parallel with their dual polyhedron; the dual of a regular polyhedron is regular, while the dual of an Archimedean solid is a Catalan solid.
The concept of uniform polyhedron is a special case of the concept of uniform polytope, which applies to shapes in higher-dimensional space. Coxeter, Longuet-Higgins & Miller define uniform polyhedra to be vertex-transitive polyhedra with regular faces, they define a polyhedron to be a finite set of polygons such that each side of a polygon is a side of just one other polygon, such that no non-empty proper subset of the polygons has the same property. By a polygon they implicitly mean a polygon in 3-dimensional Euclidean space. There are some generalizations of the concept of a uniform polyhedron. If the connectedness assumption is dropped we get uniform compounds, which can be split as a union of polyhedra, such as the compound of 5 cubes. If we drop the condition that the realization of the polyhedron is non-degenerate we get the so-called degenerate uniform polyhedra; these require a more general definition of polyhedra. Grunbaum gave a rather complicated definition of a polyhedron, while McMullen & Schulte gave a simpler and more general definition of a polyhedron: in their terminology, a polyhedron is a 2-dimensional abstract polytope with a non-degenerate 3-dimensional realization.
Here an abstract polytope is a poset of its "faces" satisfying various condition, a realization is a function from its vertices to some space, the realization is called non-degenerate if any two distinct faces of the abstract polytope have distinct realizations. Some of the ways they can be degenerate are as follows: Hidden faces; some polyhedra have faces that are hidden, in the sense that no points of their interior can be seen from the outside. These are not counted as uniform polyhedra. Degenerate compounds; some polyhedra have multiple edges and their faces are the faces of two or more polyhedra, though these are not compounds in the previous sense since the polyhedra share edges. Double covers. There are some non-orientable polyhedra that have double covers satisfying the definition of a uniform polyhedron. There double covers have doubled faces and vertices, they are not counted as uniform polyhedra. Double faces. There are several polyhedra with doubled faces produced by Wythoff's construction.
Most authors do not remove them as part of the construction. Double edges. Skilling's figure has the property that it has double edges but its faces cannot be written as a union of two uniform polyhedra; the Platonic solids date back to the classical Greeks and were studied by the Pythagoreans, Theaetetus, Timaeus of Locri and Euclid. The Etruscans discovered the regular dodecahedron before 500 BC; the cuboctahedron was known by Plato. Archimedes discovered all of the 13 Archimedean solids, his original book on the subject was lost, but Pappus of Alexandria mentioned Archimedes listed 13 polyhedra. Piero della Francesca rediscovered the five truncation of the Platonic solids: truncated tetrahedron, truncated octahedron, truncated cube, truncated dodecahedron, truncated icosahedron. Luca Pacioli republished Francesca's work in De divina proportione in 1509, adding the rhombicuboctahedron, calling it a icosihexahedron for its 26 faces, drawn by Leonardo da Vinci. Johannes Kepler was the first to publish the complete list of Archimedean solids, in 1619, as well as identified the infinite families of uniform prisms and antiprisms.
Kepler discovered two of the regular Kepler–Poinsot polyhedra and Louis Poinsot discovered the other two. The set of four were named by Arthur Cayley. Of the remaining 53, Edmund Hess discovered two, Albert Badoureau discovered 36 more, Pitsch independently discovered 18, of which 3 had not been discovered. Together these gave 41 polyhedra; the geometer H. S. M. Coxeter did not publish. M. S. Longuet-Higgins and H. C. Longuet-Higgins independently discovered eleven of these. Lesavre and Mercier rediscovered five of them in 1947. Coxeter, Longuet-Higgins & Miller published the list of uniform polyhedra. Sopov (19
In geometry, a deltoidal icositetrahedron is a Catalan solid. Its dual polyhedron is the rhombicuboctahedron; the 24 faces are kites. The short and long edges of each kite are in the ratio 1: ≈ 1:1.292893... If its smallest edges have length a, its surface area and volume are A = 6 29 − 2 2 a 2 V = 122 + 71 2 a 3 The deltoidal icositetrahedron is a crystal habit formed by the mineral analcime and garnet; the shape is called a trapezohedron in mineral contexts, although in solid geometry that name has another meaning. The deltoidal icositetrahedron has three symmetry positions, all centered on vertices: The great triakis octahedron is a stellation of the deltoidal icositetrahedron; the deltoidal icositetrahedron is topologically equivalent to a cube whose faces are divided in quadrants. It can be projected onto a regular octahedron, with kite faces, or more general quadrilaterals with pyritohedral symmetry. In Conway polyhedron notation, they represent an ortho operation to a octahedron. In crystallography a rotational variation is diploid.
The deltoidal icositetrahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron. When projected onto a sphere, it can be seen that the edges make up the edges of an octahedron and cube arranged in their dual positions; this polyhedron is topologically related as a part of sequence of deltoidal polyhedra with face figure, continues as tilings of the hyperbolic plane. These face-transitive figures have reflectional symmetry. Deltoidal hexecontahedron Tetrakis hexahedron, another 24-face Catalan solid which looks a bit like an overinflated cube. "The Haunter of the Dark", a story by H. P. Lovecraft, whose plot involves this figure Pseudo-deltoidal icositetrahedron 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, Deltoidal icositetrahedron at MathWorld.
Deltoidal Icositetrahedron – Interactive Polyhedron model
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
In geometry, a tetrakis hexahedron is a Catalan solid. Its dual is the truncated octahedron, an Archimedean solid, it can be called a disdyakis hexahedron or hexakis tetrahedron as the dual of an omnitruncated tetrahedron. The tetrakis hexahedron, dual of the truncated octahedron has 3 symmetry positions, two located on vertices and one mid-edge. Occurring formations of tetrahexahedra are observed in copper and fluorite systems. Polyhedral dice shaped like the tetrakis hexahedron are used by gamers. A 24-cell viewed under a vertex-first perspective projection has a surface topology of a tetrakis hexahedron and the geometric proportions of the rhombic dodecahedron, with the rhombic faces divided into two triangles; the tetrakis hexahedron appears as one of the simplest examples in building theory. Consider the Riemannian symmetric space associated to the group SL4, its Tits boundary has the structure of a spherical building whose apartments are 2-dimensional spheres. The partition of this sphere into spherical simplices can be obtained by taking the radial projection of a tetrakis hexahedron.
With Td, tetrahedral symmetry, the triangular faces represent the 24 fundamental domains of tetrahedral symmetry. This polyhedron can be constructed from 6 great circles on a sphere, it can be seen by a cube with its square faces triangulated by their vertices and face centers and a tetrahedron with its faces divided by vertices, mid-edges, a central point. The edges of the tetrakis hexahedron form 6 circles in the plane; each of these 6 circles represent a mirror line in tetrahedral symmetry. The 6 circles can be grouped into 3 sets of 2 pairs of orthogonal circles; these edges can be seen as a compound of 3 orthogonal square hosohedrons. If we denote the edge length of the base cube by a, the height of each pyramid summit above the cube is a/4; the inclination of each triangular face of the pyramid versus the cube face is arctan 26.565°. One edge of the isosceles triangles has length a, the other two have length 3a/4, which follows by applying the Pythagorean theorem to height and base length.
This yields an altitude of √5a/4 in the triangle. Its area is √5a/8, the internal angles are arccos and the complementary 180° − 2 arccos; the volume of the pyramid is a3/12. It can be seen as a cube with square pyramids covering each square face, it is similar to the 3D net for a 4D cubic pyramid, as the net for a square based is a square with triangles attached to each edge, the net for a cubic pyramid is a cube with square pyramids attached to each face. It is a polyhedra in a sequence defined by the face configuration V4.6.2n. This group is special for having all number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, continuing into the hyperbolic plane for any n ≥ 7. With an number of faces at every vertex, these polyhedra and tilings can be shown by alternating two colors so all adjacent faces have different colors; each face on these domains corresponds to the fundamental domain of a symmetry group with order 2,3,n mirrors at each triangle face vertex.
Disdyakis triacontahedron Disdyakis dodecahedron Kisrhombille tiling Compound of three octahedra Deltoidal icositetrahedron, another 24-face Catalan solid. 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, Tetrakis hexahedron at MathWorld. Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra VRML model Conway Notation for Polyhedra Try: "dtO" or "kC" Tetrakis Hexahedron – Interactive Polyhedron model The Uniform Polyhedra