1.
Octahedral symmetry
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A regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation. A cube has the set of symmetries, since it is the dual of an octahedron. Chiral and full octahedral symmetry are the point symmetries with the largest symmetry groups compatible with translational symmetry. They are among the point groups of the cubic crystal system. But as it is also the direct product S4 × S2, one can identify the elements of S4 as a ∈ [0,4. ). So e. g. the identity is represented as 0, the pairs can be seen in the six files below. Each file is denoted by the m ∈, and the position of each permutation in the file corresponds to the n ∈. A rotoreflection is a combination of rotation and reflection,7 ′ ∘4 =19 ′,7 ′ ∘22 =17 ′, The reflection 7 ′ applied on the 90° rotation 22 gives the 90° rotoreflection 17 ′. O,432, or + of order 24, is chiral octahedral symmetry or rotational octahedral symmetry. This group is like chiral tetrahedral symmetry T, but the C2 axes are now C4 axes, Td and O are isomorphic as abstract groups, they both correspond to S4, the symmetric group on 4 objects. Td is the union of T and the set obtained by combining each element of O \ T with inversion, O is the rotation group of the cube and the regular octahedron. Oh, *432, or m3m of order 48 - achiral octahedral symmetry or full octahedral symmetry and this group has the same rotation axes as O, but with mirror planes, comprising both the mirror planes of Td and Th. This group is isomorphic to S4. C4, and is the symmetry group of the cube. It is the group for n =3. See also the isometries of the cube, with the 4-fold axes as coordinate axes, a fundamental domain of Oh is given by 0 ≤ x ≤ y ≤ z. An object with symmetry is characterized by the part of the object in the fundamental domain, for example the cube is given by z =1. Ax + by + cz =1 gives a polyhedron with 48 faces, faces are 8-by-8 combined to larger faces for a = b =0 and 6-by-6 for a = b = c. The 9 mirror lines of full octahedral symmetry can be divided into two subgroups of 3 and 6, representing in two orthogonal subsymmetries, D2h, and Td, D2h symmetry can be doubled to D4h by restoring 2 mirrors from one of three orientations
2.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and 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, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, 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, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to 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 often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, 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, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
3.
Tessellation
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A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries, a periodic tiling has a repeating pattern. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups, a tiling that lacks a repeating pattern is called non-periodic. An aperiodic tiling uses a set of tile shapes that cannot form a repeating pattern. In the geometry of higher dimensions, a space-filling or honeycomb is called a tessellation of space. A real physical tessellation is a made of materials such as cemented ceramic squares or hexagons. Such tilings may be decorative patterns, or may have such as providing durable and water-resistant pavement. Historically, tessellations were used in Ancient Rome and in Islamic art such as in the decorative geometric tiling of the Alhambra palace, in the twentieth century, the work of M. C. Escher often made use of tessellations, both in ordinary Euclidean geometry and in geometry, for artistic effect. Tessellations are sometimes employed for decorative effect in quilting, Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs. Tessellations were used by the Sumerians in building wall decorations formed by patterns of clay tiles, decorative mosaic tilings made of small squared blocks called tesserae were widely employed in classical antiquity, sometimes displaying geometric patterns. In 1619 Johannes Kepler made a documented study of tessellations. He wrote about regular and semiregular tessellations in his Harmonices Mundi, he was possibly the first to explore and to explain the structures of honeycomb. Some two hundred years later in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the features one of seventeen different groups of isometries. Fyodorovs work marked the beginning of the mathematical study of tessellations. Other prominent contributors include Shubnikov and Belov, and Heinrich Heesch, in Latin, tessella is a small cubical piece of clay, stone or glass used to make mosaics. The word tessella means small square and it corresponds to the everyday term tiling, which refers to applications of tessellations, often made of glazed clay. Tessellation or tiling in two dimensions is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules
4.
Hexagon
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In geometry, a hexagon is a six sided polygon or 6-gon. The total of the angles of any hexagon is 720°. A regular hexagon has Schläfli symbol and can also be constructed as an equilateral triangle, t. A regular hexagon is defined as a hexagon that is both equilateral and equiangular and it is bicentric, meaning that it is both cyclic and tangential. The common length of the sides equals the radius of the circumscribed circle, all internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries and 6 reflection symmetries, the longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane, the cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral, the maximal diameter, D is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the circle, d, is twice the minimal radius or inradius. If a regular hexagon has successive vertices A, B, C, D, E, F, the regular hexagon has Dih6 symmetry, order 12. There are 3 dihedral subgroups, Dih3, Dih2, and Dih1, and 4 cyclic subgroups, Z6, Z3, Z2 and these symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order, r12 is full symmetry, and a1 is no symmetry. These two forms are duals of each other and have half the order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction and it can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites. G2 hexagons, with sides parallel are also called hexagonal parallelogons. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g6 subgroup has no degrees of freedom but can seen as directed edges. Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation, other hexagon shapes can tile the plane with different orientations
5.
Garnet
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Garnets are a group of silicate minerals that have been used since the Bronze Age as gemstones and abrasives. All species of garnets possess similar physical properties and crystal forms, the different species are pyrope, almandine, spessartine, grossular, uvarovite and andradite. The garnets make up two solid solution series, pyrope-almandine-spessartine and uvarovite-grossular-andradite, the word garnet comes from the 14th‑century Middle English word gernet, meaning dark red. It is derived from the Latin granatus, from granum, Garnet species are found in many colors including red, orange, yellow, green, purple, brown, blue, black, pink, and colorless, with reddish shades most common. Garnet species light transmission properties can range from the gemstone-quality transparent specimens to the varieties used for industrial purposes as abrasives. The minerals luster is categorized as vitreous or resinous, garnets are nesosilicates having the general formula X3Y23. The X site is occupied by divalent cations 2+ and the Y site by trivalent cations 3+ in an octahedral/tetrahedral framework with 4− occupying the tetrahedra. Garnets are most often found in the crystal habit, but are also commonly found in the trapezohedron habit. They crystallize in the system, having three axes that are all of equal length and perpendicular to each other. Garnets do not show cleavage, so when they fracture under stress, because the chemical composition of garnet varies, the atomic bonds in some species are stronger than in others. As a result, this group shows a range of hardness on the Mohs scale of about 6.5 to 7.5. The harder species like almandine are often used for abrasive purposes, for gem identification purposes, a pick-up response to a strong neodymium magnet separates garnet from all other natural transparent gemstones commonly used in the jewelry trade. Almandine, Fe3Al23 Pyrope, Mg3Al23 Spessartine, Mn3Al23 Almandine, sometimes incorrectly called almandite, is the modern gem known as carbuncle, the term carbuncle is derived from the Latin meaning live coal or burning charcoal. The name Almandine is a corruption of Alabanda, a region in Asia Minor where these stones were cut in ancient times, chemically, almandine is an iron-aluminium garnet with the formula Fe3Al23, the deep red transparent stones are often called precious garnet and are used as gemstones. Almandine occurs in metamorphic rocks like mica schists, associated with such as staurolite, kyanite, andalusite. Almandine has nicknames of Oriental garnet, almandine ruby, and carbuncle, Pyrope is red in color and chemically an aluminium silicate with the formula Mg3Al23, though the magnesium can be replaced in part by calcium and ferrous iron. The color of pyrope varies from red to black. A variety of pyrope from Macon County, North Carolina is a shade and has been called rhodolite
6.
Crystal habit
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In mineralogy, crystal habit is the characteristic external shape of an individual crystal or crystal group. A single crystals habit is a description of its shape and its crystallographic forms. Recognizing the habit may help in identifying a mineral, when the faces are well-developed due to uncrowded growth a crystal is called euhedral, one with partially developed faces is subhedral, and one with undeveloped crystal faces is called anhedral. The long axis of a quartz crystal typically has a six-sided prismatic habit with parallel opposite faces. Aggregates can be formed of individual crystals with euhedral to anhedral grains, the arrangement of crystals within the aggregate can be characteristic of certain minerals. For example, minerals used for asbestos insulation often grow in a fibrous habit, the terms used by mineralogists to report crystal habits describe the typical appearance of an ideal mineral. Recognizing the habit can aid in identification as some habits are characteristic, most minerals, however, do not display ideal habits due to conditions during crystallization. Minerals belonging to the crystal system do not necessarily exhibit the same habit. Some habits of a mineral are unique to its variety and locality, For example, while most sapphires form elongate barrel-shaped crystals, ordinarily, the latter habit is seen only in ruby. Sapphire and ruby are both varieties of the mineral, corundum. Some minerals may replace other existing minerals while preserving the originals habit, a classic example is tigers eye quartz, crocidolite asbestos replaced by silica. While quartz typically forms prismatic crystals, in tigers eye the original fibrous habit of crocidolite is preserved, the names of crystal habits are derived from, Predominant crystal faces. Abnormal grain growth Grain growth Crystallization
7.
Diamond
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Diamond is a metastable allotrope of carbon, where the carbon atoms are arranged in a variation of the face-centered cubic crystal structure called a diamond lattice. Diamond is less stable than graphite, but the rate from diamond to graphite is negligible at standard conditions. Diamond is renowned as a material with superlative physical qualities, most of which originate from the covalent bonding between its atoms. In particular, diamond has the highest hardness and thermal conductivity of any bulk material and those properties determine the major industrial application of diamond in cutting and polishing tools and the scientific applications in diamond knives and diamond anvil cells. Because of its extremely rigid lattice, it can be contaminated by very few types of impurities, such as boron, small amounts of defects or impurities color diamond blue, yellow, brown, green, purple, pink, orange or red. Diamond also has relatively high optical dispersion, most natural diamonds are formed at high temperature and pressure at depths of 140 to 190 kilometers in the Earths mantle. Carbon-containing minerals provide the source, and the growth occurs over periods from 1 billion to 3.3 billion years. Diamonds are brought close to the Earths surface through deep volcanic eruptions by magma, Diamonds can also be produced synthetically in a HPHT method which approximately simulates the conditions in the Earths mantle. An alternative, and completely different growth technique is chemical vapor deposition, several non-diamond materials, which include cubic zirconia and silicon carbide and are often called diamond simulants, resemble diamond in appearance and many properties. Special gemological techniques have developed to distinguish natural diamonds, synthetic diamonds. The word is from the ancient Greek ἀδάμας – adámas unbreakable, the name diamond is derived from the ancient Greek αδάμας, proper, unalterable, unbreakable, untamed, from ἀ-, un- + δαμάω, I overpower, I tame. Diamonds have been known in India for at least 3,000 years, Diamonds have been treasured as gemstones since their use as religious icons in ancient India. Their usage in engraving tools also dates to early human history, later in 1797, the English chemist Smithson Tennant repeated and expanded that experiment. By demonstrating that burning diamond and graphite releases the same amount of gas, the most familiar uses of diamonds today are as gemstones used for adornment, a use which dates back into antiquity, and as industrial abrasives for cutting hard materials. The dispersion of light into spectral colors is the primary gemological characteristic of gem diamonds. In the 20th century, experts in gemology developed methods of grading diamonds, four characteristics, known informally as the four Cs, are now commonly used as the basic descriptors of diamonds, these are carat, cut, color, and clarity. A large, flawless diamond is known as a paragon and these conditions are met in two places on Earth, in the lithospheric mantle below relatively stable continental plates, and at the site of a meteorite strike. The conditions for diamond formation to happen in the mantle occur at considerable depth corresponding to the requirements of temperature and pressure
8.
Archimedean solid
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In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the semi-regular convex polyhedrons composed of regular meeting in identical vertices, excluding the 5 Platonic solids. They differ from the Johnson solids, whose regular polygonal faces do not meet in identical vertices, identical vertices means that for any two vertices, there is a global isometry of the entire solid that takes one vertex to the other. Excluding these two families, there are 13 Archimedean solids. All the Archimedan solids can be made via Wythoff constructions from the Platonic solids with tetrahedral, octahedral and icosahedral symmetry, the Archimedean solids take their name from Archimedes, who discussed them in a now-lost work. Pappus refers to it, stating that Archimedes listed 13 polyhedra, kepler may have also found the elongated square gyrobicupola, at least, he once stated that there were 14 Archimedean solids. However, his published enumeration only includes the 13 uniform polyhedra, here the vertex configuration refers to the type of regular polygons that meet at any given vertex. For example, a configuration of means that a square, hexagon. Some definitions of semiregular polyhedron include one more figure, the square gyrobicupola or pseudo-rhombicuboctahedron. The number of vertices is 720° divided by the angle defect. The cuboctahedron and icosidodecahedron are edge-uniform and are called quasi-regular, the duals of the Archimedean solids are called the Catalan solids. Together with the bipyramids and trapezohedra, these are the face-uniform solids with regular vertices, the snub cube and snub dodecahedron are known as chiral, as they come in a left-handed form and right-handed form. When something comes in forms which are each others three-dimensional mirror image. The different Archimedean and Platonic solids can be related to each other using a handful of general constructions, starting with a Platonic solid, truncation involves cutting away of corners. To preserve symmetry, the cut is in a perpendicular to the line joining a corner to the center of the polyhedron and is the same for all corners. Depending on how much is truncated, different Platonic and Archimedean solids can be created, expansion or cantellation involves moving each face away from the center and taking the convex hull. Expansion with twisting also involves rotating the faces, thus breaking the rectangles corresponding to edges into triangles, the last construction we use here is truncation of both corners and edges. Ignoring scaling, expansion can also be viewed as truncation of corners and edges, note the duality between the cube and the octahedron, and between the dodecahedron and the icosahedron
9.
Platonic solid
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In three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent regular polygonal faces with the number of faces meeting at each vertex. Five solids meet those criteria, Geometers have studied the mathematical beauty and they are named for the ancient Greek philosopher Plato who theorized in his dialogue, the Timaeus, that the classical elements were made of these regular solids. The Platonic solids have been known since antiquity, dice go back to the dawn of civilization with shapes that predated formal charting of Platonic solids. The ancient Greeks studied the Platonic solids extensively, some sources credit Pythagoras with their discovery. In any case, Theaetetus gave a description of all five. The Platonic solids are prominent in the philosophy of Plato, their namesake, Plato wrote about them in the dialogue Timaeus c.360 B. C. in which he associated each of the four classical elements with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, there was intuitive justification for these associations, the heat of fire feels sharp and stabbing. Air is made of the octahedron, its components are so smooth that one can barely feel it. Water, the icosahedron, flows out of hand when picked up. By contrast, a highly nonspherical solid, the hexahedron represents earth and these clumsy little solids cause dirt to crumble and break when picked up in stark difference to the smooth flow of water. Moreover, the cubes being the regular solid that tessellates Euclidean space was believed to cause the solidity of the Earth. Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarks. the god used for arranging the constellations on the whole heaven. Aristotle added an element, aithēr and postulated that the heavens were made of this element. Euclid completely mathematically described the Platonic solids in the Elements, the last book of which is devoted to their properties, propositions 13–17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order. For each solid Euclid finds the ratio of the diameter of the sphere to the edge length. In Proposition 18 he argues there are no further convex regular polyhedra. Andreas Speiser has advocated the view that the construction of the 5 regular solids is the goal of the deductive system canonized in the Elements
10.
Honeycomb
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A honeycomb is a mass of hexagonal prismatic wax cells built by honey bees in their nests to contain their larvae and stores of honey and pollen. Beekeepers may remove the entire honeycomb to harvest honey, honey bees consume about 8.4 lb of honey to secrete 1 lb of wax, so it makes economic sense to return the wax to the hive after harvesting the honey. The structure of the comb may be left intact when honey is extracted from it by uncapping and spinning in a centrifugal machine—the honey extractor. If the honeycomb is too worn out, the wax can be reused in a number of ways, such foundation sheets allow the bees to build the comb with less effort, and the hexagonal pattern of worker-sized cell bases discourages the bees from building the larger drone cells. Fresh, new comb is sometimes sold and used intact as comb honey, broodcomb becomes dark over time, because of the cocoons embedded in the cells and the tracking of many feet, called travel stain by beekeepers when seen on frames of comb honey. Honeycomb in the supers that are not allowed to be used for brood stays light-coloured, however, the term honeycomb is not often used for such structures. The axes of honeycomb cells are always quasihorizontal, and the rows of honeycomb cells are always horizontally aligned. Thus, each cell has two vertically oriented walls, with the upper and lower parts of the composed of two angled walls. The open end of a cell is referred to as the top of the cell. The cells slope slightly upwards, between 9 and 14°, towards the open ends, two possible explanations exist as to why honeycomb is composed of hexagons, rather than any other shape. First, the hexagonal tiling creates a partition with equal-sized cells, known in geometry as the honeycomb conjecture, this was given by Jan Brożek and proved much later by Thomas Hales. Thus, a hexagonal structure uses the least material to create a lattice of cells within a given volume, in support of this, he notes that queen cells, which are constructed singly, are irregular and lumpy with no apparent attempt at efficiency. The closed ends of the cells are also an example of geometric efficiency. The ends are trihedral sections of rhombic dodecahedra, with the angles of all adjacent surfaces measuring 120°. The shape of the cells is such that two opposing honeycomb layers nest into each other, with each facet of the ends being shared by opposing cells. Individual cells do not show this geometric perfection, in a regular comb, in transition zones between the larger cells of drone comb and the smaller cells of worker comb, or when the bees encounter obstacles, the shapes are often distorted. Cells are also angled up about 13° from horizontal to prevent honey from dripping out, in 1965, László Fejes Tóth discovered the trihedral pyramidal shape used by the honeybee is not the theoretically optimal three-dimensional geometry. A cell end composed of two hexagons and two smaller rhombuses would actually be. 035% more efficient, bees used their antennae, mandibles and legs to manipulate the wax during comb construction, while actively warming the wax
11.
Cuboctahedron
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In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, as such, it is a quasiregular polyhedron, i. e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. Its dual polyhedron is the rhombic dodecahedron, the cuboctahedron was probably known to Plato, Herons Definitiones quotes Archimedes as saying that Plato knew of a solid made of 8 triangles and 6 squares. Heptaparallelohedron Fuller applied the name Dymaxion to this shape, used in a version of the Dymaxion map. He also called it the Vector Equilibrium and he called a cuboctahedron consisting of rigid struts connected by flexible vertices a jitterbug. With Oh symmetry, order 48, it is a cube or rectified octahedron With Td symmetry, order 24. With D3d symmetry, order 12, it is a triangular gyrobicupola. The area A and the volume V of the cuboctahedron of edge length a are, the cuboctahedron has four special orthogonal projections, centered on a vertex, an edge, and the two types of faces, triangular and square. The last two correspond to the B2 and A2 Coxeter planes, the skew projections show a square and hexagon passing through the center of the cuboctahedron. The cuboctahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. The cuboctahedrons 12 vertices can represent the vectors of the simple Lie group A3. With the addition of 6 vertices of the octahedron, these represent the 18 root vectors of the simple Lie group B3. The cuboctahedron can be dissected into two triangular cupolas by a common hexagon passing through the center of the cuboctahedron, if these two triangular cupolas are twisted so triangles and squares line up, Johnson solid J27, the triangular orthobicupola, is created. The cuboctahedron can also be dissected into 6 square pyramids and 8 tetrahedra meeting at a central point and this dissection is expressed in the alternated cubic honeycomb where pairs of square pyramids are combined into octahedra. A cuboctahedron can be obtained by taking a cross section of a four-dimensional 16-cell. Its first stellation is the compound of a cube and its dual octahedron, the cuboctahedron is a rectified cube and also a rectified octahedron. It is also a cantellated tetrahedron, with this construction it is given the Wythoff symbol,33 |2
12.
Catalan solid
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In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, the Catalan solids are all convex. They are face-transitive but not vertex-transitive and this is because the dual Archimedean solids are vertex-transitive and not face-transitive. Note that unlike Platonic solids and Archimedean solids, the faces of Catalan solids are not regular polygons, however, the vertex figures of Catalan solids are regular, and they have constant dihedral angles. Being face-transitive, Catalan solids are isohedra, additionally, two of the Catalan solids are edge-transitive, the rhombic dodecahedron and the rhombic triacontahedron. These are the duals of the two quasi-regular Archimedean solids, just as prisms and antiprisms are generally not considered Archimedean solids, so bipyramids and trapezohedra are generally not considered Catalan solids, despite being face-transitive. Two of the Catalan solids are chiral, the pentagonal icositetrahedron and these each come in two enantiomorphs. Not counting the enantiomorphs, bipyramids, and trapezohedra, there are a total of 13 Catalan solids, the Catalan solids, along with their dual Archimedean solids, can be grouped by their symmetry, tetrahedral, octahedral, and icosahedral. There are 6 forms per symmetry, while the self-symmetric tetrahedral group only has three forms and two of those are duplicated with octahedral symmetry. J. lÉcole Polytechnique 41, 1-71,1865, alan Holden Shapes, Space, and Symmetry. Wenninger, Magnus, Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR730208 Williams, the Geometrical Foundation of Natural Structure, A Source Book of Design. California, University of California Press Berkeley, chapter 4, Duals of the Archimedean polyhedra, prisma and antiprisms Weisstein, Eric W. Catalan Solids. Archived from the original on 4 February 2007, Archimedean duals – at Virtual Reality Polyhedra Interactive Catalan Solid in Java