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In the study of abstract polytopes, a chiral polytope is a polytope that is as symmetric as possible without being mirror-symmetric, formalized in terms
Chiral_polytope
Property of an object that is not congruent to its mirror image
chiral knot. For example, the unknot and the figure-eight knot are achiral, whereas the trefoil knot is chiral. Asymmetry Chiral polytope Chirality (chemistry)
Chirality_(mathematics)
Polytope with highest degree of symmetry
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. In
Regular_polytope
Regular object in four dimensional geometry
In four-dimensional geometry, the 24-cell is a convex regular 4-polytope, a four-dimensional analogue of a Platonic solid. It is named for the 24 octahedra
24-cell
Class of 4-dimensional polytopes
In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra
Uniform_4-polytope
Polyhedron associated with another by swapping vertices for faces
of a polytope's dual will be the topological duals of the polytope's vertex figures. For the polar reciprocals of the regular and uniform polytopes, the
Dual_polyhedron
Five-dimensional geometric shape
5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets
Uniform_5-polytope
Poset representing certain properties of a polytope
mathematics, an abstract polytope is an algebraic partially ordered set which captures certain combinatorial properties of a traditional polytope without specifying
Abstract_polytope
Flat-sided three-dimensional shape
two-dimensional polygons and to be the three-dimensional specialization of polytopes (a more general concept in any number of dimensions). Polyhedra have several
Polyhedron
Aspect of geometry
improper faces. A polytope is regular if, and only if, its symmetry group is transitive on its flags. This definition excludes chiral polytopes. Two flags are
Flag_(geometry)
In the latter paper, Pasteur sketches from natural concrete reality chiral polytopes quite possibly for the first time. The optical property of tartaric
Racemic_acid
Four-dimensional analog of the octahedron
convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}. It is one of the six regular convex 4-polytopes first described
16-cell
Polyhedron with two kinds of faces
colors. Their vertex figures are quasiregular triangular tilings, . Chiral polytope Rectification (geometry) Coxeter, H.S.M., Longuet-Higgins, M.S. and
Quasiregular_polyhedron
Four-dimensional analog of the icosahedron
In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,5}. It is also known
600-cell
3D shape made of polyhedra sharing a common center
regular polytopes. Coxeter lists a few of these in his book Regular Polytopes. McMullen added six in his paper New Regular Compounds of 4-Polytopes. Self-duals:
Polytope_compound
In 4-dimensional geometry, there are 9 uniform 4-polytopes with F4 symmetry, and one chiral half symmetry, the snub 24-cell. There is one self-dual regular
List_of_F4_polytopes
Polyhedron with four faces
tetrahedron of the cube is an example of a Heronian tetrahedron. Every regular polytope, including the regular tetrahedron, has its characteristic orthoscheme
Tetrahedron
Extending the elements of a polytope to form a new figure
in two dimensions, a polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. Starting with an original figure
Stellation
Group of geometric symmetries with at least one fixed point
polyhedral groups of 3D, it can be named by its related convex regular 4-polytope. Related pure rotational groups exist for each with half the order, and
Point_group
a runcinated 120-cell (or runcinated 600-cell) is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular 120-cell.
Runcinated_120-cells
Four-dimensional analog of the dodecahedron
In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {5,3,3}. It is also called
120-cell
the rectified 600-cell or rectified hexacosichoron is a convex uniform 4-polytope composed of 600 regular octahedra and 120 icosahedra cells. Each edge has
Rectified_600-cell
Polyhedron with 12 faces
pentagon-tritetrahedron, and tetrahedric pentagon dodecahedron) is a dodecahedron with chiral tetrahedral symmetry (T). Like the regular dodecahedron, it has twelve identical
Dodecahedron
Prism with a 5-sided base
through a right angle without changing its chirality. It exists as cells of four nonprismatic uniform 4-polytopes in four dimensions: Weisstein, Eric W. "Pentagonal
Pentagonal_prism
Four-dimensional geometrical object
In four-dimensional geometry, a runcinated 5-cell is a convex uniform 4-polytope, being a runcination (a 3rd order truncation, up to face-planing) of the
Runcinated_5-cell
Simplex formed from a right-angled path
it. The characteristic simplex is chiral (it comes in two mirror-image forms which are different), and the polytope is dissected into an equal number
Schläfli_orthoscheme
Overview of and topical guide to geometry
triangulation Quasicrystal Parallelogram law Polytope Schläfli symbol Regular polytope Regular Polytopes Sphere Quadric Hypersphere, sphere Spheroid Ellipsoid
Outline_of_geometry
Convex polyhedron with regular faces
by gyration, diminishment, or dissection. Near-miss Johnson solid Blind polytope Araki, Yoshiaki; Horiyama, Takashi; Uehara, Ryuhei (2015). "Common Unfolding
Johnson_solid
Compound polyhedron
There are two enantiomorphous forms (the same figure but having opposite chirality) of this compound polyhedron. Both forms together create the reflection
Compound_of_five_tetrahedra
Geometric operation applied to a polyhedron
simus) and snub dodecahedron (dodecaedron simum). In general, snubs have chiral symmetry with two forms: with clockwise or counterclockwise orientation
Snub_(geometry)
regular polytopes. Extended symmetries exist in uniform polychora with symmetric ring-patterns within the Coxeter diagram construct. Chiral symmetries
Point groups in four dimensions
Point_groups_in_four_dimensions
Infinite regular skew polyhedron
resulting polytope, select one connected component. For regular polytopes the last step is guaranteed to produce a unique result. This new polytope is called
Regular_skew_apeirohedron
Linear stacking of regular tetrahedra that form helices
belong to only one tetrahedron form three intertwined helices. There are two chiral forms, with either right-handed or left-handed windings. Unlike any other
Boerdijk–Coxeter_helix
Belgian mathematician (born 1972)
geometry. He has made major contributions in the study of regular and chiral polytopes whose automorphism groups are finite almost simple groups. He has published
Dimitri_Leemans
a runcinated tesseract (or runcinated 16-cell) is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular tesseract
Runcinated_tesseracts
Slovenian mathematician (born 1949)
1007/s00454-005-1224-9 Conder, M., I. Hubard, T. Pisanski. Constructions for chiral polytopes, Journal of the London Mathematical Society 77 (1), 2007, 115-129.
Tomaž_Pisanski
Mexican mathematician
Bracho, Javier; Hubard, Isabel; Pellicer, Daniel (2014). "A finite chiral 4-Polytope in R 4 {\displaystyle \mathbb {R} ^{4}} ". Discrete & Computational
Isabel_Hubard_Escalera
Solid objects based on tetrahedrons
distinct types of objects, all based on the tetrahedron: A regular convex 4-polytope made up of 600 tetrahedral cells. It is more commonly known as a 600-cell
Polytetrahedron
On lattices and sphere packing in Euclidean space
der euklidischen Räume in kongruente Polytope [On the decomposition of Euclidean spaces into congruent polytopes] (PDF). Sitzungsberichte der Preussischen
Hilbert's_eighteenth_problem
Archimedean solid with 38 faces
{\begin{Bmatrix}4\\3\end{Bmatrix}}} . The snub cube, like the snub dodecahedron, is chiral, which means it does not equal its mirror image; it has two equally valid
Snub_cube
Polyhedron with 6 faces
hexahedra, the cuboid and six others, which are depicted below. One of these is chiral, in the sense that it cannot be deformed into its mirror image. Three further
Hexahedron
Infinite polyhedron with non-planar faces
Coxeter and Petrie found three of these that filled 3-space: There also exist chiral skew apeirohedra of types {4,6}, {6,4}, and {6,6}. These skew apeirohedra
Skew_apeirohedron
Solid with eight equal triangular faces
segments. More generally, every cross-polytope and its dual, hypercube, in any higher-dimensional space are Hanner polytope. The polyhedral compounds, in which
Regular_octahedron
Cycle graph with all opposite nodes linked
configurations within these problems can be used to define facets of the polytope describing a linear programming relaxation of the problem; these facets
Möbius_ladder
Isohedral and isogonal polyhedron
eds. Springer, New York 2003, pp. 461–488. List of noble polyhedra at Polytope Wiki [1] Regeneron awards announcement of winner enumerating all noble
Noble_polyhedron
Tiling of euclidean or hyperbolic space of three or more dimensions
non-Euclidean spaces, such as hyperbolic honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical
Honeycomb_(geometry)
Solid with four equal triangular faces
stacked face-to-face in a chiral aperiodic chain called the Boerdijk–Coxeter helix. The pentachoron is a four-dimensional polytope, a generalization of a
Regular_tetrahedron
Mathematical coincidence
diagrams, yielding isomorphisms of the corresponding Coxeter groups and of polytopes realizing the symmetries, as well as isomorphisms of Lie algebras whose
Exceptional_isomorphism
tiling of the hyperbolic plane. It has Schläfli symbol of sr{6,6}. Drawn in chiral pairs, with edges missing between black triangles: A higher symmetry coloring
Snub_hexahexagonal_tiling
78-dimensional exceptional simple Lie group
maximal subgroups of E6 up to dimension 78 are shown to the right. The E6 polytope is the convex hull of the roots of E6. It therefore exists in 6 dimensions;
E6_(mathematics)
Symmetric tessellation of a closed surface
vertex-edge-vertex in straight lines. Topological graph theory Abstract polytope Planar graph Toroidal graph Graph embedding Regular tiling Platonic solid
Regular_map_(graph_theory)
Only regular space-filling tessellation of the cube
non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical
Cubic_honeycomb
Polytope or tiling with one type of edge
In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal (from Greek τόξον 'arc') or edge-transitive if its symmetries
Isotoxal_figure
Geometric polyhedral group
Wythoff symbol List of spherical symmetry groups Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. (The Polyhedral Groups. §3.5, pp. 46–47)
Polyhedral_group
Regular tiling of a two-dimensional space
symmetry. Type 2 contains glide reflections, and is 2-isohedral keeping chiral pairs distinct. There are also 15 monohedral convex pentagonal tilings,
Hexagonal_tiling
Archimedean solid with 26 faces
is interchangeable by the appearance of inversion center. It is also non-chiral; that is, it is congruent to its own mirror image. The rhombicuboctahedron
Rhombicuboctahedron
Polyhedron with 8 triangles and 6 squares
equilateral polytopes are those that can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing
Cuboctahedron
Polyhedral compound
arrangement as a dodecahedron. The compound of five tetrahedra represents two chiral halves of this compound (it can therefore be seen as a "compound of two
Compound_of_ten_tetrahedra
face are seen from the same side). In the 1948 first edition of Regular Polytopes, H. S. M. Coxeter describes the stellation process as the reciprocal action
List of polyhedral stellations
List_of_polyhedral_stellations
Solid with 12 equal pentagonal faces
Configurations". Regular Polytopes (3rd ed.). New York: Dover Publications. Coxeter, H. S. M. (1991). Regular Complex Polytopes (2nd ed.). Cambridge: Cambridge
Regular_dodecahedron
Notation for a polyhedron's vertex figure
type and therefore the vertex configuration fully defines the polyhedron. (Chiral polyhedra exist in mirror-image pairs with the same vertex configuration
Vertex_configuration
Concept in mathematics
hyperbolic plane. It has Schläfli symbols of s{(3,4,3)} and s{3,8}. Drawn in chiral pairs: The alternated construction from the truncated order-8 triangular
Snub order-8 triangular tiling
Snub_order-8_triangular_tiling
pentagons and three equilateral triangles around every vertex. Drawn in chiral pairs, with edges missing between black triangles: A double symmetry coloring
Snub_pentapentagonal_tiling
Canadian geometer (1907–2003)
author of 12 books, including The Fifty-Nine Icosahedra (1938) and Regular Polytopes (1947). Many concepts in geometry and group theory are named after him
H.S.M._Coxeter
Polygon with 12 edges
The regular dodecagon is the Petrie polygon for many higher-dimensional polytopes, seen as orthogonal projections in Coxeter planes. Examples in 4 dimensions
Dodecagon
heptagons and three equilateral triangles around every vertex. Drawn in chiral pairs, with edges missing between black triangles: A double symmetry coloring
Snub_heptaheptagonal_tiling
Tessellation of convex uniform polyhedron cells
generated. If a hole has two branches, a Vinberg polytope is generated, although only Vinberg polytope with mirror symmetry are related to the simplex
Paracompact uniform honeycombs
Paracompact_uniform_honeycombs
Groups of point isometries in 3 dimensions
object is equal to its full symmetry group if and only if the object is chiral. The point groups that are generated purely by a finite set of reflection
Point groups in three dimensions
Point_groups_in_three_dimensions
Classification system for symmetry groups in geometry
elements can be seen in ringed nodes Coxeter-Dynkin diagram for uniform polytopes and honeycomb are related to hole nodes around the + elements, empty circles
Coxeter_notation
Pattern in hyperbolic geometry
tiling of the hyperbolic plane. It has Schläfli symbol of sr{6,4}. Drawn in chiral pairs, with edges missing between black triangles: The snub tetrahexagonal
Snub_tetrahexagonal_tiling
Polygon with 14 edges
skew tetradecagons exist as Petrie polygon for many higher-dimensional polytopes, shown in these skew orthogonal projections, including: Wantzel, Pierre
Tetradecagon
demiregular tilings. Note that there are two mirror image (enantiomorphic or chiral) forms of 34.6 (snub hexagonal) tiling, only one of which is shown in the
List_of_k-uniform_tilings
Tetrahedron whose faces are all congruent
disphenoid, the rhombic disphenoid has no reflection symmetry, so it is chiral. Both tetragonal disphenoids and rhombic disphenoids are isohedra: as well
Disphenoid
Abstraction of ordered linear algebra
either study. Günter M. Ziegler introduces oriented matroids via convex polytopes. A standard matroid is called orientable if its circuits are the supports
Oriented_matroid
Right-angled non-convex polyhedron
solids". Regular Polytopes (3rd ed.). New York: Dover.; 1st ed., Methuen, 1947 Coxeter, H.S.M. (1940). "Regular and semi-regular polytopes. I". Mathematische
Jessen's_icosahedron
Archimedean solid with 32 faces
(2013). "Coxeter groups, quaternions, symmetries of polyhedra and 4D polytopes". Mathematical Physics: Proceedings of the 13th Regional Conference, Antalya
Truncated_dodecahedron
Polygon with 20 edges
regular icosagon is the Petrie polygon for a number of higher-dimensional polytopes, shown in orthogonal projections in Coxeter planes: It is also the Petrie
Icosagon
248-dimensional exceptional simple Lie group
are the vertices of a semi-regular polytope discovered by Thorold Gosset in 1900, sometimes known as the 421 polytope. In the so-called even coordinate
E8_(mathematics)
Tiling of a plane by regular hexagons and equilateral triangles
pattern of a trihexagonal tiling. The woven process gives the Kagome a chiral wallpaper group symmetry, p6 (632). The term kagome lattice was coined by
Trihexagonal_tiling
Nonabelian group of order 120
4-dimensional space match the 120 vertices of the 600-cell, a regular 4-polytope. The binary icosahedral group, denoted by 2I, is the universal perfect
Binary_icosahedral_group
Polygon with 16 edges
regular hexadecagon is the Petrie polygon for many higher-dimensional polytopes, shown in these skew orthogonal projections, including: A hexadecagram
Hexadecagon
Uniform tiling of the hyperbolic plane
tiling of the hyperbolic plane. It has Schläfli symbol of sr{5,4}. Drawn in chiral pairs, with edges missing between black triangles: The dual is called an
Snub_tetrapentagonal_tiling
Pictorial representation of symmetry
hexagonal lattice. An associated polytope – for example Gosset 421 polytope may be referred to as "the E8 polytope", as its vertices are derived from
Dynkin_diagram
Polyhedra in which all vertices are the same
dodecahedron. The resulting construction of these solids gives the property of chirality, meaning they are not identical when reflected in a mirror. However, not
Archimedean_solid
Polygon with 18 edges
octadecagon is the Petrie polygon for a number of higher-dimensional polytopes, shown in these skew orthogonal projections from Coxeter planes: Kinsey
Octadecagon
Spatial tiling of convex uniform polyhedra
zonohedra. 1900: Thorold Gosset enumerated the list of semiregular convex polytopes with regular cells (Platonic solids) in his publication On the Regular
Convex_uniform_honeycomb
Hexahedron with parallelogram faces
outside, the mirror image of the opposite face. The faces are in general chiral, but the parallelepiped is not. A space-filling tessellation is possible
Parallelepiped
Periodic spatial graph
1016/j.carbon.2014.04.077 Lanier, Jaron (2009), "From planar patterns to polytopes", American Scientist, 97: 73, doi:10.1511/2009.76.73. Séquin, Carlo H
Laves_graph
Covering by shapes without overlaps or gaps
pioneered this by defining polyschemes, which mathematicians nowadays call polytopes. These are the analogues to polygons and polyhedra in spaces with more
Tessellation
24-dimensional repeating pattern of points
244823040, or 8315553613086720000. Leech lattice, and by extension, Leech polytope, still possesses a lower symmetry order than 24-dimensional regular simplex
Leech_lattice
Tiling of hyperbolic 3-space by uniform polyhedra
vertices at infinity. Other paracompact Coxeter groups exists as Vinberg polytope fundamental domains, including these triangular bipyramid fundamental domains
Uniform honeycombs in hyperbolic space
Uniform_honeycombs_in_hyperbolic_space
Mathematical invariance under transformations
nature (i.e., via the interaction of natural and human-made chiral molecules with inherently chiral biological systems). The control of the symmetry of molecules
Symmetry
tiling of the hyperbolic plane. It has Schläfli symbol of sr{6,5}. Drawn in chiral pairs, with edges missing between black triangles: John H. Conway, Heidi
Snub_pentahexagonal_tiling
Subdivision of the plane into polygons that are all regular
semiregular tilings. Note that there are two mirror image (enantiomorphic or chiral) forms of 34.6 (snub hexagonal) tiling, only one of which is shown in the
Euclidean tilings by convex regular polygons
Euclidean_tilings_by_convex_regular_polygons
133-dimensional exceptional simple Lie group
Weyl spinors of spin(12) of opposite chirality, and their chirality generator, and two other generators of chiralities ± 2 {\displaystyle \pm {\sqrt {2}}}
E7_(mathematics)
3-3-3-3-3-i. Square tiling Uniform tilings in hyperbolic plane List of regular polytopes Weisstein, Eric W. "Hyperbolic tiling". MathWorld. Weisstein, Eric W.
Snub infinite-order triangular tiling
Snub_infinite-order_triangular_tiling
tetrahemihexahedron in Coxeter et al. 1954, pp. 415–6 Skilling, 1974 Coxeter, Regular Polytopes, p. 114 Grünbaum, Branko; Miller, J. C. P.; Shephard, G. C. (1981). "Uniform
List of uniform polyhedra by Schwarz triangle
List_of_uniform_polyhedra_by_Schwarz_triangle
Theory proposed by Roger Penrose
of scattering amplitudes in terms of Grassmann integral formulae and polytopes. These ideas have evolved more recently into the positive Grassmannian
Twistor_theory
Group of rotations in 3 dimensions
^{n}} expressed in its standard basis. Coxeter, H. S. M. (1973). Regular polytopes (Third ed.). New York: Dover Publications, Inc. p. 53. ISBN 0-486-61480-8
3D_rotation_group
Tilings of regular polygons List of uniform planar tilings List of regular polytopes John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of
Snub_apeiroapeirogonal_tiling
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