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solid figure or to its boundary surface. The terms solid polyhedron and polyhedral surface are commonly used to distinguish the two concepts. Also, the term...

In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron...

In mathematics, a polyhedral complex is a set of polyhedra in a real vector space that fit together in a specific way. Polyhedral complexes generalize...

In computational geometry, a polyhedral terrain in three-dimensional Euclidean space is a polyhedral surface that intersects every line parallel to some...

surface, with each value being equally likely. Dice may also have other polyhedral or irregular shapes, may have faces marked with numerals or symbols instead...

mathematics, the centered polyhedral numbers are a class of figurate numbers, each formed by a central dot, surrounded by polyhedral layers with a constant...

A polyhedral map projection is a map projection based on a spherical polyhedron. Typically, the polyhedron is overlaid on the globe, and each face of the...

Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the...

The polyhedral symbol is sometimes used in coordination chemistry and crystallography to indicate the approximate coordination geometry around the central...

The polyhedral model (also called the polytope method) is a mathematical framework for programs that perform large numbers of operations -- too large to...

Polyhedral space is a certain metric space. A (Euclidean) polyhedral space is a (usually finite) simplicial complex in which every simplex has a flat metric...

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré...

Use of the polyhedral model (also called the polytope model) within a compiler requires software to represent the objects of this framework (sets of integer-valued...

In geometry, the polyhedral groups are the symmetry groups of the Platonic solids. There are three polyhedral groups: The tetrahedral group of order 12...

topological tori, with Euler characteristic of zero. The topological polyhedral net can be cut from two rows of a square tiling (with vertex configuration...

finitely generated cone is a polyhedral cone, and every polyhedral cone is a finitely generated cone. Every polyhedral cone has a unique representation...

numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The ancient Greek mathematicians already considered triangular...

that can be folded (along edges) to become the faces of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and solid geometry in...

that a longer strip would be. The Möbius strip can also be embedded as a polyhedral surface in space or flat-folded in the plane, with only five triangular...

Hierarchical organic-inorganic (hybrid) polysilsesquioxane (PSQ) materials using polyhedral oligomeric silsesquioxanes (POSS) cages as singular building blocks of...

Dymaxion map projection, also called the Fuller projection, is a kind of polyhedral map projection of the Earth's surface onto the unfolded net of an icosahedron...

cylinder with non-parallel bases is called a truncated cylinder. From a polyhedral perspective, a cylinder may also be viewed as the dual of a bicone, interpreted...

and not based on a particular projection Polyhedral maps Polyhedral maps can be folded up into a polyhedral approximation to the sphere, using particular...

11 vertices and 18 edges. It is a polyhedral graph (the graph of a convex polyhedron), and is the smallest polyhedral graph that does not have a Hamiltonian...

Unsolved problem in mathematics Is every cubic bipartite polyhedral graph Hamiltonian? More unsolved problems in mathematics Barnette's conjecture is an...

locations closer than at a distance equal to the constant d0 are not shown. Polyhedral map projections use a polyhedron to subdivide the globe into faces, and...

such as star polyhedra, when we seek to rigorously define this form of polyhedral duality in terms of projective polarity, various problems appear. Because...

equivalent polyhedra can be thought of as one of many embeddings of a polyhedral graph on the sphere. Unless otherwise specified, in this article (and...

In graph theory, a nested triangles graph with n vertices is a planar graph formed from a sequence of n/3 triangles, by connecting pairs of corresponding...

wedge defined by two non-parallel half-spaces, a polyhedral cylinder (infinite prism), and a polyhedral cone (infinite cone) defined by three or more half-spaces...

In mathematics, a polymatroid is a polytope associated with a submodular function. The notion was introduced by Jack Edmonds in 1970. It is also a generalization...

In chemistry the polyhedral skeletal electron pair theory (PSEPT) provides electron counting rules useful for predicting the structures of clusters such...

John Flinders Petrie (26 April 1907 – 1972) was an English mathematician. Petrie was the great grandson of the explorer and navigator, Matthew Flinders...

diagonals", Acta Sci. Math. Szeged, 13: 140–142. Ziegler, Günter M. (2008), "Polyhedral Surfaces of High Genus", in Bobenko, A. I.; Schröder, P.; Sullivan, J...

The compound of five cubes is one of the five regular polyhedral compounds. It was first described by Edmund Hess in 1876. Its vertices are those of a...

The Boerdijk–Coxeter helix, named after H. S. M. Coxeter and Arie Hendrick Boerdijk [es], is a linear stacking of regular tetrahedra, arranged so that...

star polyhedra in 14th century Europe, a proper mathematical account of polyhedral stellations was given by Johannes Kepler in his 1619 classic work, Harmonices...

In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of...

subgroups of S U ( 2 ) {\displaystyle SU(2)} , the binary polyhedral groups; properly, binary polyhedral groups correspond to the simply laced affine Dynkin...

explain the structures of condensed polyhedral boranes such as B20H16, which are obtained by condensing polyhedral boranes by sharing a triangular face...

Barnette's conjecture, an open problem on Hamiltonicity of cubic bipartite polyhedral graphs Eulerian path, a path through all edges in a graph Fleischner's...

In mathematics, and more specifically in polyhedral combinatorics, a Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They...

B. C.; Sülzle, D.; Hauer, H. "Onion-Like Inorganic Fullerenes from a Polyhedral Perspective". In Sattler, Klaus D. (ed.). 21st Century Nanoscience: A...

and one celestial), a shepherd's dial, a quadrant, a torquetum, and a polyhedral sundial, as well as various textiles. The floor mosaic is based on a design...

cover an even number of black squares) to fit a rectangle of size 360. A polyhedral net for the cube is necessarily a hexomino, with 11 hexominoes (shown...

together include each edge of the graph exactly twice. For instance, for any polyhedral graph, the faces of a convex polyhedron that represents the graph provide...

Nepovirus is a genus of viruses in the order Picornavirales, in the family Secoviridae, in the subfamily Comovirinae. Plants serve as natural hosts. There...

In mathematics, a piecewise linear or segmented function is a real-valued function of a real variable, whose graph is composed of straight-line segments...

figures by Stella4D software. Other convex 4-polytopes: Polyhedral pyramid Polyhedral bipyramid Polyhedral prism Infinite uniform 4-polytopes of Euclidean 3-space...

An n-sided antiprism has 2n vertices and 4n edges. They are regular, polyhedral (and therefore by necessity also 3-vertex-connected, vertex-transitive...

area proportions, and unfolding it in the form of a rectangle: it is a polyhedral map projection. The map reduces the distortion of sizes and shapes of...

2-fold axes. D2 is a subgroup of all the polyhedral symmetries (see below), and D2h is a subgroup of the polyhedral groups Th and Oh. D2 occurs in molecules...

shortest path problem is a problem in computational geometry: given a set of polyhedral obstacles in a Euclidean space, and two points, find the shortest path...

self-intersections) in 3-space. Boy's surface is an example of an immersion. Polyhedral examples must have at least nine faces. Steiner's Roman surface is a more...

Thomas Penyngton Kirkman FRS (31 March 1806 – 3 February 1895) was a British mathematician and ordained minister of the Church of England. Despite being...

prisms. 5 are polyhedral prisms based on the Platonic solids (1 overlap with regular since a cubic hyperprism is a tesseract) 13 are polyhedral prisms based...

area of a triangle and the volume of a pyramid. A hyperpyramid with a polyhedral base with f-vector (v,e,f) (vertices, edges, and faces), will have a new...

Organoboron chemistry or organoborane chemistry studies organoboron compounds, also called organoboranes. These chemical compounds combine boron and carbon;...

chromatic index 3, girth 4 and diameter 8. The Tutte graph is a cubic polyhedral graph, but is non-hamiltonian. Therefore, it is a counterexample to Tait's...

normal fan of a convex polytope P is a polyhedral fan that is dual to P. Normal fans have applications to polyhedral combinatorics, linear programming, tropical...

Octahedral pyramid Schlegel diagram Type Polyhedral pyramid Schläfli symbol ( ) ∨ {3,4} ( ) ∨ r{3,3} ( ) ∨ s{2,6} ( ) ∨ [{4} + { }] ( ) ∨ [{ } + { } +...

Snyder equal-area projection is a polyhedral map projection used in the ISEA (Icosahedral Snyder Equal Area) discrete global grids. It is named for John...

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...

{\displaystyle G} . A stronger form of the circle packing theorem applies to any polyhedral graph and its dual graph, and proves the existence of a primal–dual packing...

In geometry, this polyhedron can be seen as either a polyhedral stellation or a compound. It can be seen as the compound of an icosahedron and dodecahedron...

In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices...

the Dutch artist M. C. Escher in 1948, depicting two chameleons in a polyhedral cage floating through space. The compound of three octahedra used for...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

Lectures in Geometric Combinatorics is a textbook on polyhedral combinatorics. It was written by Rekha R. Thomas, based on a course given by Thomas at...

The Cahill–Keyes projection is a polyhedral compromise map projection first proposed by Gene Keyes in 1975. The projection is a refinement of an earlier...

In geometry, the Perles configuration is a system of nine points and nine lines in the Euclidean plane for which every combinatorially equivalent realization...

explicit representation in terms of integer divisions. It is used as backend polyhedral library in the GCC Graphite framework and in the LLVM Polly framework...

A molecular model is a physical model of an atomistic system that represents molecules and their processes. They play an important role in understanding...

stellation and may also be applied to higher-dimensional polytopes. In polyhedral combinatorics and in the general theory of polytopes, a face that has...

perpendicular to each other. With this construction, the bidiakis cube is a polyhedral graph, and can be realized as a convex polyhedron. Therefore, by Steinitz's...

In microbiology, a pentamer is one of the proteins that compose the polyhedral protein shell that encloses the bacterial micro-compartments known as...

compilers. Research subsequently continued into techniques including polyhedral compilation, runtime dependence analysis, automatic vectorization and...

2 or higher (n may be 1 as a special case). There are three types of polyhedral symmetry: tetrahedral symmetry, octahedral symmetry, and icosahedral symmetry...

these shapes. However, these balls have rounded knobs rather than being polyhedral. The number of knobs frequently differed from the numbers of vertices...

gives a polyhedral representation of G or of its dual; in the case that the dual graph is the one with the triangle, polarization gives a polyhedral representation...

MR 0062446 Wenninger, Magnus J. (1979), "An introduction to the notion of polyhedral density", Spherical models, CUP Archive, pp. 132–134, ISBN 978-0-521-22279-2...

Wings with more than one angle change along the full span are said to be polyhedral. Dihedral angle has important stabilizing effects on flying bodies because...

degenerate polyhedral compounds eleven isotoxal polyhedra which are neither isohedral nor isogonal There are at least five isotoxal polyhedral compounds:...

characterizations. They are the chordal maximal planar graphs, the chordal polyhedral graphs, and the planar 3-trees. They are the uniquely 4-colorable planar...

three-dimensional motion of the surface of the polyhedron, cut to form a polyhedral net, from the polyhedron into a flat and non-self-overlapping placement...

blue, as the base {2n/d}-gon has been withdrawn. The hypercupolae or polyhedral cupolae are a family of convex nonuniform polychora (here four-dimensional...

\arccos(-{\tfrac {1}{3}})\approx 109.47^{\circ }} . The angles of the polyhedral structure are different; for instance, its edges meet at angles of 90...

The Yoshimoto Cube is a polyhedral mechanical puzzle toy invented in 1971 by Naoki Yoshimoto (吉本直貴, Yoshimoto Naoki), who discovered that two stellated...

tetrahedra, also known as the icosicosahedron, is one of the five regular polyhedral compounds. This polyhedron can be seen as either a stellation of the icosahedron...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

Look up dihedral or polyhedral in Wiktionary, the free dictionary. Dihedral or polyhedral may refer to: Dihedral angle, the angle between two mathematical...

1992. His dissertation, "Visibility Computations in Densely Occluded Polyhedral Environments," was supervised by Carlo H. Séquin. He was a post-doctoral...

William N. Lipscomb Jr. Hoffman worked on the molecular orbital theory of polyhedral molecules. Under Lipscomb's direction the Extended Hückel method was developed...

plus one artist's proof. The Minimalist work comprises five different polyhedral elements painted black. The five elements of the sculpture have different...

polychoron (four-dimensional polytope). It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of Platonic solids...

lighter members of the group, lead can bond with itself, forming chains and polyhedral structures. Since lead is easily extracted from its ores, prehistoric...

2-sphere in R3 can be approximated by polyhedral 2-spheres. Bing, R. H. (1957), "Approximating surfaces with polyhedral ones", Annals of Mathematics, Second...

In graph theory, the Golomb graph is a polyhedral graph with 10 vertices and 18 edges. It is named after Solomon W. Golomb, who constructed it (with a...

deltahedra, which can be used in the applications of chemistry as in the polyhedral skeletal electron pair theory and chemical compounds. There are infinitely...

fixed m × {\displaystyle \times } n matrix A, we consider the family of polyhedral sets X b = { x | A x ≥ b } , ␣ b ∈ R m {\displaystyle X_{b}=\{x|Ax\geq...

at the general public via toy stores. The boxed set included a set of polyhedral dice and supplemental materials. In that same year, Games Workshop (U...

of cube and octahedron is a polyhedron which can be seen as either a polyhedral stellation or a compound. The 14 Cartesian coordinates of the vertices...

unusually stable nature of some spherical compounds such as fullerenes and polyhedral boranes. In 2000, Andreas Hirsch and coworkers in Erlangen, Germany, formulated...

information on the program structure in the process of optimization including polyhedral compilation. At version 16, LLVM supports many instruction sets, including...

ISBN 9781799883760. Avis, David; Bremner, David; Deza, Antoine, eds. (2009). Polyhedral Computation. CRM Proceedings and Lecture Notes. Vol. 48. American Mathematical...

tetrahedra, also known as the chiricosahedron, is one of the five regular polyhedral compounds. This compound polyhedron is also a stellation of the regular...

H3 Coxeter plane projections with overlapping vertices colored. These polyhedral sections are solids in the sense that they are 3-dimensional, but of course...

tetrahedra. In origami, a polypolyhedron is "a compound of multiple linked polyhedral skeletons with uniform nonintersecting edges". There exist two topologically...

In geometry, a flexible polyhedron is a polyhedral surface without any boundary edges, whose shape can be continuously changed while keeping the shapes...

In algebraic combinatorics, the h-vector of a simplicial polytope is a fundamental invariant of the polytope which encodes the number of faces of different...

theory, the Barnette–Bosák–Lederberg graph is a cubic (that is, 3-regular) polyhedral graph with no Hamiltonian cycle, the smallest such graph possible. It...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

angles between its adjacent faces). A cuboid is a convex polyhedron whose polyhedral graph is the same as that of a cube. General cuboids have many different...

combinatorics in particular. It includes a number of subareas such as polyhedral combinatorics (the study of faces of convex polyhedra), convex geometry...

order greater than 2. These polyhedral groups are characterized by not having a C5 proper rotation axis. These polyhedral groups are characterized by...

Chloriridovirus. Viruses in Iridovirus are enveloped, with icosahedral and polyhedral geometries, and T=147 symmetry. Their diameter is around 185 nm. Genomes...

tetrahedral prism is a convex uniform 4-polytope. This 4-polytope has 6 polyhedral cells: 2 tetrahedra connected by 4 triangular prisms. It has 14 faces:...

computation time. Another reason is that it led to a linear programming polyhedral description of the matching polytope, yielding an algorithm for min-weight...

Lymphocystivirus Polyhedral T=189–217 Linear Monopartite Megalocytivirus Polyhedral T=189–217 Linear Monopartite Ranavirus Polyhedral T=133 or 147 Linear...

The Lee conformal world in a tetrahedron is a polyhedral, conformal map projection that projects the globe onto a tetrahedron using Dixon elliptic functions...

In 3D computer graphics and modeling, a volumetric mesh is a polyhedral representation of the interior region of an object. It is unlike polygon meshes...

complementary chord pairs corresponds to a distinct pair of opposing polyhedral sections of the 120-cell, beginning with a vertex, the 00 section. The...

by a group of operations. Many such puzzles are mechanical puzzles of polyhedral shape, consisting of multiple layers of pieces along each axis which can...

In geometry, the great icosahedron is one of four Kepler–Poinsot polyhedra (nonconvex regular polyhedra), with Schläfli symbol {3,5⁄2} and Coxeter-Dynkin...

to Ars Technica, he is considered to be one of the "best designers of polyhedral interlocking puzzles in the world." Coffin majored in electrical engineering...

with John Moore and Joseph Neisendorfer. Late in his life, Cohen studied polyhedral products in a series of articles with Bahri, Bendersky, and Gitler. Cohen...

dodecahedral prism is a convex uniform 4-polytope. This 4-polytope has 14 polyhedral cells: 2 dodecahedra connected by 12 pentagonal prisms. It has 54 faces:...

Viruses in the genus Chloriridovirus are enveloped, with icosahedral and polyhedral geometries, and T=189-217 symmetry. The diameter is around 180 nm. Genomes...

topologically but not geometrically the t0,1{4, 3, 3} tiling, and can be (polyhedrally) immersed in Euclidean 3-space as the small cubicuboctahedron (which...

Thomas More and Holbein himself. In his hand he holds a half-finished polyhedral sundial, whilst on the shelves behind him are a semi-circular star quadrant...

In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of...

Pehau-Arnaudet G, Krupovic M, Prangishvili D (9 July 2017). "A Novel Type of Polyhedral Viruses Infecting Hyperthermophilic Archaea". J Virol. 91 (13): e00589-17...

quadrilateralized spherical cube, or quad sphere for short, is an equal-area polyhedral map projection and discrete global grid scheme for data collected on a...

the standard ball of association football. IUPAC defines fullerenes as "polyhedral closed cages made up entirely of n three-coordinate carbon atoms and having...

polyhedral approximation of a cyclide-offset....

this shape form the simplest of the five regular polyhedral compounds, and the only regular polyhedral compound composed of only two polyhedra. They form...

the polyhedral graphs is log 3 ⁡ 2 ≈ 0.631 {\displaystyle \log _{3}2\approx 0.631} . A construction based on kleetopes shows that some polyhedral graphs...

The prismatic uniform 4-polytopes consist of two infinite families: Polyhedral prisms: products of a line segment and a uniform polyhedron. This family...

such infinitesimal arguments – unlike the 2-dimensional formulae for polyhedral area, though similar to the area of the circle – and hence admitted less...

was reported by Collatz and Sinogowitz in 1957. The smallest pair of polyhedral cospectral mates are enneahedra with eight vertices each. Almost all trees...

Cubic pyramid Type Polyhedral pyramid Schläfli symbol ( ) ∨ {4,3} ( ) ∨ [{4} × { }] ( ) ∨ [{ } × { } × { }] Cells 1 cube 6 square pyramids Faces 12 triangles...

Look up dice in Wiktionary, the free dictionary. Dice are polyhedral objects used in games for generating random numbers. Dice or DICE may also refer to:...

octahedral prism is a convex uniform 4-polytope. This 4-polytope has 10 polyhedral cells: 2 octahedra connected by 8 triangular prisms. Octahedral dyadic...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

point not in the base plane as the apex. There is a third topological polyhedral figure with 5 faces, degenerate as a polyhedron: it exists as a spherical...

can be used to construct a cube. This showed the requirements for the polyhedral equivalence conditions in terms of the Dehn invariant. Orthogonal polyhedra...

In geometry, the great stellated dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol {5⁄2, 3}. It is one of four nonconvex regular polyhedra...

equilateral triangles on each regular octahedron's face. In the perspective of polyhedral compounds, this figure comprises two such dual tetrahedra, a self-dual...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second? Based on earlier...

Polyhedral representation of spinel MgAl2O4...

– 1 November 1934) was a German geometer, known for his collection of polyhedral models. Brückner was born in Hartau, in the Kingdom of Saxony, a town...

type of asymptotic behavior useful in number theory Normal polytopes, in polyhedral geometry and computational commutative algebra Normal ring, a reduced...

three-dimensional figurate numbers. Said by some scholars to prefigure Euler's polyhedral formula. Unpublished; discovered in Descartes's estate in Stockholm in...

produce testosterone in the presence of luteinizing hormone (LH). They are polyhedral in shape and have a large, prominent nucleus, an eosinophilic cytoplasm...

Homotopy homotopy group fundamental group Simplicial complex CW complex Polyhedral complex Manifold topological smooth Bundle (mathematics) Cobordism Metrics...

Genus Structure Symmetry Capsid Genomic arrangement Genomic segmentation Mastadenovirus Polyhedral Pseudo T=59 Non-enveloped Linear Monopartite...

In graph theory and polyhedral combinatorics, areas of mathematics, Kotzig's theorem is the statement that every polyhedral graph has an edge whose two...

version of polyominoes. Two or more cubes can have the same centre, forming polyhedral compounds. Cubes can form a honeycomb by attaching face-to-face, filling...

tubes/handles yields such a shape), which have been dubbed "tetruses", or by polyhedral approximations, which have been dubbed "tetroids"; in both cases this...

professor in mathematics. Her research concerns algebraic combinatorics and polyhedral combinatorics. Novik earned her Ph.D. from the Hebrew University of Jerusalem...

_{+}^{n},} hence the closedness condition holds automatically. Indeed, for polyhedral convex cone, i.e., there exists a B ∈ R n × k {\displaystyle \mathbf {B}...

Cubic cupola Schlegel diagram Type Polyhedral cupola Schläfli symbol {4,3} v rr{4,3} Cells 28 1 rr{4,3} 1+6 {4,3} 12 {}×{3} 8 {3,3} Faces 80 32 triangles...

molecular compound featuring Bi bound to six oxygen ligands in a distorted polyhedral sphere. According to X-ray crystallography, the acetate ligands are bound...

normally angled upwards in a polyhedral wing configuration, increasing the local dihedral near the wing tip, with polyhedral wing designs themselves having...

are the rulebooks, a character sheet for each player, and a number of polyhedral dice. Many players also use miniature figures on a grid map as a visual...

Publishing. pp. 635–640. ISBN 978-1-938664-19-9. Gailiunas, Paul (2001). "A Polyhedral Byway" (PDF). In Sarhangi, Reza; Jablan, Slavik (eds.). Bridges: Mathematical...

dual polyhedral decomposition. The dual polyhedral decomposition is a cell decomposition of the manifold such that the k-cells of the dual polyhedral decomposition...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

surface of genus 3), and thus can be interpreted as a (polyhedral) immersion of a genus 3 polyhedral surface, in the complement of its 24 vertices, into...

In geometry, a uniform polyhedron compound is a polyhedral compound whose constituents are identical (although possibly enantiomorphous) uniform polyhedra...

is an irreducible tropical variety if it is the support of a weighted polyhedral complex of pure dimension d that satisfies the zero-tension condition...

solids. There are 13 Archimedean graphs, and all of them are regular, polyhedral (and therefore by necessity also 3-vertex-connected planar graphs), and...

"Fabaviruses". In Murant, A.F.; Harrison, B.D. (eds.). The Plant Viruses Polyhedral Virions and Bipartite RNA Genomes. Boston, MA: Springer US. pp. 229–250...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

concerns the rigidity of polyhedra in 3 {\displaystyle 3} -dimensions, see polyhedral systems for a definition of this type of GCS. A polyhedron is rigid if...

"Card Crafting System" Randomizers such as: Coins Dice, which can include polyhedral dice Dice tower or Dice cup Playing cards, which can be collectible cards...

{\displaystyle \sigma } .[further explanation needed] A (polyhedral) fan is a collection of (polyhedral) cones closed under taking intersections and faces....

structure of fangite along the a-axis, it is apparent that fangite exhibits a polyhedral arrangement. Arsenic (As) atoms in this orientation display perfect tetrahedral...

map in this arrangement in 1996. The arrangement is an unfolding of a polyhedral globe with the shape of a truncated octahedron, evoking the butterfly...

Homotopy homotopy group fundamental group Simplicial complex CW complex Polyhedral complex Manifold topological smooth Bundle (mathematics) Cobordism Metrics...

Ghost Coast to Coast), a television episode Cahill–Keyes projection, a polyhedral map projection Cahill ministry (disambiguation) McCahill, a surname All...

as (n) or [n]+. It is the binary polyhedral group corresponding to the cyclic group. In terms of binary polyhedral groups, the binary cyclic group is...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

non-Hamiltonicity of some counterexamples to Tait's conjecture that cubic polyhedral graphs are Hamiltonian. Grinberg's theorem is named after Latvian mathematician...

song by Tool from their 2000 box set Salival The stellated octahedron, a polyhedral compound and stellation named "merkaba" by some contemporary mystics Markaba...

be drawn in a graph, named Archimedean graph. Such graphs are regular, polyhedral (and therefore by necessity also 3-vertex-connected planar graphs), and...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

graphics and morphing. Physically, the Bit was represented by a blue polyhedral shape that alternated between the compound of dodecahedron and icosahedron...

A Kokotsakis polyhedron is a polyhedral surface in three-dimensional space consisting of any polygon as its base and quadrilaterals alternating with triangles...

Institutions Massachusetts Institute of Technology Thesis Spatial Planning with Polyhedral Models  (1980) Doctoral advisor Berthold K.P. Horn Notable students Matthew...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

Homotopy homotopy group fundamental group Simplicial complex CW complex Polyhedral complex Manifold topological smooth Bundle (mathematics) Cobordism Metrics...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

In polyhedral combinatorics (a branch of mathematics), a stacked polytope is a polytope formed from a simplex by repeatedly gluing another simplex onto...

salmon-pink basidiospores which colour the gills at maturity and are angular (polyhedral) under a microscope. The genus is large, with almost 2000 species worldwide...

watershed, where other lanceolate points, small prismatic blades, and small polyhedral blade cores have been recovered. According to Adovasio et al., this complex...

Yoshimoto, Japanese poet, literary critic, and philosopher Yoshimoto Cube, a polyhedral mechanical puzzle toy invented in 1971 This disambiguation page lists...

endmember of the coronadite group. Hollandite star in rare quartz inclusion Polyhedral representation of the 2 x 2 tunnel structure of hollandite. The black...

In solid geometry, a prismatic surface is a polyhedral surface generated by all the lines that are parallel to a given line and that intersect a polygonal...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

2006; her dissertation, Liquid crystals and tangent unit-vector fields in polyhedral geometries, was jointly supervised by Jonathan Robbins and Maxim Zyskin...

Genus Structure Symmetry Capsid Genomic arrangement Genomic segmentation Aviadenovirus Polyhedral Pseudo T=50 Non-enveloped Linear Monopartite...

gaming hobbyist, game distributor and publisher, and maker and seller of polyhedral game dice. In 1986, he was elected to the Charles Roberts Awards Hall...

way: the trees do not, for example. Steinitz's theorem says that the polyhedral graphs formed from convex polyhedra are precisely the finite 3-connected...

icosahedron, like that of any convex polyhedron, can be represented as a polyhedral graph, meaning a planar graph (one that can be drawn without crossing...

diagonals of a polygon. The vertices are defined as the point locations. Polyhedral graph is an undirected graph that forms the vertices and edges of a three-dimensional...

(in French). 6: 793–800. Fukuda, Komei. "Frequently Asked Questions in Polyhedral Computation". www.cs.mcgill.ca. Retrieved 29 October 2018. Seidel, Raimund...

In geometry, the excavated dodecahedron is a star polyhedron that looks like a dodecahedron with concave pentagonal pyramids in place of its faces. Its...

Viruses in the genus Lymphocystivirus are enveloped, with icosahedral and polyhedral geometries, and T=189-217 symmetry. The diameter is around 120-350 nm...

theory — density functional theory — Frontier molecular orbital theory — Polyhedral skeletal electron pair theory — Baeyer strain theory — Quantum theory...

five-dimensional space, bounded by (4-polytope) facets, pairs of which share a polyhedral cell. A 5-polytope is a closed five-dimensional figure with vertices,...

arsenic equivalent of freieslebenite (PbAgSbS3), but also displays close polyhedral characteristics with sicherite and diaphorite. Marrite was first described...

and consists of two types of cells: parenchyma, which have an oval or polyhedral shape, and strongly elongated tracheids. Tracheids make up more than 90%...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

icosahedron vertices (12)   Apex vertices (2)   Connecting edges (24) Type Polyhedral bipyramid Schläfli symbol {3,5} + { } dt{2,5,3} Coxeter diagram Cells...

In mathematics, a unique sink orientation is an orientation of the edges of a polytope such that, in every face of the polytope (including the whole polytope...

polygonal number theorem to three-dimensional figurate numbers, also called polyhedral numbers. Pollock tetrahedral numbers conjecture: Every positive integer...

Octahedral cupola Schlegel diagram Type Polyhedral cupola Schläfli symbol {3,4} v rr{3,4} Cells 28 1 {3,4} 1 rr{4,3} 8+12 {}×{3} 6 {}v{4} Faces 82 40 triangles...

(1991), "Cubic Block Puzzles: The 3 x 3 x 3 Cube", The Puzzling World of Polyhedral Dissections, Oxford University Press, archived from the original on 2006-10-31...

braunite, chalcophanite, manganite and various other manganese oxides. Polyhedral representation of the 2 × 2 tunnel structure of cryptomelane. The black...

elements such as silicon, germanium, or tin, that enclose guest atoms in polyhedral cages. The properties of a clathrate depend on both the host framework...

roleplaying, this solid is known as a 4-sided die, one of the more common polyhedral dice, with the number rolled appearing around the bottom or on the top...

arrangement as in the compound of 5 icosahedra. It is one of only five polyhedral compounds (along with the compound of six tetrahedra, the compound of...

a new field of chemistry dedicated to the synthesis and properties of polyhedral organic compounds. Adamantane derivatives have found practical application...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

There is a tendency in the polyhedral and chemical literature to refer to geometric graphs as nets (contrast with polyhedral nets), and the nomenclature...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

make central tetrahedron. 2 yellow vertices are bipyramid apexes. Type Polyhedral bipyramid Schläfli symbol {3,3} + { } dt{2,3,3} Coxeter diagram Cells...

Carboxysomes are bacterial microcompartments (BMCs) consisting of polyhedral protein shells filled with the enzymes ribulose-1,5-bisphosphate carboxylase/oxygenase...

chairman of its Department of Chemistry. Wade's Rules, also known as polyhedral skeletal electron pair theory, are a set of electron counting rules to...

uniform polyhedra, including the five regular convex polyhedra, five polyhedral compounds, four Kepler-Poinsot polyhedra, and thirteen Archimedean solids...

In mathematics, and more particularly in polyhedral combinatorics, Eberhard's theorem partially characterizes the multisets of polygons that can form the...

to an underlying basement membrane. Then follow two or three layers of polyhedral cells, commonly known as wing cells. The majority of these are prickle...

dialects such as affine, which supports affine loop nests suitable for polyhedral optimization, and scf, which provides structured control flow using constructs...

{\displaystyle Z} , given by a zonotopal tiling of Z {\displaystyle Z} , i.e., a polyhedral complex with support Z {\displaystyle Z} : the union of all zonotopes...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

Polyhedral compound...

uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells. In 3-dimensional hyperbolic space there are nine Coxeter group...

Inc. p. 86. ISBN 978-0-486-23729-9. Diudea, M. V. (2018). Multi-shell Polyhedral Clusters. Carbon Materials: Chemistry and Physics. Vol. 10. Springer....

Weisstein, Eric W. "Szekeres Snark". MathWorld. Szekeres, G. (1973). "Polyhedral decompositions of cubic graphs". Bull. Austral. Math. Soc. 8 (3): 367–387...

1007/978-94-017-1687-1. ISBN 978-94-017-1687-1. Diudea, M. V. (2018). Multi-shell Polyhedral Clusters. Carbon Materials: Chemistry and Physics. Vol. 10. Springer....

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

Polyhedral representation of the 2 x 3 tunnel structure of romanèchite. The red atoms represent Ba and H2O....

specifically in polytope theory, Kalai's 3d conjecture is a conjecture on the polyhedral combinatorics of centrally symmetric polytopes, made by Gil Kalai in 1989...

of order 168, known as PSL(2,7). The resulting surface can in turn be polyhedrally immersed into Euclidean 3-space, yielding the small cubicuboctahedron...

work on the Hirsch conjecture on the diameter of convex polytopes and in polyhedral combinatorics more generally. Kalai is a noted skeptic of quantum computing...

rule for the π-electrons of aromatic compounds, Polyhedral skeletal electron pair theory for polyhedral cluster compounds, including transition metals...

topological equivalence to spheres (in the bounded case) and Euler's polyhedral formula. After a lemma of Augustin Cauchy on the impossibility of labeling...

vertices (0-faces), and the empty set. In some areas of mathematics, such as polyhedral combinatorics, a polytope is by definition convex. In this setting, there...

of Button Men spans multiple rounds. Players begin by rolling a set of polyhedral dice then proceed through several turns. During each turn, their goal...

diborane, B2H6. The larger boranes all consist of boron clusters that are polyhedral, some of which exist as isomers. For example, isomers of B20H26 are based...

Operations Research. His research concerns mathematical programming, polyhedral combinatorics, and algorithmic game theory, including interior-point methods...

{R} ^{m\times n}} , b ∈ R m {\displaystyle b\in \mathbb {R} ^{m}} and a polyhedral convex ordering cone C {\displaystyle C} having nonempty interior and...

concept of a combinatorial map was introduced informally by J. Edmonds for polyhedral surfaces which are planar graphs. It was given its first definite formal...

and computational geometry of mechanical linkages, paper folding, and polyhedral nets, by Erik Demaine and Joseph O'Rourke. It was published in 2007 by...

flattened square pyramid that can become part of a quilt or a larger polyhedral face, e.g. in 12 and 24 modules large cubes. Such a square lacks structural...

data structure to describe polygons in computer graphics Fan, a type of polyhedral complex One of several types of fan-shaped deposits of sediment caused...

forms the tetrahedron, the simplest possible Platonic solid, no other polyhedral phosphorus clusters are known. White phosphorus converts to the thermodynamically...

Polyhedral compound...

7 additional polyhedral groups (also called Platonic). In Schoenflies notation, Axial groups: Cn, S2n, Cnh, Cnv, Dn, Dnd, Dnh Polyhedral groups: T, Td...

the structural chemistry of boron by the Jemmis mno rules which relates polyhedral and macropolyhedral boranes to allotropes of boron and boron-rich solids...

installed in the water at the cooling water pipe inlet in the form of a polyhedral housing with plain filter surfaces. It is cleaned by pressurized air blast...

Homotopy homotopy group fundamental group Simplicial complex CW complex Polyhedral complex Manifold topological smooth Bundle (mathematics) Cobordism Metrics...

100-sided die invented by Lou Zocchi, which debuted in 1985. Unlike other polyhedral dice, it takes the appearance of a ball with 100 flattened spots. It is...

In geometry and polyhedral combinatorics, an integral polytope is a convex polytope whose vertices all have integer Cartesian coordinates. That is, it...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

much recent research has been devoted to extending known results from polyhedral combinatorics, such as various restrictions on f-vectors of convex simplicial...

erythrocytes (small round and biconcave) and epithelial cells (large and polyhedral). The test for leukocyte esterase is indicative and does not replace microscopic...

mathematics, the compound of three octahedra or octahedron 3-compound is a polyhedral compound formed from three regular octahedra, all sharing a common center...

A compound of tetrahedra might be: Compound of two tetrahedra – stellated octahedron Compound of two truncated tetrahedra Compound of three tetrahedra...

a collection of vertices, edges and faces that define the shape of a polyhedral object's surface. It simplifies rendering, as in a wire-frame model. The...

In mathematics, economics, and computer science, the stable matching polytope or stable marriage polytope is a convex polytope derived from the solutions...

Cyclic symmetry Cnv, (*nn) [n] = Dihedral symmetry Dnh, (*n22) [n,2] = Polyhedral group, [n,3], (*n32) Tetrahedral symmetry Td, (*332) [3,3] = Octahedral...

this honeycomb a cubille. A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example...

included the world's first commercially available polyhedral meshing algorithm. The use of a polyhedral mesh has proven to be more accurate for fluid-flow...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

"a kind of anthology of my view of reality and it has turned out to be polyhedral, because I wanted to tell many things". The series is a Producciones Mandarina...

Genus Structure Symmetry Capsid Genomic arrangement Genomic segmentation Ranavirus Polyhedral T=133 or 147 Linear Monopartite...

 455, ISBN 978-1-61564-698-2. Wohlleben, Eva (2019), "Duality in Non-Polyhedral Bodies Part I: Polyliner", in Cocchiarella, Luigi (ed.), ICGG 2018 - Proceedings...

Genus Structure Symmetry Capsid Genomic arrangement Genomic segmentation Siadenovirus Polyhedral Pseudo T=25 Non-enveloped Linear Monopartite...

back to top List of uniform polyhedra The Fifty-Nine Icosahedra List of polyhedral stellations Wenninger, Magnus (1974). Polyhedron Models. Cambridge University...

JSTOR 3610702, MR 0026309, S2CID 250434686 Szekeres, George (1973), "Polyhedral decompositions of cubic graphs", Bulletin of the Australian Mathematical...

Theodore Motzkin, Victor Klee, and others. They play an important role in polyhedral combinatorics: according to the upper bound theorem, proved by Peter McMullen...

hydrocarbons are typically chains or rings. Like for other electron-delocalized polyhedral clusters, the electronic structure of these cluster compounds can be described...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

and being abelian, is itself solvable. The dicyclic group is a binary polyhedral group — it is one of the classes of subgroups of the Pin group Pin−(2)...

form the structure of a cuboctahedron. The four hexagonal planes form a polyhedral surface when eight triangles are added. Thus, the resulting polyhedron...

all 4-polytopes is zero, we have the 4-dimensional analogue of Euler's polyhedral formula: N 0 − N 1 + N 2 − N 3 = 0 {\displaystyle N_{0}-N_{1}+N_{2}-N_{3}=0\...

doi:10.1007/s00253-017-8381-7. PMID 28664324. Bobik TA (May 2006). "Polyhedral organelles compartmenting bacterial metabolic processes". Applied Microbiology...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

hexagonal ice crystals. 1996 Ray tracing Andreas Macke source codes Fortran90 Ray-tracing codes for light scattering by polyhedral shaped ice crystals....

to drop in a number of stages Compound bow, a type of bow for archery Polyhedral compound, a polyhedron composed of multiple polyhedra sharing the same...

congruent corresponding faces must be congruent to each other. That is, any polyhedral net formed by unfolding the faces of the polyhedron onto a flat surface...

In geometry, the small stellated dodecahedron is a Kepler–Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol {5⁄2, 5}. It is one of four...

factories and mausoleums like the Columbarium of San Francisco. His polyhedral Butterfly World Map, like Buckminster Fuller's later Dymaxion map of 1943...

employing a pressure flaking-like technique that removed blades from a polyhedral core, was ubiquitous throughout Mesoamerica. Modern attempts to redesign...

graph with a polyhedral embedding on any two-dimensional oriented manifold such as a torus must be of class one. In this context, a polyhedral embedding...

Salmonella cells grown on propanediol or ethanolamine showed the presence of polyhedral bodies similar to carboxysomes. The term metabolosome is used to refer...

types of combinatorial optimization problems, under the framework of polyhedral combinatorics. The related branch and cut method combines the cutting...

tissue is present. Cells of this type of tissue are roughly spherical or polyhedral to rectangular in shape, with thin cell walls. New cells produced by meristem...

Polyhedral compound...

represented by {4,3}. A regular 4-dimensional polytope, with r {p,q} regular polyhedral cells around each edge is represented by {p,q,r}. For example, a tesseract...

normal anatomical relationship in chorionic villi. Cytotrophoblasts are polyhedral, mononuclear cells with hyperchromatic nuclei and a clear or pale cytoplasm...

Diamond et al. from 1972 to 1979. In 1972, they hypothesized two separate polyhedral and filamentous viral strains within E. histolytica that caused cell lysis...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

Lorenz; Smessaert, Hans (2017). "Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation". Symmetry. 9 (10):...

triangles. The primary face of the subdivision is called a principal polyhedral triangle (PPT) or the breakdown structure. Calculating a single PPT allows...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

on the possibility of prescribing the structure of negatively-curved polyhedral surfaces in three-dimensional Euclidean space. He proved that any such...

In geometry, the great triambic icosahedron and medial triambic icosahedron (or midly triambic icosahedron) are visually identical dual uniform polyhedra...

ISBN 978-0-521-29432-4, MR 0552023 Dover 1999 ISBN 978-0-486-40921-4 Pentakis snub dodecahedron VTML polyhedral generator Try "k5sD" (Conway polyhedron notation) v t e...

numbers of faces of different dimensions in polytopes through Euler's polyhedral formula, the Dehn–Sommerville equations, and the extremal combinatorics...

Amanatides, John; Thibault, William (August 1990). "Merging BSP Trees Yields Polyhedral Set Operations". ACM SIGGRAPH Computer Graphics. 24 (4). Association of...

Entertainment) Best Dice/Dice Related Accessory Gemini Purple-Teal/Gold Mini Polyhedral 7-Die Set (Chessex Manufacturing) Puzzle Cube Dice (Foam Brain Games)...

is not divisible by 3 {\displaystyle 3} . The Frucht graph is a cubic polyhedral nut graph. Every nut graph is connected, non-bipartite, and without leaf...

plumbides and bismuth polycations. The term is usually used in the context of polyhedral skeletal electron pair theory or systematic nomenclature in inorganic...

Dean Winchester and Sam Winchester. It also requires a coin and multiple polyhedral dice ranging from four to twelve sides. The Supernatural Role Playing...

In geometry and polyhedral combinatorics, the Kleetope of a polyhedron or higher-dimensional convex polytope P is another polyhedron or polytope PK formed...

harnesses the dimensional principles of an airframe externally covered by polyhedral shapes for the Lampyridae. These shapes deliberately avoid both conventional...

MultiLineStrings, MultiPolygons, GeometryCollections, 3D types TINS and polyhedral surfaces, including solids. Spheroidal types under the geography datatype...

candidates for the carrier of the diffuse interstellar bands: C+ 60 and other polyhedral carbon ions". Astron. Astrophys. 203 (1): 145. Bibcode:1988A&A...203....

abilities and flaws, as well as seeing players roll different kinds of polyhedral dice depending on how competent their character is at performing the task...

121–160, doi:10.1007/BF01580897, MR 1183645, S2CID 18981099. This paper in polyhedral combinatorics describes some of the facets of a polytope that encodes...

Genus Structure Symmetry Capsid Genomic arrangement Genomic segmentation Barthadenovirus Polyhedral Pseudo T=25 Non-enveloped Linear Monopartite...

Simons Science News. Fourth class of convex equilateral polyhedron with polyhedral symmetry related to fullerenes and viruses, Stan Schein and James Maurice...

order-4 dodecahedral honeycomb. A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

the tetrahedron, the simplest possible Platonic hydrocarbon, no other polyhedral phosphorus clusters are known. White phosphorus converts to the...

vertices. The smallest possible number of vertices for a non-Hamiltonian polyhedral graph is 11. Therefore, the Goldner–Harary graph is a minimal example...

In geometry, the great dodecahedron is one of four Kepler–Poinsot polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagons), intersecting...

The polyhedral compounds, in which the regular octahedra share the same centre, are uniform polyhedron compounds, meaning they are polyhedral compounds...

on polyhedra. Central to the book is the disputed priority for Euler's polyhedral formula between Leonhard Euler, who published an explicit version of the...

modified in light of his later work, Kepler never relinquished the Platonist polyhedral-spherical cosmology of Mysterium Cosmographicum. His subsequent main astronomical...

per player in backgammon. Role-playing games typically use one or more polyhedral dice. Games such as Pachisi and chaupur traditionally use cowrie shells...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

As in Fantasy Flight Games' Star Wars RPG, the system requires custom polyhedral dice, or dice modified with stickers to play. However, this game eliminated...

as well as cryptomelane, hollandite, manjiroite and strontiomelane. Polyhedral representation of the 2 × 2 tunnel structure of coronadite. The black...

In the mathematical discipline of polyhedral combinatorics, the Gale transform turns the vertices of any convex polytope into a set of vectors or points...

Polyhedral compound...

if on two. Discoid tools are roughly circular with a peripheral edge. Polyhedral tools are edged in the shape of a polyhedron. In addition there are spheroidal...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

finds an optimal solution by traversing the edges between vertices on a polyhedral set, interior-point methods move through the interior of the feasible...

Polyhedrosis is a viral disease in which polyhedral 'occlusion bodies' are formed, which carry the virus. Polyhedrosis may refer to: Cytoplasmic polyhedrosis...

nucleophilic additions will occur on pi ligands. Mingos' 1984 paper on the polyhedral skeletal electron pair theory develops Wade's electron counting rules...

Sciences Conference in Reno, Nevada. A unique feature was a generalized polyhedral cell formulation, allowing the solver to handle any mesh type imported...

fundamental contributions to the fields of combinatorial optimization, polyhedral combinatorics, discrete mathematics and the theory of computing. He was...

257 combinatorially distinct convex polyhedra with eight vertices (or polyhedral graphs with eight nodes). It is the 2nd Mersenne prime exponent that Mersenne...

Genus Structure Symmetry Capsid Genomic arrangement Genomic segmentation Ichtadenovirus Polyhedral Pseudo T=58 Non-enveloped Linear Monopartite...

a flexible multi-class system. It only uses 6-sided dice, since other polyhedral dice were uncommon in 1989, especially in rural Japan. The paperback (bunkobon)...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

eta conventions sequence the ligands specify coordination geometry i.e polyhedral symbol, configuration index (using CIP rules and absolute configuration...

K. J. (May 2004), "Dimensions of intersections and distance sets for polyhedral norms", Real Analysis Exchange, 30 (2): 719–726, doi:10.14321/realanalexch...

Homotopy homotopy group fundamental group Simplicial complex CW complex Polyhedral complex Manifold topological smooth Bundle (mathematics) Cobordism Metrics...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

used the Hajós construction to generate an infinite set of 4-critical polyhedral graphs, each having more than twice as many edges as vertices. Similarly...

representing planar graphs by systems of tangent circles, states that every polyhedral graph can be represented by the vertices and edges of a polyhedron with...

graph drawing date back to the work of Tutte (1963), who showed that polyhedral graphs may be drawn in the plane with all faces convex by fixing the vertices...

octahedron in the compound of three octahedra. It is one of only five polyhedral compounds (along with the compound of two great dodecahedra, the compound...

Homotopy homotopy group fundamental group Simplicial complex CW complex Polyhedral complex Manifold topological smooth Bundle (mathematics) Cobordism Metrics...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

they had them for sale." Some sources claim that certain of the original polyhedral dice used in the Dungeons & Dragons role-playing game system were obtained...

its earliest edition (1974), D&D had no standardized way to call for polyhedral die rolls or to refer to the results of such rolls. In some places the...

notwithstanding, Frederick II still commands a lasting popular reputation as a polyhedral monarch who transcended his time. Indeed, despite the innate controversy...

ZnO 10–100 Irregular and hexagonal Banana pseudo stem TiO2 9.98–24.56 Polyhedral Agro-waste durva grass ZrO2 15-35 Spherical Agricultural waste Hibiscus...

M.; Papadimitriou, Christos H. (1985). "The complexity of recognizing polyhedral scenes". 26th Annual Symposium on Foundations of Computer Science (SFCS...

stimulated. They have thin and flexible cellulose cell walls and are generally polyhedral when close-packed, but can be roughly spherical when isolated from their...

Polyhedral compound...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

craftsman surrounded by the tools of his trade, and with an unfinished polyhedral sundial. His close relationship with Holbein and More also may be observed...

rectification is still possible in this case: every polyhedron has a polyhedral graph as its 1-skeleton, and from that graph one may form the medial graph...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

rhombicuboctahedron vertex figure. A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example...

pyramids, which is essentially the content of Hilbert's third problem – polyhedral pyramids and cones cannot be cut and rearranged into a standard shape...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

dodecahedron, although that name is usually associated with the regular polyhedral form with pentagonal faces. The hexagonal bipyramid has a plane of symmetry...

stand-up reference screen, a brief but informative player's guide, seven polyhedral dice, three pewter miniatures, and a shiny red pencil. The case is designed...

three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells. Twenty-eight such honeycombs are known: the familiar cubic honeycomb...

the context of geodesy include spherical harmonics, mascon models, and polyhedral gravity representations. The type of gravity model used for the Earth...

polyhedron onto which the Laves graph can be inscribed. The edges of the Laves graph are diagonals of some of the squares of this polyhedral surface....

In mathematics, the Schwarz lantern is a polyhedral approximation to a cylinder, used as a pathological example of the difficulty of defining the area...

halides include those with B-B bonding, such as B2F4 and B4Cl4. Larger polyhedral boron(I) halides also exist, e.g. B7Br7, B8Cl8 and B9I9. Boron-containing...

In geometry, the small complex icosidodecahedron is a degenerate uniform star polyhedron. Its edges are doubled, making it degenerate. The star has 32...

The small cubicuboctahedron is a polyhedral immersion of the tiling of the Klein quartic by 56 triangles, meeting at 24 vertices....

is that surface area cannot be defined simply as the limit of areas of polyhedral shapes approximating a given smooth surface. It was demonstrated by Hermann...

Polyhedral compound...

transformations (such as tiling) do not fit easily into this framework. The polyhedral model handles a wider class of programs and transformations than the unimodular...

List of regular polytopes Uniform polyhedron Uniform star polyhedron Polyhedral compound Regular star 4-polytope – the ten regular star 4-polytopes, 4-dimensional...

are embedded in a metric space other than Euclidean space, such as a polyhedral terrain or some Euclidean space with obstacles, the distance between two...

engravings depict additional polyhedral forms, including polyhedral compounds such as the stella octangula, polyhedral variations of spheres and cones...

the placenta to the maternal tissue. Villous intermediate trophoblast polyhedral and uniform nuclei prominent cell border; abundant eosinophilic to clear...

1145/12130.12172, ISBN 0-89791-193-8, S2CID 8342016. Weisstein, Eric W., "Polygonal edge", MathWorld Weisstein, Eric W., "Polyhedral edge", MathWorld...

Cyclic symmetry Cnv, (*nn) [n] = Dihedral symmetry Dnh, (*n22) [n,2] = Polyhedral group, [n,3], (*n32) Tetrahedral symmetry Td, (*332) [3,3] = Octahedral...

cannot be dissected into each other. Two polyhedra have a dissection into polyhedral pieces that can be reassembled into either one, if and only if their volumes...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

is by now also capable of dealing with simplicial complexes, matroids, polyhedral fans, graphs, tropical objects, toric varieties and other objects. In...

November 21, 2017. Snyder, J.P. (1992). "An equal-area map projection for polyhedral globes". Cartographica. 29 (1): 10–21. doi:10.3138/27h7-8k88-4882-1752...

slightly convex; bottom side convex Elsewhere is stated: Tetrathecidae: polyhedral cross-section; shell axially twisted Novitatidae: bottom side concave...

Princeton: Van Nostrand. Coffin, Stewart T. (1990). The Puzzling World of Polyhedral Dissections. Oxford University Press. ISBN 0-19-853207-5. Frederickson...

units, FK crystallographic structures are classified into low and high polyhedral groups denoted by their coordination numbers (CN) referring to the number...

German mathematician known for his research on combinatorial optimization, polyhedral combinatorics, and operations research. From 1991 to 2012 he was Vice...

Polyhedral compound...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

frequent mitoses. The seminoma component (upper center) shows large, round-polyhedral cells with distinct cell membranes, abundant clear/watery cytoplasm, large...

formed Polyhedral Developments (a private company) in partnership with architectural designer Hayward Hill; together they pioneered the use of polyhedral domes...

Polyhedral compound...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

Interval Computation. Walter, E.; Piet-Lahanier, H. (1989). "Exact Recursive Polyhedral Description of the Feasible Parameter Set for Bounded-Error Models". IEEE...

Pattenden 1979, pp. 30–31. Pattenden 1979, p. 12. Pattenden 1979, pp. 11–12. "Polyhedral Dial, by Nicolaus Kratzer?, English, 1518–30". MHS Collection Database...

scissors-congruent to a cube, meaning that it can be sliced into smaller polyhedral pieces that can be rearranged to form a solid cube. It is star-shaped...

pathogenic to this pest. An absolutely specific virus with refractile polyhedral inclusion bodies, staining blue in Giemsa and thick blue in Buffalo Black...

nanostructured tessellation all over the terraces looking like a space-filling polyhedral foam reduced to dimension 2. Plumbene's electronic and optical properties...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

included transportation scheduling, logistics, facility location, and polyhedral combinatorics. Leung studied applied mathematics in Radcliffe College...

virus, which are formed by the self-assembly of protein subunits into a polyhedral symmetry. Large organic cages were reported by Tozawa and Cooper in 2009...

Slothouber-Graatsma puzzle in Stewart Coffin's "The Puzzling World of Polyhedral Dissections" Jan Slothouber and William Graatsma: Cubic constructs William...

map framework is included, allowing users to make their own. Instead of polyhedral dice, the game uses 6 Attack dice, 1 reveal traps die, 1 disarm traps...

divination Divination: ascertaining information by supernatural means Dice: polyhedral objects used to randomize decisions Oracle: person or object used to obtain...

In polyhedral combinatorics, a branch of mathematics, Balinski's theorem is a statement about the graph-theoretic structure of three-dimensional convex...

ball (which may actually be a disc or a non-spherical object similar to polyhedral dice) and drive it to the goal between the opposing player's figures....

Robert Scott, A Greek-English Lexicon, on Perseus Sarhangi, Reza (2012), "Polyhedral Modularity in a Special Class of Decagram Based Interlocking Star Polygons"...

already been studied over a century earlier by Kirkman. Halin graphs are polyhedral graphs, meaning that every Halin graph can be used to form the vertices...

MultiPoint LineString, MultiLineString Polygon, MultiPolygon, Triangle PolyhedralSurface TIN (Triangulated irregular network) GeometryCollection Coordinates...

Andrzej (1986). "A regularized decomposition method for minimizing a sum of polyhedral functions". Mathematical Programming. 35: 309–333. doi:10.1007/BF01580883...

The convex conjugate of a polyhedral convex function (a convex function with polyhedral epigraph) is again a polyhedral convex function. Declare that...

Chavín pottery with respect to shape reveals two kinds of vessels: a polyhedral carved type and a globular painted type. Stylistically, Chavín art forms...

this binary tetrahedral flavor symmetry, agrees with experiment. Binary polyhedral group Binary cyclic group, ⟨n⟩, order 2n Binary dihedral group, ⟨2,2,n⟩...

Thompson analyses the polyhedral forms of Radiolaria from the Challenger expedition drawn by Ernst Haeckel, 1904....

In (polyhedral) geometry, a flag is a sequence of faces of a polytope, each contained in the next, with exactly one face from each dimension. More formally...

Igor (April 20, 2010). "40.4 Folding a napkin". Lectures on Discrete and Polyhedral Geometry. pp. 354–355. Tabachnikov, Sergei (2007). "Book review of Arnold's...

Archimedean solids leading to vertex-transitive polyhedra Livio Zefiro VRML polyhedral generator (Conway polyhedron notation) VRML model Chamfered cube 3.2.7...

rudder is commonly used in model aircraft where if sufficient dihedral or polyhedral is included in the wing design, primary roll control such as ailerons...

right-handed. There are three regular skew apeirohedra, which look rather like polyhedral sponges: 6 squares around each vertex, Coxeter symbol {4,6|4} 4 hexagons...

with a clay content of at least 30% and a polyhedral structure or a blocky structure, breaking into a polyhedral or a flat-edged structure. The soil aggregates...

round to form a closed surface. Skew apeirohedra have also been called polyhedral sponges. Many are directly related to a convex uniform honeycomb, being...

In mathematics, the vertex enumeration problem for a polytope, a polyhedral cell complex, a hyperplane arrangement, or some other object of discrete geometry...

the Pacific Ocean minimising scale errors; and computed new conformal polyhedral map projections using elliptic functions, building on the work of Oscar...

Cyclic symmetry Cnv, (*nn) [n] = Dihedral symmetry Dnh, (*n22) [n,2] = Polyhedral group, [n,3], (*n32) Tetrahedral symmetry Td, (*332) [3,3] = Octahedral...

molecular geometry, VSEPR theory, and stereochemistry. At this level, regular polyhedral and symmetric forms, such as the capsids of many icosahedral or helical...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

pseudocylindric projection. The octant projection is the first known polyhedral map projection. It is neither conformal nor equal-area. In it, the spherical...

André; Vogtenhuber, Birgit; Wolff, Alexander (2024), "Adjacency Graphs of Polyhedral Surfaces", Discrete & Computational Geometry, 71 (4): 1429–1455, doi:10...

mathematician whose research interests include combinatorial optimization, polyhedral combinatorics, packing problems, and graph algorithms. She is a professor...

polyhedron has a dissection into any other polyhedron of the same volume using polyhedral pieces (see Dehn invariant). This process is possible, however, for any...

geometric perspective for problems in linear programming. Many traditional polyhedral forms are n-dimensional polyhedra. Other examples include: A half-space...

generally in every polyhedral graph, the peripheral cycles are exactly the faces of a planar embedding of the graph, so a polyhedral graph is strangulated...

following are some of the aspects of polytopes studied in discrete geometry: Polyhedral combinatorics Lattice polytopes Ehrhart polynomials Pick's theorem Hirsch...

conforming or non-conforming, and may rely on very general polygonal or polyhedral meshes (or may even be meshless). Some core properties are required to...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

edge-truncation of the Platonic and Archimedean solids leading to vertex-transitive polyhedra Livio Zefiro VRML polyhedral generator (Conway polyhedron notation)...

combinatorics, there are 3n signed subsets of a set of n elements. In polyhedral combinatorics, the hypercube and all other Hanner polytopes have a number...

construction or GC operation) is a graph operation defined on regular polyhedral graphs with degree 3 or 4. It also applies to the dual graph of these...

shared among three or more Al-O polyhedra, and the majority of inter-polyhedral connections are corner-sharing, with the remaining 10–20% being edge-sharing...

orthogonal direction. This gives rise for the notion of polyhedral terrain in three dimensions: a polyhedral surface with the property that each vertical (i.e...

isosceles triangles Concave Rhombic hexecontahedron - rhombi 27 uniform star-polyhedral duals: (self-intersecting) Small dodecicosacron, Great dodecicosacron...

about 50 radially grooved holes in the fruit body, which are more or less polyhedral to spherical near the top, but more elongated near the base. The spongy...

Afwillite is composed of double chains that consist of calcium and silicon polyhedral connected to each other by sharing corners and edges. This causes continuous...

dichloroborane(3) and a polyhedral dichloroborane. Heat induces disproportionation back to boron trichloride and a polyhedral boron(I) chloride. P. L...

source of meteor showers. The Herschel graph, the smallest non-Hamiltonian polyhedral graph, is named after him due to his pioneering work on William Rowan...

The decidual cells become filled with lipids and glycogen and take the polyhedral shape characteristic of decidual cells. Factors from the blastocyst also...

Side four No. Title Length 1. "Mortegage" 5:04 2. "Wild Animals and Polyhedral Garden" 18:20...

involving a group of students who discuss proposed proofs of Euler's polyhedral formula. A central theme is that definitions are not carved in stone,...

belongs to at most two blocks, then it is called a Christmas cactus. Every polyhedral graph has a Christmas cactus subgraph that includes all of its vertices...

earn a mathematics PhD Margaret Bayer, American mathematician working in polyhedral combinatorics Pilar Bayer (born 1946), Spanish number theorist Eva Bayer-Fluckiger...

proportione. Every Platonic graph, including the icosahedral graph, is a polyhedral graph: they can be drawn in the plane without crossing its edges, and...

polyhedral structures", Mathematics Magazine, 51 (3): 165–170, doi:10.2307/2689996, JSTOR 2689996, MR 0498579 Ziegler, Günter M. (2008), "Polyhedral surfaces...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

In mathematics, the Dehn–Sommerville equations are a complete set of linear relations between the numbers of faces of different dimension of a simplicial...

through techniques such as counting edges incident to mapped vertices. The polyhedral approach underlying these integer programming methods involves identifying...

the theory of double groups. Finite double groups include the binary polyhedral groups. In physical chemistry, double groups are used in the treatment...

rhombicosidodecahedron vertex figure. A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example...

with a panelled tapered bowl. On the top is a later pyramidal cap with a polyhedral finial, and on the bowl is an inscription. Gravenhunger Hall and outbuilding...

perpendicularly with two lateral galleries that are accessed by wide polyhedral lobes and end in separate pointed, multilobed arches whose alfiz is decorated...

degree-three vertices. Triakis triangular tiling for other "triakis" polyhedral forms. Great triakis icosahedron Conway, John H.; Burgiel, Heidi; Goodman-Strauss...

Margaret M. Bayer is an American mathematician working in polyhedral combinatorics. She is a professor of mathematics at the University of Kansas. Bayer...

Faithful Permutation Degrees for Irreducible Coxeter Groups and Binary Polyhedral Groups". arXiv:0812.0182 [math.GR]. Adams, J. Frank (1996). Lectures on...

Troy, New York, U.S. Hirsch conjecture, in mathematical programming and polyhedral combinatorics h-index, or Hirsch index, an author-level metric All pages...

Polyhedral compound...

space with amenable holonomy by a different technique using nonstandard polyhedral Gauss–Bonnet theorem developed by Ethan Bloch and Kim and Lee. In 2002...

elements introduced a century earlier by Empedocles were composed of small polyhedral forms: tetrahedron (fire), octahedron (air), icosahedron (water), and...

Icosahedral pyramid Schlegel diagram Type Polyhedral pyramid Schläfli symbol ( ) ∨ {3,5} Cells 21 1 {3,5} 20 ( ) ∨ {3} Faces 50 20+30 {3} Edges 12+30 Vertices...

blue, and yellow. These Monkeys have also been used for modeling of polyhedral structures, including virus particles and other protein structures. In...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...

Reinelt, G. (1987), "Calculating exact ground states of spin glasses: a polyhedral approach", Heidelberg colloquium on glassy dynamics (Heidelberg, 1986)...

canaliculi empty directly into the canals of Hering. Hepatocytes are polyhedral in shape, therefore having no set shape or design, although they are made...

Oxford: Clarendon Press, MR 0124167. Diudea, M. V. (2018), Multi-shell Polyhedral Clusters, Carbon Materials: Chemistry and Physics, vol. 10, Springer,...

number of triangles in this way, the proof concludes by using Euler's polyhedral formula to relate the number of triangles to the number of grid points...

Polyhedral compound...

markings are generally along leaf keels or margins, giving a sort of polyhedral appearance. Marginal teeth are usually lacking, while the terminus of...

consists of aquatic, coccoid algae. The cells are spherical or with complex polyhedral shapes. Chloroplasts have pyrenoids that are surrounded by continuous...

connected rigid body translate along its axis line. They model edges of the polyhedral objects used in many task planners or sensor processing modules. They...

the defect at a vertex is a discrete version of the curvature of the polyhedral surface concentrated at that point. Negative defect indicates that the...

Polyhedral dice and a character sheet...

by its two regular cells. A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example...

including extraction of edges from images, labeling of lines, non-polyhedral and polyhedral modeling, representation of objects as interconnections of smaller...

from highland sources in Guatemala and Mexico, probably as preformed polyhedral cores. One-handed use of macanas enables the user to hold a shield in...

Perspective Planar Gnomonic Orthographic Stereographic Central cylindrical Polyhedral AuthaGraph Cahill Butterfly Cahill–Keyes M-shape Dymaxion ISEA Quadrilateralized...