Hexagonal tiling
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which three hexagons meet at each vertex. It has Schläfli symbol of or t. English mathematician John Conway calls it a hextille; the internal angle of the hexagon is 120 degrees so three hexagons at a point make a full 360 degrees. It is one of three regular tilings of the plane; the other two are the square tiling. The hexagonal tiling is the densest way to arrange circles in two dimensions; the Honeycomb conjecture states that the hexagonal tiling is the best way to divide a surface into regions of equal area with the least total perimeter. The optimal three-dimensional structure for making honeycomb was investigated by Lord Kelvin, who believed that the Kelvin structure is optimal. However, the less regular Weaire–Phelan structure is better; this structure exists in the form of graphite, where each sheet of graphene resembles chicken wire, with strong covalent carbon bonds. Tubular graphene sheets have been synthesised.
They have many potential applications, due to electrical properties. Silicene is similar. Chicken wire consists of a hexagonal lattice of wires; the hexagonal tiling appears in many crystals. In three dimensions, the face-centered cubic and hexagonal close packing are common crystal structures, they are the densest known sphere packings in three dimensions, are believed to be optimal. Structurally, they comprise parallel layers of hexagonal tilings, similar to the structure of graphite, they differ in the way that the layers are staggered from each other, with the face-centered cubic being the more regular of the two. Pure copper, amongst other materials, forms a face-centered cubic lattice. There are three distinct uniform colorings of a hexagonal tiling, all generated from reflective symmetry of Wythoff constructions; the represent the periodic repeat of one colored tile, counting hexagonal distances as h first, k second. The same counting is used in the Goldberg polyhedra, with a notation h,k, can be applied to hyperbolic tilings for p>6.
The 3-color tiling is a tessellation generated by the order-3 permutohedrons. A chamferred hexagonal tiling replacing edges with new hexagons and transforms into another hexagonal tiling. In the limit, the original faces disappear, the new hexagons degenerate into rhombi, it becomes a rhombic tiling; the hexagons can be dissected into sets of 6 triangles. This process leads to two 2-uniform tilings, the triangular tiling: The hexagonal tiling can be considered an elongated rhombic tiling, where each vertex of the rhombic tiling is stretched into a new edge; this is similar to the relation of the rhombic dodecahedron and the rhombo-hexagonal dodecahedron tessellations in 3 dimensions. It is possible to subdivide the prototiles of certain hexagonal tilings by two, four or nine equal pentagons: This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol, Coxeter diagram, progressing to infinity; this tiling is topologically related to regular polyhedra with vertex figure n3, as a part of sequence that continues into the hyperbolic plane.
It is related to the uniform truncated polyhedra with vertex figure n.6.6. This tiling is a part of a sequence of truncated rhombic polyhedra and tilings with Coxeter group symmetry; the cube can be seen as a rhombic hexahedron. The truncated forms have regular n-gons at the truncated vertices, nonregular hexagonal faces. Like the uniform polyhedra there are eight uniform tilings that can be based from the regular hexagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, blue along the original edges, there are 8 forms, 7 which are topologically distinct. There are 3 types of monohedral convex hexagonal tilings, they are all isohedral. Each has parametric variations within a fixed symmetry. Type 2 contains glide reflections, is 2-isohedral keeping chiral pairs distinct. Hexagonal tilings can be made with the identical topology as the regular tiling. With isohedral faces, there are 13 variations. Symmetry given assumes all faces are the same color. Colors here represent the lattice positions.
Single-color lattices are parallelogon hexagons. Other isohedrally-tiled topological hexagonal tilings are seen as quadrilaterals and pentagons that are not edge-to-edge, but interpreted as colinear adjacent edges: The 2-uniform and 3-uniform tessellations have a rotational degree of freedom which distorts 2/3 of the hexagons, including a colinear case that can be seen as a non-edge-to-edge tiling of hexagons and larger triangles, it can be distorted into a chiral 4-colored tri-directional weaved pattern, distorting some hexagons into parallelograms. The weaved pattern with 2 colored faces have rotational 632 symmetry. A chevron pattern has pmg symmetry, lowered to p1 with 3 or 4 colored tiles; the hexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing; the gap inside each hexagon allows for one circle, creating the densest packing from the triangular tiling, with each circle contact with the maximum of 6 circles.
There are 2 regular complex apeirogons, sharing the vertices of the
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
Rhombicosidodecahedron
In geometry, the rhombicosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces. It has 20 regular triangular faces, 30 square faces, 12 regular pentagonal faces, 60 vertices, 120 edges. Johannes Kepler in Harmonices Mundi named this polyhedron a rhombicosidodecahedron, being short for truncated icosidodecahedral rhombus, with icosidodecahedral rhombus being his name for a rhombic triacontahedron. There are different truncations of a rhombic triacontahedron into a topological rhombicosidodecahedron: Prominently its rectification, the one that creates the uniform solid, the core of the compound with its dual, it can be called an expanded or cantellated dodecahedron or icosahedron, from truncation operations on either uniform polyhedron. If you expand an icosahedron by moving the faces away from the origin the right amount, without changing the orientation or size of the faces, do the same to its dual dodecahedron, patch the square holes in the result, you get a rhombicosidodecahedron.
Therefore, it has the same number of triangles as an icosahedron and the same number of pentagons as a dodecahedron, with a square for each edge of either. The rhombicosidodecahedron shares the vertex arrangement with the small stellated truncated dodecahedron, with the uniform compounds of six or twelve pentagrammic prisms; the Zometool kits for making geodesic domes and other polyhedra use slotted balls as connectors. The balls are "expanded" rhombicosidodecahedra, with the squares replaced by rectangles; the expansion is chosen. Twelve of the 92 Johnson solids are derived from the rhombicosidodecahedron, four of them by rotation of one or more pentagonal cupolae: the gyrate, parabigyrate and trigyrate rhombicosidodecahedron. Eight more can be constructed by removing up to three cupolae, sometimes rotating one or more of the other cupolae. Cartesian coordinates for the vertices of a rhombicosidodecahedron with an edge length of 2 centered at the origin are all permutations of:,where φ = 1 + √5/2 is the golden ratio.
Therefore, the circumradius of this rhombicosidodecahedron is the common distance of these points from the origin, namely √φ6+2 = √8φ+7 for edge length 2. For unit edge length, R must be halved, giving R = √8φ+7/2 = √11+4√5/2 ≈ 2.233. The rhombicosidodecahedron has six special orthogonal projections, centered, on a vertex, on two types of edges, three types of faces: triangles and pentagons; the last two correspond to the A2 and H2 Coxeter planes. The rhombicosidodecahedron 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
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
Dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, they play an important role in group theory and chemistry; the notation for the dihedral group differs in abstract algebra. In geometry, Dn or Dihn refers to the symmetries of a group of order 2n. In abstract algebra, D2n refers to this same dihedral group; the geometric convention is used in this article. A regular polygon with n sides has 2 n different symmetries: n rotational symmetries and n reflection symmetries. We take n ≥ 3 here; the associated rotations and reflections make up the dihedral group D n. If n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is there are n/2 axes of symmetry connecting the midpoints of opposite sides and n / 2 axes of symmetry connecting opposite vertices. In either case, there are 2 n elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes.
The following picture shows the effect of the sixteen elements of D 8 on a stop sign: The first row shows the effect of the eight rotations, the second row shows the effect of the eight reflections, in each case acting on the stop sign with the orientation as shown at the top left. As with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, this gives the symmetries of a polygon the algebraic structure of a finite group; the following Cayley table shows the effect of composition in the group D3. R0 denotes the identity. For example, s2s1 = r1, because the reflection s1 followed by the reflection s2 results in a rotation of 120°; the order of elements denoting the composition is right to left, reflecting the convention that the element acts on the expression to its right. The composition operation is not commutative. In general, the group Dn has elements r0, …, rn−1 and s0, …, sn−1, with composition given by the following formulae: r i r j = r i + j, r i s j = s i + j, s i r j = s i − j, s i s j = r i − j.
In all cases and subtraction of subscripts are to be performed using modular arithmetic with modulus n. If we center the regular polygon at the origin elements of the dihedral group act as linear transformations of the plane; this lets us represent elements of Dn with composition being matrix multiplication. This is an example of a group representation. For example, the elements of the group D4 can be represented by the following eight matrices: r 0 =, r 1 =, r 2 =, r 3 =, s 0 =, s 1 =, s 2 =
Cuboctahedron
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, 24 identical edges, each separating a triangle from a square; as such, it is a quasiregular polyhedron, i.e. an Archimedean solid, not only vertex-transitive but edge-transitive. It is the only radially equilateral convex polyhedron, its dual polyhedron is the rhombic dodecahedron. The cuboctahedron was known to Plato: Heron's 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 an early version of the Dymaxion map, he called it the "Vector Equilibrium" because of its radial equilateral symmetry. He called a cuboctahedron consisting of rigid struts connected by flexible vertices a "jitterbug". With Oh symmetry, order 48, it is a rectified cube or rectified octahedron With Td symmetry, order 24, it is a cantellated tetrahedron or rhombitetratetrahedron.
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: A = a 2 ≈ 9.464 1016 a 2 V = 5 3 2 a 3 ≈ 2.357 0226 a 3. The cuboctahedron has four special orthogonal projections, centered on a vertex, an edge, the two types of faces and square; the last two correspond to the B2 and A2 Coxeter planes. The skew projections show a hexagon passing through the center of the cuboctahedron; the cuboctahedron 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; the Cartesian coordinates for the vertices of a cuboctahedron centered at the origin are: An alternate set of coordinates can be made in 4-space, as 12 permutations of: This construction exists as one of 16 orthant facets of the cantellated 16-cell. The cuboctahedron's 12 vertices can represent the root vectors of the simple Lie group A3.
With the addition of 6 vertices of the octahedron, these vertices 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 be dissected into 6 square pyramids and 8 tetrahedra meeting at a central point. This dissection is expressed in the alternated cubic honeycomb where pairs of square pyramids are combined into octahedra; the cuboctahedron is the unique convex polyhedron in which the long radius is the same as the edge length. This radial equilateral symmetry is a property of only a few polytopes, including the two-dimensional hexagon, the three-dimensional cuboctahedron, the four-dimensional 24-cell and 8-cell. Radially equilateral polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.
Therefore, all the interior elements which meet at the center of these polytopes have equilateral triangle inward faces, as in the dissection of the cuboctahedron into 6 square pyramids and 8 tetrahedra. Each of these radially equilateral polytopes occurs as cells of a characteristic space-filling tessellation: the tiling of regular hexagons, the rectified cubic honeycomb, the 24-cell honeycomb and the tesseractic honeycomb, respectively; each tessellation has a dual tessellation. The densest known regular sphere-packing in two and four dimensions uses the cell centers of one of these tessellations as sphere centers. A cuboctahedron has octahedral symmetry, its first stellation is the compound of a cube and its dual octahedron, with the vertices of the cuboctahedron located at the midpoints of the edges of either. A cuboctahedron can be obtained by taking an equatorial cross section of a four-dimensional 24-cell or 16-cell. A hexagon can be obtained by taking an equatorial cross section of a cuboctahedron.
The cuboctahedron is a rectified cube and a rectified octahedron. It is a cantellated tetrahedron. With this construction it is given the Wythoff symbol: 3 3 | 2. A skew cantellation of the tetrahedron produces a solid with faces parallel to those of the cuboctahedron, namely eight triangles of two sizes, six rectangles. While its edges are unequal, this solid remains vertex-uniform: the solid has the full tetrahedral symmet