Disdyakis dodecahedron
In geometry, a disdyakis dodecahedron, is a Catalan solid with 48 faces and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons, it superficially resembles an inflated rhombic dodecahedron—if one replaces each face of the rhombic dodecahedron with a single vertex and four triangles in a regular fashion one ends up with a disdyakis dodecahedron. More formally, the disdyakis dodecahedron is the Kleetope of the rhombic dodecahedron, it is the net of a rhombic dodecahedral pyramid. It has Oh octahedral symmetry, its collective edges represent the reflection planes of the symmetry. It can be seen in the corner and mid-edge triangulation of the regular cube and octahedron, rhombic dodecahedron. Seen in stereographic projection the edges of the disdyakis dodecahedron form 9 circles in the plane; the 9 circles can be divided into two groups of 3 and 6, representing in two orthogonal subgroups:, and: If its smallest edges have length a, its surface area and volume are A = 6 7 783 + 436 2 a 2 V = 1 7 3 a 3 The truncated cuboctahedron and its dual, the disdyakis dodecahedron can be drawn in a number of symmetric orthogonal projective orientations.
Between a polyhedron and its dual and faces are swapped in positions, edges are perpendicular. The disdyakis dodecahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron, 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. First stellation of rhombic dodecahedron Disdyakis triacontahedron Kisrhombille tiling Great rhombihexacron—A uniform dual polyhedron with the same surface topology Williams, Robert; the Geometrical Foundation of Natural Structure: A Source Book of Design.
Dover Publications, Inc. ISBN 0-486-23729-X; the Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 Eric W. Weisstein, Disdyakis dodecahedron at MathWorld. Disdyakis Dodecahedron Interactive Polyhedron Model
Hexagon
In geometry, a hexagon is a six-sided polygon or 6-gon. The total of the internal angles of any simple hexagon is 720°. A regular hexagon has Schläfli symbol and can be constructed as a truncated equilateral triangle, t, which alternates two types of edges. A regular hexagon is defined as a hexagon, both equilateral and equiangular, it is bicentric, meaning that it is both tangential. The common length of the sides equals the radius of the circumscribed circle, which equals 2 3 times the apothem. All internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries and 6 reflection symmetries, making up the dihedral group D6; the longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. From this it can be seen that a triangle with a vertex at the center of the regular hexagon and sharing one side with the hexagon is equilateral, that the regular hexagon can be partitioned into six equilateral triangles. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane, so are useful for constructing tessellations.
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 regular triangular lattice is the honeycomb tessellation of hexagons, it is not 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 inscribed circle, d, is twice the minimal radius or inradius, r. The maxima and minima are related by the same factor: 1 2 d = r = cos R = 3 2 R = 3 2 t and d = 3 2 D; the area of a regular hexagon A = 3 3 2 R 2 = 3 R r = 2 3 r 2 = 3 3 8 D 2 = 3 4 D d = 3 2 d 2 ≈ 2.598 R 2 ≈ 3.464 r 2 ≈ 0.6495 D 2 ≈ 0.866 d 2. For any regular polygon, the area can be expressed in terms of the apothem a and the perimeter p. For the regular hexagon these are given by a = r, p = 6 R = 4 r 3, so A = a p 2 = r ⋅ 4 r 3 2 = 2 r 2 3 ≈ 3.464 r 2. The regular hexagon fills the fraction 3 3 2 π ≈ 0.8270 of its circumscribed circle.
If a regular hexagon has successive vertices A, B, C, D, E, F and if P is any point on the circumscribing circle between B and C PE + PF = PA + PB + PC + PD. The regular hexagon has Dih6 symmetry, order 12. There are 3 dihedral subgroups: Dih3, Dih2, Dih1, 4 cyclic subgroups: Z6, Z3, Z2, Z1; these symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a group order. R12 is full symmetry, a1 is no symmetry. P6, an isogonal hexagon constructed by three mirrors can alternate long and short edges, d6, an isotoxal hexagon constructed with equal edge lengths, but vertices alternating two different internal angles; these two forms have half the symmetry order of the regular hexagon. The
Disdyakis triacontahedron
In geometry, a disdyakis triacontahedron, hexakis icosahedron, decakis dodecahedron or kisrhombic triacontahedron is a Catalan solid with 120 faces and the dual to the Archimedean truncated icosidodecahedron. As such it is face uniform but with irregular face polygons, it resembles an inflated rhombic triacontahedron—if one replaces each face of the rhombic triacontahedron with a single vertex and four triangles in a regular fashion one ends up with a disdyakis triacontahedron. That is, the disdyakis triacontahedron is the Kleetope of the rhombic triacontahedron, it has the most faces among the Archimedean and Catalan solids, with the snub dodecahedron, with 92 faces, in second place. If the bipyramids and the trapezohedra are excluded, the disdyakis triacontahedron has the most faces of any other convex polyhedron where every face of the polyhedron has the same shape. Projected into a sphere, the edges of a disdyakis triacontahedron define 15 great circles. Buckminster Fuller used these 15 great circles, along with 10 and 6 others in two other polyhedra to define his 31 great circles of the spherical icosahedron.
The edges of the polyhedron projected onto a sphere form 15 great circles, represent all 15 mirror planes of reflective Ih icosahedral symmetry, as shown in this image. Combining pairs of light and dark triangles define the fundamental domains of the nonreflective icosahedral symmetry; the edges of a compound of five octahedra represent the 10 mirror planes of icosahedral symmetry. The disdyakis triacontahedron has three types of vertices which can be centered in orthogonally projection: The disdyakis triacontahedron, as a regular dodecahedron with pentagons divided into 10 triangles each, is considered the "holy grail" for combination puzzles like the Rubik's cube; this unsolved problem called the "big chop" problem has no satisfactory mechanism. It is the most significant unsolved problem in mechanical puzzles; this shape was used to create d120 dice using 3D printing. More the Dice Lab has used the Disdyakis triacontahedron to mass market an injection moulded 120 sided die, it is claimed that the d120 is the largest number of possible faces on a fair dice, aside from infinite families that would be impractical in reality due to the tendency to roll for a long time.
It is topologically related to a polyhedra 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. This is *n32 in orbifold notation, in Coxeter notation. 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, Disdyakis triacontahedron at MathWorld.
Disdyakis triacontahedron – Interactive Polyhedron Model
Zonohedron
A zonohedron is a convex polyhedron with point symmetry, every face of, a polygon with point symmetry. Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in three-dimensional space, or as the three-dimensional projection of a hypercube. Zonohedra were defined and studied by E. S. Fedorov, a Russian crystallographer. More in any dimension, the Minkowski sum of line segments forms a polytope known as a zonotope; the original motivation for studying zonohedra is that the Voronoi diagram of any lattice forms a convex uniform honeycomb in which the cells are zonohedra. Any zonohedron formed in this way can tessellate 3-dimensional space and is called a primary parallelohedron; each primary parallelohedron is combinatorially equivalent to one of five types: the rhombohedron, hexagonal prism, truncated octahedron, rhombic dodecahedron, the rhombo-hexagonal dodecahedron. Let be a collection of three-dimensional vectors. With each vector vi we may associate a line segment.
The Minkowski sum forms a zonohedron, all zonohedra that contain the origin have this form. The vectors from which the zonohedron is formed are called its generators; this characterization allows the definition of zonohedra to be generalized to higher dimensions, giving zonotopes. Each edge in a zonohedron is parallel to at least one of the generators, has length equal to the sum of the lengths of the generators to which it is parallel. Therefore, by choosing a set of generators with no parallel pairs of vectors, by setting all vector lengths equal, we may form an equilateral version of any combinatorial type of zonohedron. By choosing sets of vectors with high degrees of symmetry, we can form in this way, zonohedra with at least as much symmetry. For instance, generators spaced around the equator of a sphere, together with another pair of generators through the poles of the sphere, form zonohedra in the form of prism over regular 2k-gons: the cube, hexagonal prism, octagonal prism, decagonal prism, dodecagonal prism, etc.
Generators parallel to the edges of an octahedron form a truncated octahedron, generators parallel to the long diagonals of a cube form a rhombic dodecahedron. The Minkowski sum of any two zonohedra is another zonohedron, generated by the union of the generators of the two given zonohedra. Thus, the Minkowski sum of a cube and a truncated octahedron forms the truncated cuboctahedron, while the Minkowski sum of the cube and the rhombic dodecahedron forms the truncated rhombic dodecahedron. Both of these zonohedra are simple, as is the truncated small rhombicuboctahedron formed from the Minkowski sum of the cube, truncated octahedron, rhombic dodecahedron; the Gauss map of any convex polyhedron maps each face of the polygon to a point on the unit sphere, maps each edge of the polygon separating a pair of faces to a great circle arc connecting the corresponding two points. In the case of a zonohedron, the edges surrounding each face can be grouped into pairs of parallel edges, when translated via the Gauss map any such pair becomes a pair of contiguous segments on the same great circle.
Thus, the edges of the zonohedron can be grouped into zones of parallel edges, which correspond to the segments of a common great circle on the Gauss map, the 1-skeleton of the zonohedron can be viewed as the planar dual graph to an arrangement of great circles on the sphere. Conversely any arrangement of great circles may be formed from the Gauss map of a zonohedron generated by vectors perpendicular to the planes through the circles. Any simple zonohedron corresponds in this way to a simplicial arrangement, one in which each face is a triangle. Simplicial arrangements of great circles correspond via central projection to simplicial arrangements of lines in the projective plane, which were studied by Grünbaum, he listed three infinite families of simplicial arrangements, one of which leads to the prisms when converted to zonohedra, the other two of which correspond to additional infinite families of simple zonohedra. There are many known examples that do not fit into these three families. Any prism over a regular polygon with an number of sides forms a zonohedron.
These prisms can be formed so that all faces are regular: two opposite faces are equal to the regular polygon from which the prism was formed, these are connected by a sequence of square faces. Zonohedra of this type are the cube, hexagonal prism, octagonal prism, decagonal prism, dodecagonal prism, etc. In addition to this infinite family of regular-faced zonohedra, there are three Archimedean solids, all omnitruncations of the regular forms: The truncated octahedron, with 6 square and 8 hexagonal faces; the truncated cuboctahedron, with 12 squares, 8 hexagons, 6 octagons. The truncated icosidodecahedron, with 30 squares, 20 hexagons and 12 decagons. In addition, certain Catalan solids are again zonohedra: Kepler’s rhombic dodecahedron is the dual of the cuboctahedron; the rhombic triacontahedron is the dual of the icosidodecahedron. Others with congruent rhombic faces: Bilinski’s rhombic dodecahedron. Rhombic icosahedron RhombohedronThere are infinitely many zonohedra with rhombic faces that are not all congruent to each other.
They include: Rhombic enneacontahedron Although it is not true that any polyhedron has a dissection into any other polyhedron of the same volume, it is known that any two zonohedra of equal volumes can be dissected into each other. Zonohedrification is a process defined by George W. Hart for creating a zonohedron from another polyhedron. First the vertices of an
Truncated octahedron
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
Truncated triapeirogonal tiling
In geometry, the truncated triapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of tr. The dual of this tiling represents the fundamental domains of, *∞32 symmetry. There are 3 small index subgroup constructed from by mirror alternation. In these images fundamental domains are alternately colored black and white, mirrors exist on the boundaries between colors. A special index 4 reflective subgroup, is, its direct subgroup +, semidirect subgroup. Given with generating mirrors its index 4 subgroup has generators. An index 6 subgroup constructed as, becomes; this tiling can be considered a member of a sequence of uniform patterns with vertex figure and Coxeter-Dynkin diagram. For p < 6, the members of the sequence are omnitruncated polyhedra, shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling. List of uniform planar tilings Tilings of regular polygons Uniform tilings in hyperbolic plane
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