163 relations: Abel–Ruffini theorem, Abelian group, Algebraic number field, Allotropes of boron, Alternating group, American Mathematical Monthly, Angular defect, Antiprism, Apollonius of Perga, BMC domain, Borromean rings, Brooks' theorem, Buckminster Fuller, Capsid, Capsomere, Carborane, Cartography, Chirality (mathematics), Circular shift, Compound of five octahedra, Compound of two icosahedra, Conformal map, Conway polyhedron notation, Coxeter element, Coxeter group, Coxeter–Dynkin diagram, Crystal structure of boron-rich metal borides, Crystal twinning, D20 System, Digon, Dihedral angle, Dihedral symmetry in three dimensions, Dimension, Distance (graph theory), Distance-regular graph, Distance-transitive graph, Dodecahedron, Dual graph, Dual polyhedron, Dungeons & Dragons, Dymaxion map, Edge-contracted icosahedron, Eigenvalues and eigenvectors, Equiangular lines, Ernst Haeckel, Euclidean space, Face (geometry), Facet (geometry), Faceting, Felix Klein, ..., Fibonacci, Galois group, Genome, Geodesic grid, Geographic coordinate system, Geometry, Golden ratio, Golden rectangle, Graph (discrete mathematics), Graph automorphism, Graph coloring, Great dodecahedron, Great icosahedron, Group representation, Gyrobifastigium, Gyroelongated bipyramid, Gyroelongated pentagonal pyramid, Hamiltonian path, Hero of Alexandria, Herpesviridae, Hyperbolic space, Icosahedral 120-cell, Icosahedral honeycomb, Icosahedral symmetry, Icosahedral twins, Icosahedron, Invariant (mathematics), Inverse trigonometric functions, Isogonal figure, Isometry, Isomorphism, Jessen's icosahedron, Johnson solid, K-vertex-connected graph, Kepler–Poinsot polyhedron, Kernel (linear algebra), Kirby 64: The Crystal Shards, List of finite spherical symmetry groups, Magic 8-Ball, Matrix (mathematics), Metabidiminished icosahedron, N-skeleton, Nanoparticle, Net (polyhedron), Normal subgroup, Octahedron, Orbifold notation, Orthogonality, Orthographic projection, Pappus of Alexandria, Pentagonal antiprism, Pentagonal bipyramid, Pentagonal pyramid, Planar graph, Platonic graph, Platonic solid, Poincaré disk model, Polyhedral skeletal electron pair theory, Polyhedron, Polytope, Polytope compound, Projection (linear algebra), Protein, Quintic function, Quotient space (linear algebra), Radiolaria, Radius, Regular 4-polytope, Regular dodecahedron, Regular graph, Regular polyhedron, Rhombic triacontahedron, Role-playing game, Rotation, Scattergories, Schläfli symbol, Shoji Sadao, Simple group, Skew apeirohedron, Small stellated dodecahedron, Snub (geometry), Snub 24-cell, Snub cube, Snub dodecahedron, Sphere, Spherical coordinate system, Spherical polyhedron, Stellation, Stellation diagram, Stereographic projection, Symmetric graph, Symmetric group, Symmetric matrix, Symmetry group, Tangent, Tetrahedral symmetry, Tetrahedron, The Fifty-Nine Icosahedra, Trace (linear algebra), Tridiminished icosahedron, Truncated icosahedron, Truncation (geometry), Uniform coloring, Uniform polyhedron compound, Vertex arrangement, Vertex figure, Virus, Volume, Yes–no question, 4-polytope, 6-cube, 6-demicube, 6-orthoplex. Expand index (113 more) » « Shrink index
In algebra, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no algebraic solution—that is, solution in radicals—to the general polynomial equations of degree five or higher with arbitrary coefficients.
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.
In mathematics, an algebraic number field (or simply number field) F is a finite degree (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.
Boron can be prepared in several crystalline and amorphous forms.
In mathematics, an alternating group is the group of even permutations of a finite set.
The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894.
In geometry, the (angular) defect (or deficit or deficiency) means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the Euclidean plane would.
In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles.
Apollonius of Perga (Ἀπολλώνιος ὁ Περγαῖος; Apollonius Pergaeus; late 3rdearly 2nd centuries BC) was a Greek geometer and astronomer known for his theories on the topic of conic sections.
In molecular biology the Bacterial Microcompartment (BMC) domain is a protein domain found in a variety of shell proteins, including CsoS1A, CsoS1B and CsoS1C of Thiobacillus neapolitanus (Halothiobacillus neapolitanus) and their orthologs from other bacteria.
In mathematics, the Borromean rings consist of three topological circles which are linked and form a Brunnian link (i.e., removing any ring results in two unlinked rings).
In graph theory, Brooks' theorem states a relationship between the maximum degree of a graph and its chromatic number.
Richard Buckminster "Bucky" Fuller (July 12, 1895 – July 1, 1983) was an American architect, systems theorist, author, designer, inventor and futurist.
A capsid is the protein shell of a virus.
The capsomere is a subunit of the capsid, an outer covering of protein that protects the genetic material of a virus.
A carborane is a cluster composed of boron, carbon and hydrogen atoms.
Cartography (from Greek χάρτης chartēs, "papyrus, sheet of paper, map"; and γράφειν graphein, "write") is the study and practice of making maps.
In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone.
In combinatorial mathematics, a circular shift is the operation of rearranging the entries in a tuple, either by moving the final entry to the first position, while shifting all other entries to the next position, or by performing the inverse operation.
The compound of five octahedra is one of the five regular polyhedron compounds.
This uniform polyhedron compound is a composition of 2 icosahedra.
In mathematics, a conformal map is a function that preserves angles locally.
In geometry, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations.
In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group.
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors).
In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes).
Metals, and specifically rare-earth elements (RE), form numerous chemical complexes with boron.
Crystal twinning occurs when two separate crystals share some of the same crystal lattice points in a symmetrical manner.
The d20 System is a role-playing game system published in 2000 by Wizards of the Coast originally developed for the third edition of Dungeons & Dragons.
In geometry, a digon is a polygon with two sides (edges) and two vertices.
A dihedral angle is the angle between two intersecting planes.
In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as abstract group is a dihedral group Dihn (n ≥ 2).
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.
In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them.
In mathematics, a distance-regular graph is a regular graph such that for any two vertices v and w, the number of vertices at distance j from v and at distance k from w depends only upon j, k, and i.
In the mathematical field of graph theory, a distance-transitive graph is a graph such that, given any two vertices v and w at any distance i, and any other two vertices x and y at the same distance, there is an automorphism of the graph that carries v to x and w to y.
In geometry, a dodecahedron (Greek δωδεκάεδρον, from δώδεκα dōdeka "twelve" + ἕδρα hédra "base", "seat" or "face") is any polyhedron with twelve flat faces.
In the mathematical discipline of graph theory, the dual graph of a plane graph is a graph that has a vertex for each face of.
In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other.
Dungeons & Dragons (abbreviated as D&DMead, Malcomson; ''Dungeons & Dragons'' FAQ or DnD) is a fantasy tabletop role-playing game (RPG) originally designed by Gary Gygax and Dave Arneson.
The Dymaxion map or Fuller map is a projection of a world map onto the surface of an icosahedron, which can be unfolded and flattened to two dimensions.
In geometry, an edge-contracted icosahedron is a polyhedron with 18 triangular faces, 27 edges, and 11 vertices with C2v symmetry, order 4.
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.
In geometry, a set of lines is called equiangular if all the lines intersect at a single point, and every pair of lines makes the same angle.
Ernst Heinrich Philipp August Haeckel (16 February 1834 – 9 August 1919) was a German biologist, naturalist, philosopher, physician, professor, marine biologist, and artist who discovered, described and named thousands of new species, mapped a genealogical tree relating all life forms, and coined many terms in biology, including anthropogeny, ecology, phylum, phylogeny, and Protista. Haeckel promoted and popularised Charles Darwin's work in Germany and developed the influential but no longer widely held recapitulation theory ("ontogeny recapitulates phylogeny") claiming that an individual organism's biological development, or ontogeny, parallels and summarises its species' evolutionary development, or phylogeny.
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.
In solid geometry, a face is a flat (planar) surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by flat faces is a polyhedron.
In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself.
Stella octangula as a faceting of the cube In geometry, faceting (also spelled facetting) is the process of removing parts of a polygon, polyhedron or polytope, without creating any new vertices.
Christian Felix Klein (25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group theory.
Fibonacci (c. 1175 – c. 1250) was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages".
In mathematics, more specifically in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension.
In the fields of molecular biology and genetics, a genome is the genetic material of an organism.
A geodesic grid is a spatial grid based on a geodesic polyhedron or Goldberg polyhedron.
A geographic coordinate system is a coordinate system used in geography that enables every location on Earth to be specified by a set of numbers, letters or symbols.
Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities.
In geometry, a golden rectangle is a rectangle whose side lengths are in the golden ratio, 1: \tfrac, which is 1:\varphi (the Greek letter phi), where \varphi is approximately 1.618.
In mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related".
In the mathematical field of graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge–vertex connectivity.
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints.
In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol and Coxeter–Dynkin diagram of.
In geometry, the great icosahedron is one of four Kepler-Poinsot polyhedra (nonconvex regular polyhedra), with Schläfli symbol and Coxeter-Dynkin diagram of.
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication.
In geometry, the gyrobifastigium is the 26th Johnson solid (J26).
In geometry, the gyroelongated bipyramids are an infinite set of polyhedra, constructed by elongating an n-gonal bipyramid by inserting an n-gonal antiprism between its congruent halves.
In geometry, the gyroelongated pentagonal pyramid is one of the Johnson solids (J11).
In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once.
Hero of Alexandria (ἭρωνGenitive: Ἥρωνος., Heron ho Alexandreus; also known as Heron of Alexandria; c. 10 AD – c. 70 AD) was a mathematician and engineer who was active in his native city of Alexandria, Roman Egypt.
Herpesviridae is a large family of DNA viruses that cause diseases in animals, including humans.
In mathematics, hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature.
In geometry, the icosahedral 120-cell, polyicosahedron, faceted 600-cell or icosaplex is a regular star 4-polytope with Schläfli symbol.
The icosahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space.
A regular icosahedron has 60 rotational (or orientation-preserving) symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation.
An icosahedral twin is a nanostructure appearing for atomic clusters.
In geometry, an icosahedron is a polyhedron with 20 faces.
In mathematics, an invariant is a property, held by a class of mathematical objects, which remains unchanged when transformations of a certain type are applied to the objects.
In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains).
In geometry, a polytope (a polygon, polyhedron or tiling, for example) is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure.
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.
In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.
Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron, is a non-convex polyhedron with the same number of vertices, edges and faces as the regular icosahedron.
In geometry, a Johnson solid is a strictly convex polyhedron, which is not uniform (i.e., not a Platonic solid, Archimedean solid, prism, or antiprism), and each face of which is a regular polygon.
In graph theory, a connected graph G is said to be k-vertex-connected (or k-connected) if it has more than k vertices and remains connected whenever fewer than k vertices are removed.
In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra.
In mathematics, and more specifically in linear algebra and functional analysis, the kernel (also known as null space or nullspace) of a linear map between two vector spaces V and W, is the set of all elements v of V for which, where 0 denotes the zero vector in W. That is, in set-builder notation,.
is a side-scrolling platform game in the Kirby series developed by HAL Laboratory and published by Nintendo for the their Nintendo 64 home video game console.
Finite spherical symmetry groups are also called point groups in three dimensions.
The Magic 8-Ball is a toy used for fortune-telling or seeking advice, developed in the 1950s and manufactured by Mattel.
In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
In geometry, the metabidiminished icosahedron is one of the Johnson solids (J62).
In mathematics, particularly in algebraic topology, the of a topological space X presented as a simplicial complex (resp. CW complex) refers to the subspace Xn that is the union of the simplices of X (resp. cells of X) of dimensions In other words, given an inductive definition of a complex, the is obtained by stopping at the.
Nanoparticles are particles between 1 and 100 nanometres (nm) in size with a surrounding interfacial layer.
In geometry a net of a polyhedron is an arrangement of edge-joined polygons in the plane which can be folded (along edges) to become the faces of the polyhedron.
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part.
In geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces, twelve edges, and six vertices.
In geometry, orbifold notation (or orbifold signature) is a system, invented by William Thurston and popularized by the mathematician John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature.
In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms.
Orthographic projection (sometimes orthogonal projection), is a means of representing three-dimensional objects in two dimensions.
Pappus of Alexandria (Πάππος ὁ Ἀλεξανδρεύς; c. 290 – c. 350 AD) was one of the last great Greek mathematicians of Antiquity, known for his Synagoge (Συναγωγή) or Collection (c. 340), and for Pappus's hexagon theorem in projective geometry.
In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.
In geometry, the pentagonal bipyramid (or dipyramid) is third of the infinite set of face-transitive bipyramids.
In geometry, a pentagonal pyramid is a pyramid with a pentagonal base upon which are erected five triangular faces that meet at a point (the vertex).
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints.
In the mathematical field of graph theory, a Platonic graph is a graph that has one of the Platonic solids as its skeleton.
In three-dimensional space, a Platonic solid is a regular, convex polyhedron.
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which the points of the geometry are inside the unit disk, and the straight lines consist of all segments of circles contained within that disk that are orthogonal to the boundary of the disk, plus all diameters of the disk.
In chemistry the polyhedral skeletal electron pair theory (PSEPT) provides electron counting rules useful for predicting the structures of clusters such as borane and carborane clusters.
In geometry, a polyhedron (plural polyhedra or polyhedrons) is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices.
In elementary geometry, a polytope is a geometric object with "flat" sides.
A polyhedral compound is a figure that is composed of several polyhedra sharing a common centre.
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that.
Proteins are large biomolecules, or macromolecules, consisting of one or more long chains of amino acid residues.
In algebra, a quintic function is a function of the form where,,,, and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero.
In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero.
The Radiolaria, also called Radiozoa, are protozoa of diameter 0.1–0.2 mm that produce intricate mineral skeletons, typically with a central capsule dividing the cell into the inner and outer portions of endoplasm and ectoplasm.The elaborate mineral skeleton is usually made of silica.
In classical geometry, a radius of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length.
In mathematics, a regular 4-polytope is a regular four-dimensional polytope.
A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of twelve regular pentagonal faces, three meeting at each vertex.
In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency.
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags.
In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces.
A role-playing game (sometimes spelled roleplaying game and abbreviated to RPG) is a game in which players assume the roles of characters in a fictional setting.
A rotation is a circular movement of an object around a center (or point) of rotation.
Scattergories is a creative-thinking category-based party game originally published by Parker Brothers in 1988.
In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.
Shoji Sadao (born 1927) is a Japanese American architect, best known for his work and collaborations with R. Buckminster Fuller and Isamu Noguchi.
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself.
In geometry, a skew apeirohedron is an infinite skew polyhedron consisting of nonplanar faces or nonplanar vertex figures, allowing the figure to extend indefinitely without folding round to form a closed surface.
In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol.
In geometry, a snub is an operation applied to a polyhedron.
In geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells.
In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles.
In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.
A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane.
In mathematics, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons.
In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure.
In geometry, a stellation diagram or stellation pattern is a two-dimensional diagram in the plane of some face of a polyhedron, showing lines where other face planes intersect with this one.
In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane.
In the mathematical field of graph theory, a graph G is symmetric (or arc-transitive) if, given any two pairs of adjacent vertices u1—v1 and u2—v2 of G, there is an automorphism such that In other words, a graph is symmetric if its automorphism group acts transitively upon ordered pairs of adjacent vertices (that is, upon edges considered as having a direction).
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions.
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose.
In group theory, the symmetry group of an object (image, signal, etc.) is the group of all transformations under which the object is invariant with composition as the group operation.
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point.
A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners.
The Fifty-Nine Icosahedra is a book written and illustrated by H. S. M. Coxeter, P. Du Val, H. T. Flather and J. F. Petrie.
In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e., where aii denotes the entry on the ith row and ith column of A. The trace of a matrix is the sum of the (complex) eigenvalues, and it is invariant with respect to a change of basis.
In geometry, the tridiminished icosahedron is one of the Johnson solids (J63).
In geometry, the truncated icosahedron is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose faces are two or more types of regular polygons.
In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex.
In geometry, a uniform coloring is a property of a uniform figure (uniform tiling or uniform polyhedron) that is colored to be vertex-transitive.
A uniform polyhedron compound is a polyhedral compound whose constituents are identical (although possibly enantiomorphous) uniform polyhedra, in an arrangement that is also uniform: the symmetry group of the compound acts transitively on the compound's vertices.
In geometry, a vertex arrangement is a set of points in space described by their relative positions.
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
A virus is a small infectious agent that replicates only inside the living cells of other organisms.
Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains.
In linguistics, a yes–no question, formally known as a polar question or a general question, is a question whose expected answer is either "yes" or "no".
In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope.
In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces.
In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a 6-cube (hexeract) with alternated vertices removed.
In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell 4-faces, and 64 5-faces.