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| This page lists the [[regular polytope]]s in [[Euclidean geometry|Euclidean]], [[spherical geometry|spherical]] and [[hyperbolic geometry|hyperbolic]] spaces. Clicking on any picture will magnify it.
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| The [[Schläfli symbol]] notation describes every regular polytope, and is used widely below as a compact reference name for each.
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| The regular polytopes are grouped by dimension and subgrouped by convex, nonconvex and infinite forms. Nonconvex forms use the same vertices as the convex forms, but have intersecting [[Facet (mathematics)|facets]]. Infinite forms [[Tessellation|tessellate]] a one lower dimensional Euclidean space.
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| Infinite forms can be extended to tessellate a [[hyperbolic space]]. Hyperbolic space is like normal space at a small scale, but parallel lines diverge at a distance. This allows vertex figures to have negative [[defect (geometry)|angle defects]], like making a vertex with 7 [[equilateral triangle]]s and allowing it to lie flat. It cannot be done in a regular plane, but can be at the right scale of a hyperbolic plane.
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| ==Overview==
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| This table shows a summary of regular polytope counts by dimension.
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| {| class="wikitable"
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| |-
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| !Dimension
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| !Convex
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| !Nonconvex
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| !Convex<BR>Euclidean<BR>tessellations
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| !Convex<BR>hyperbolic<BR>tessellations
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| !Nonconvex<BR>hyperbolic<BR>tessellations
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| !Hyperbolic Tessellations<BR>with infinite cells<BR>and/or vertex figures
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| !Abstract<br>Polytopes
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| |- align=center
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| |1||1 [[List of regular polytopes#One-dimensional regular polytope|line segment]]||0||1||0||0||0||1
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| |- align=center
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| |2||∞ [[#Two-dimensional regular polytopes|polygon]]s||∞ [[#Non-convex|star polygon]]s||3 [[#Euclidean tilings|tilings]]||1||0||0||∞
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| |- align=center
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| |3||5 [[#Three-dimensional regular polytopes|Platonic solids]]||4 [[#Non-convex 2|Kepler–Poinsot solid]]s||1 [[#Tessellations of Euclidean 3-space|honeycomb]]|| ∞|| ∞||0||∞
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| |- align=center
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| |4||6 [[#Four-dimensional regular polytopes|convex polychora]]||10 [[#Non-convex 3|Schläfli–Hess polychora]]||3 [[#Tessellations of Euclidean 4-space|tessellations]]|| 4 || 0||11||∞
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| |- align=center
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| |5 ||3 [[#Five-dimensional regular polytopes and higher|convex 5-polytopes]]||0||1 [[#Tessellations of Euclidean 5-space and higher|tessellation]]||5 || 4||2||∞
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| |- align=center
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| |6||3 [[#Five-dimensional regular polytopes and higher|convex 6-polytopes]]||0||1 [[#Tessellations of Euclidean 5-space and higher|tessellation]]||0||0||5||∞
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| |- align=center
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| |7+||3||0||1||0||0||0||∞
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| |}
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| There are no nonconvex Euclidean regular tessellations in any number of dimensions.
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| ===Tessellations===
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| The classical convex polytopes may be considered [[tessellation]]s, or tilings, of spherical space. Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. For example, the (three dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere.
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| ==One-dimensional regular polytope==
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| There is only one polytope in 1 dimensions, whose boundaries are the two endpoints of a [[line segment]], represented by the empty [[Schläfli symbol]] {}.
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| == Two-dimensional regular polytopes ==
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| The two dimensional polytopes are called [[polygon]]s. Regular polygons are [[equilateral]] and [[cyclic polygon|cyclic]]. A p-gonal regular polygon is represented by [[Schläfli symbol]] {p}.
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| Usually only [[convex polygon]]s are considered regular, but [[star polygon]]s, like the [[pentagram]], can also be considered regular. They use the same vertices as the convex forms, but connect in an alternate connectivity which passes around the circle more than once to complete.
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| Star polygons should be called ''nonconvex'' rather than ''concave'' because the intersecting edges do not generate new vertices and all the vertices exist on a circle boundary.
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| ===Convex===
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| The Schläfli symbol {p} represents a [[Regular polygon|regular ''p''-gon]].
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| {| class="wikitable" style="text-align:center;"
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| |- bgcolor="#e0e0e0" valign="top"
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| !Name
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| ![[Equilateral triangle|Triangle]]<br />([[Simplex|2-simplex]])
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| ![[Square (geometry)|Square]]<br />([[Cross-polytope|2-orthoplex]])<br />([[Hypercube|2-cube]])
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| ![[Pentagon]]
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| ![[Hexagon]]
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| ![[Heptagon]]
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| ![[Octagon]]
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| |- bgcolor="#ffe0e0"
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| ![[Schläfli symbol|Schläfli]]
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| |{3}
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| |{4}
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| |{5}
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| |{6}
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| |{7}
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| |{8}
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| |-
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| ![[Coxeter-Dynkin diagram|Coxeter]]
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| |{{CDD|node_1|3|node}}
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| |{{CDD|node_1|4|node}}
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| |{{CDD|node_1|5|node}}
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| |{{CDD|node_1|6|node}}
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| |{{CDD|node_1|7|node}}
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| |{{CDD|node_1|8|node}}
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| |-
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| !Image
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| |[[Image:Regular triangle.svg|75px]]
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| |[[Image:Regular quadrilateral.svg|75px]]
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| |[[Image:Regular pentagon.svg|75px]]
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| |[[Image:Regular hexagon.svg|75px]]
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| |[[Image:Regular heptagon.svg|75px]]
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| |[[Image:Regular octagon.svg|75px]]
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| |-
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| !Name
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| ![[Enneagon]]
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| ![[Decagon]]
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| ![[Hendecagon]]
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| ![[Dodecagon]]
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| ![[Tridecagon]]
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| ![[Tetradecagon]]
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| |- bgcolor="#ffe0e0"
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| !Schläfli
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| |{9}
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| |{10}
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| |{11}
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| |{12}
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| |{13}
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| |{14}
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| |-
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| !Dynkin
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| |{{CDD|node_1|9|node}}
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| |{{CDD|node_1|10|node}}
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| |{{CDD|node_1|11|node}}
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| |{{CDD|node_1|12|node}}
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| |{{CDD|node_1|13|node}}
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| |{{CDD|node_1|14|node}}
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| |-
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| !Image
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| |[[Image:Regular nonagon.svg|75px]]
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| |[[Image:Regular decagon.svg|75px]]
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| |[[Image:Regular hendecagon.svg|75px]]
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| |[[Image:Regular dodecagon.svg|75px]]
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| |[[Image:Regular tridecagon.svg|75px]]
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| |[[Image:Regular tetradecagon.svg|75px]]
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| |-
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| !Name
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| ![[Pentadecagon]]
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| ![[Hexadecagon]]
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| ![[Heptadecagon]]
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| ![[Octadecagon]]
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| ![[Enneadecagon]]
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| ![[Icosagon]]
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| |...[[Regular polygon|p-gon]]
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| |- bgcolor="#ffe0e0"
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| !Schläfli
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| |{15}
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| |{16}
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| |{17}
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| |{18}
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| |{19}
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| |{20}
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| |{''p''}
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| |-
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| !Dynkin
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| |{{CDD|node_1|15|node}}
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| |{{CDD|node_1|16|node}}
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| |{{CDD|node_1|17|node}}
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| |{{CDD|node_1|18|node}}
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| |{{CDD|node_1|19|node}}
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| |{{CDD|node_1|20|node}}
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| |{{CDD|node_1|p|node}}
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| |-
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| !Image
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| |[[Image:Regular pentadecagon.svg|75px]]
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| |[[Image:Regular hexadecagon.svg|75px]]
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| |[[Image:Regular heptadecagon.svg|75px]]
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| |[[Image:Regular octadecagon.svg|75px]]
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| |[[Image:Regular enneadecagon.svg|75px]]
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| |[[Image:Regular icosagon.svg|75px]]
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| |}
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| ==== Degenerate (spherical) ====
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| The regular [[henagon]] {1} and regular [[digon]] {2} can be considered degenerate regular polygons. They can exist nondegenerately in non-Euclidean spaces like on the surface of a [[sphere]] or [[torus]].
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| {| class="wikitable" style="text-align:center;"
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| |- bgcolor="#e0e0e0" valign="top"
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| !Name
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| |[[Henagon]]
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| |[[Digon]]
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| |- bgcolor="#ffe0e0"
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| ![[Schläfli symbol]]
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| |{1}
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| |{2}
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| |-
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| ![[Coxeter-Dynkin diagram|Coxeter diagram]]
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| |{{CDD|node}}
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| |{{CDD|node_1}}
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| |-
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| !Image
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| |[[Image:Henagon.svg|75px]]
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| |[[Image:Digon.svg|75px]]
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| |}
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| ===Non-convex===
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| There exist infinitely many non-convex regular polytopes in two dimensions, whose Schläfli symbols consist of rational numbers {n/m}. They are called [[star polygon]]s and share the same [[vertex arrangement]]s of the convex regular polygons.
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| In general, for any natural number n, there are n-pointed non-convex regular polygonal stars with Schläfli symbols {n/m} for all m such that m < n/2 (strictly speaking {n/m}={n/(n−m)}) and m and n are [[coprime]].
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| {| class="wikitable" style="text-align:center;"
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| |- bgcolor="#e0e0e0"
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| !Name
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| |[[Pentagram]]
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| | colspan="2" | [[Heptagram]]s
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| |[[Octagram]]
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| | colspan="2" | [[Enneagram (geometry)|Enneagrams]]
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| |[[Decagram (geometry)|Decagram]]
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| |...[[star polygon|n-agrams]]
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| |- bgcolor="#ffe0e0"
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| ![[Schläfli symbol|Schläfli]]
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| |{5/2}
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| |{7/2}
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| |{7/3}
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| |{8/3}
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| |{9/2}
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| |{9/4}
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| |{10/3}
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| |{''p/q''}
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| |-
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| ![[Coxeter-Dynkin diagram|Coxeter]]
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| |{{CDD|node_1|5|rat|d2|node}}
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| |{{CDD|node_1|7|rat|d2|node}}
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| |{{CDD|node_1|7|rat|d3|node}}
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| |{{CDD|node_1|8|rat|d3|node}}
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| |{{CDD|node_1|9|rat|d2|node}}
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| |{{CDD|node_1|9|rat|d4|node}}
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| |{{CDD|node_1|10|rat|d3|node}}
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| |{{CDD|node_1|p|rat|dq|node}}
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| |-
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| !Image
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| |[[Image:Star polygon 5-2.svg|75px]]
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| |[[Image:Star polygon 7-2.svg|75px]]
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| |[[Image:Star polygon 7-3.svg|75px]]
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| |[[Image:Star polygon 8-3.svg|75px]]
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| |[[Image:Star polygon 9-2.svg|75px]]
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| |[[Image:Star polygon 9-4.svg|75px]]
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| |[[Image:Star polygon 10-3.svg|75px]]
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| |}
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| ===Tessellation===
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| There is one tessellation of the line, giving one polytope, the (two-dimensional) [[apeirogon]]. This has infinitely many vertices and edges. Its [[Schläfli symbol]] is {∞}, and Coxeter diagram {{CDD|node_1|infin|node}}.
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| ...[[Image:Regular apeirogon.png|320px]]...
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| == Three-dimensional regular polytopes ==
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| In three dimensions, polytopes are called [[polyhedron|polyhedra]]:
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| A regular polyhedron with [[Schläfli symbol]] <math>\{p,q\}</math> has a regular face type <math>\{p\}</math>, and regular [[vertex figure]] <math>\{q\}</math>.
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| A [[vertex figure]] (of a polyhedron) is a polygon, seen by connecting those vertices which are one edge away from a given vertex. For regular polyhedra, this vertex figure is always a regular (and planar) polygon.
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| Existence of a regular polyhedron <math>\{p,q\}</math> is constrained by an inequality, related to the vertex figure's [[defect (geometry)|angle defect]]:
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| : <math>1/p + 1/q > 1/2</math> : Polyhedron (existing in Euclidean 3-space)
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| : <math>1/p + 1/q = 1/2</math> : ''Euclidean'' plane tiling
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| : <math>1/p + 1/q < 1/2</math> : Hyperbolic plane tiling
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| By enumerating the [[permutation]]s, we find 5 convex forms, 4 nonconvex forms and 3 plane tilings, all with polygons <math>\{p\}</math> and <math>\{q\}</math> limited to:
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| <math>\{3\}, \{4\}, \{5\}, \{\frac{5}{2}\}</math>, and <math>\{6\}</math>.
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| Beyond Euclidean space, there is an infinite set of regular hyperbolic tilings.
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| ===Convex===
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| The convex regular [[polyhedron|polyhedra]] are called the 5 [[Platonic solid]]s. The [[vertex figure]] is given with each vertex count. All these polyhedra have an [[Euler characteristic]] (χ) of 2.
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| {| class="wikitable" style="text-align:center;"
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| |-
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| !Name
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| ![[Schläfli symbol|Schläfli]]<br />{p,q}
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| ![[Coxeter-Dynkin diagram|Coxeter]]<br />{{CDD|node_1|p|node|q|node}}
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| !Image<BR>(transparent)
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| !Image<BR>(solid)
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| !Image<BR>(sphere)
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| ![[Face (geometry)|Faces]]<br />{p}
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| ![[Edge (geometry)|Edges]]
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| ![[Vertex (geometry)|Vertices]]<br />{q}
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| ![[Symmetry group|Symmetry]]
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| ![[Dual polyhedron|Dual]]
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| |- bgcolor="#e0e0e0"
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| |[[Tetrahedron]]<br />([[Simplex|3-simplex]])<br />(Triangular pyramid)
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| |{3,3}
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| |{{CDD|node_1|3|node|3|node}}
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| |[[Image:Tetrahedron.svg|75px]]
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| |[[Image:Tetrahedron.png|75px]]
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| |[[File:Uniform tiling 332-t0-1-.png|75px]]
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| |4<br />{3}
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| |6
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| |4<br />{3}
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| |T<sub>d</sub>
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| |(self)
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| |- bgcolor="#ffe0e0"
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| |[[Cube]] <br />([[hypercube|3-cube]])<br />(Hexahedron)
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| |{4,3}
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| |{{CDD|node_1|4|node|3|node}}
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| |[[Image:Hexahedron.svg|75px]]
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| |[[Image:Hexahedron.png|75px]]
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| |[[File:Uniform tiling 432-t0.png|75px]]
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| |6<br />{4}
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| |12
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| |8<br />{3}
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| |O<sub>h</sub>
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| |Octahedron
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| |- bgcolor="#e0e0ff"
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| |[[Octahedron]]<br />([[Cross-polytope|3-orthoplex]])
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| |{3,4}
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| |{{CDD|node_1|3|node|4|node}}
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| |[[Image:Octahedron.svg|75px]]
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| |[[Image:Octahedron.png|75px]]
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| |[[File:Uniform tiling 432-t2.png|75px]]
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| |8<br />{3}
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| |12
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| |6<br />{4}
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| |O<sub>h</sub>
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| |Cube
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| |- bgcolor="#ffe0e0"
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| |[[Dodecahedron]]
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| |{5,3}
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| |{{CDD|node_1|5|node|3|node}}
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| |[[Image:POV-Ray-Dodecahedron.svg|75px]]
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| |[[Image:Dodecahedron.png|75px]]
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| |[[File:Uniform tiling 532-t0.png|75px]]
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| |12<br />{5}
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| |30
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| |20<br />{3}
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| |I<sub>h</sub>
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| |Icosahedron
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| |- bgcolor="#e0e0ff"
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| |[[Icosahedron]]
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| |{3,5}
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| |{{CDD|node_1|3|node|5|node}}
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| |[[Image:Icosahedron.svg|75px]]
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| |[[Image:Icosahedron.png|75px]]
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| |[[File:Uniform tiling 532-t2.png|75px]]
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| |20<br />{3}
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| |30
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| |12<br />{5}
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| |I<sub>h</sub>
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| |Dodecahedron
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| |}
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| ==== Degenerate (spherical) ====
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| In [[spherical geometry]], the [[hosohedron|hosohedra]] {2,n}, [[dihedron|dihedra]] {n,2} and [[Henagon#In_spherical_geometry|henagonal henahedron]] {1,1} can be considered regular polyhedra ([[tessellation|tiling]]s of the [[sphere]]).
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| Some include:
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| {| class="wikitable" style="text-align:center;"
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| |- valign="top"
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| !Name
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| ![[Schläfli symbol|Schläfli]]<br />{p,q}
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| ![[Coxeter-Dynkin diagram|Coxeter<BR>diagram]]
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| !Image<BR>(sphere)
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| ![[Face (geometry)|Faces]]<br />{p}
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| ![[Edge (geometry)|Edges]]
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| ![[Vertex (geometry)|Vertices]]<br />{q}
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| ![[List of spherical symmetry groups|Symmetry]]
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| ![[Dual polyhedron|Dual]]
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| |- bgcolor="#e0e0e0"
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| |Henagonal henahedron
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| |{1,1}
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| |{{CDD|node}}
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| |[[File:Spherical henagonal henahedron.png|75px]]
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| |1<BR>{1}
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| |0
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| |1<BR>{1}
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| |C<sub>1</sub><BR>(*1)
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| |Self
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| |- bgcolor="#e0e0ff"
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| |Henagonal dihedron
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| |{1,2}
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| |{{CDD|node_1|2|node}}
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| |[[File:Hengonal dihedron.png|75px]]
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| |2<BR>{1}
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| |1
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| |1<BR>{2}
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| |C<sub>1v</sub><BR>(*22)
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| |Henagonal hosohedron
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| |- bgcolor="#ffe0e0"
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| |Henagonal hosohedron
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| |{2,1}
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| |{{CDD|node|2|node}}
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| |[[File:Henagonal hosohedron.png|75px]]
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| |1<BR>{2}
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| |1
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| |2<BR>{1}
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| |C<sub>1v</sub><BR>(*22)
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| |Henagonal dihedron
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| |- bgcolor="#e0e0e0"
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| |Digonal dihedron<BR>Digonal hosohedron
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| |{2,2}
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| |{{CDD|node_1|2|node|2|node}}
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| |[[File:Digonal dihedron.png|75px]]
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| |2<BR>{2}
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| |2
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| |2<BR>{2}
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| |D<sub>2h</sub><BR>(*222)
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| |Self
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| |- bgcolor="#ffe0e0"
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| |Trigonal hosohedron
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| |{2,3}
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| |{{CDD|node_1|2|node|3|node}}
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| |[[File:Trigonal hosohedron.png|75px]]
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| |3<BR>{2}
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| |3
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| |2<BR>{3}
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| |D<sub>3h</sub><BR>(*322)
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| |Trigonal dihedron
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| |- bgcolor="#e0e0ff"
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| |Trigonal dihedron
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| |{3,2}
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| |{{CDD|node_1|3|node|2|node}}
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| |[[File:Trigonal dihedron.png|75px]]
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| |2<BR>{3}
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| |3
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| |3<BR>{2}
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| |D<sub>3h</sub><BR>(*322)
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| |Trigonal hosohedron
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| |- bgcolor="#ffe0e0"
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| |Hexagonal hosohedron
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| |{2,6}
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| |{{CDD|node_1|2|node|6|node}}
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| |[[File:Hexagonal hosohedron.png|75px]]
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| |6<BR>{2}
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| |6
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| |2<BR>{6}
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| |D<sub>6h</sub><BR>(*622)
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| |Hexagonal dihedron
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| |- bgcolor="#e0e0ff"
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| |Hexagonal dihedron
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| |{6,2}
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| |{{CDD|node_1|6|node|2|node}}
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| |[[File:Hexagonal dihedron.png|75px]]
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| |2<BR>{6}
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| |6
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| |6<BR>{2}
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| |D<sub>6h</sub><BR>(*622)
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| |Hexagonal hosohedron
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| |}
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| ===Non-convex===
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| The regular [[Star polyhedron|star polyhedra]] are called the [[Kepler–Poinsot polyhedra]] and there are four of them, based on the [[vertex arrangement]]s of the [[dodecahedron]] {5,3} and [[icosahedron]] {3,5}:
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| As [[spherical tiling]]s, these nonconvex forms overlap the sphere multiple times, called its ''density'', being 3 or 7 for these forms. The tiling images show a single [[spherical polygon]] face in yellow.
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| {| class="wikitable"
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| |-
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| !Name
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| !Image<BR>(transparent)
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| !Image<BR>(solid)
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| !Image<BR>(sphere)
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| ![[Stellation]]<BR>diagram
| |
| ![[Schläfli symbol|Schläfli]]<br />{p,q} and<br />[[Coxeter-Dynkin diagram|Coxeter-Dynkin]]
| |
| !Faces<br />{p}
| |
| !Edges
| |
| !Vertices<br />{q}<br />[[Vertex figure|verf.]]
| |
| ![[Euler characteristic|χ]]
| |
| ![[Density (polytope)|Density]]
| |
| ![[Symmetry group|Symmetry]]
| |
| ![[dual polyhedron|Dual]]
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| |[[Small stellated dodecahedron]]
| |
| |[[Image:SmallStellatedDodecahedron.jpg|80px]]
| |
| |[[Image:Small stellated dodecahedron.png|80px]]
| |
| |[[Image:Small stellated dodecahedron tiling.png|80px]]
| |
| |[[File:First stellation of dodecahedron facets.svg|80px]]
| |
| |{5/2,5}<br />{{CDD|node|5|node|5|rat|d2|node_1}}
| |
| |12<br />{5/2}<br />[[Image:Pentagram.svg|30px]]
| |
| |30||12<br />{5}<br />[[Image:Pentagon.svg|30px]]
| |
| | −6||3||I<sub>h</sub>||Great dodecahedron
| |
| |- BGCOLOR="#e0e0ff" align=center
| |
| |[[Great dodecahedron]]
| |
| |[[Image:GreatDodecahedron.jpg|80px]]
| |
| |[[Image:Great dodecahedron.png|80px]]
| |
| |[[Image:Great dodecahedron tiling.png|80px]]
| |
| |[[File:Second stellation of dodecahedron facets.svg|80px]]
| |
| |{5,5/2}<br />{{CDD|node_1|5|node|5|rat|d2|node}}
| |
| |12<br />{5}<br />[[Image:Pentagon.svg|30px]]
| |
| |30||12<br />{5/2}<br />[[Image:Pentagram.svg|30px]]
| |
| | −6||3||I<sub>h</sub>||Small stellated dodecahedron
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| |[[Great stellated dodecahedron]]
| |
| |[[Image:GreatStellatedDodecahedron.jpg|80px]]
| |
| |[[Image:Great stellated dodecahedron.png|80px]]
| |
| |[[Image:Great stellated dodecahedron tiling.png|80px]]
| |
| |[[File:Third stellation of dodecahedron facets.svg|80px]]
| |
| |{5/2,3}<br />{{CDD|node|3|node|5|rat|d2|node_1}}
| |
| |12<br />{5/2}<br />[[Image:Pentagram.svg|30px]]
| |
| |30||20<br />{3}<br />[[Image:Triangle.Equilateral.svg|30px]]
| |
| |2||7||I<sub>h</sub>||Great icosahedron
| |
| |- BGCOLOR="#e0e0ff" align=center
| |
| |[[Great icosahedron]]
| |
| |[[Image:GreatIcosahedron.jpg|80px]]
| |
| |[[Image:Great icosahedron.png|80px]]
| |
| |[[Image:Great icosahedron tiling.png|80px]]
| |
| |
| |
| |{3,5/2}<br />{{CDD|node_1|3|node|5|rat|d2|node}}
| |
| |20<br />{3}<br />[[Image:Triangle.Equilateral.svg|30px]]
| |
| |30||12<br />{5/2}<br />[[Image:Pentagram.svg|30px]]
| |
| |2||7||I<sub>h</sub>||Great stellated dodecahedron
| |
| |}
| |
| | |
| ===Tessellations===
| |
| ==== Euclidean tilings ====
| |
| | |
| There are three regular tessellations of the plane. All three have an [[Euler characteristic]] (χ) of 0.
| |
| {| class=wikitable
| |
| !Name
| |
| ![[Square tiling]]<BR>(Quadrille)
| |
| ![[Triangular tiling]]<BR>(Deltille)
| |
| ![[Hexagonal tiling]]<BR>(Hextille)
| |
| |- align=center
| |
| ![[Schläfli symbol|Schläfli]] {p,q}
| |
| |{4,4}
| |
| |{3,6}
| |
| |{6,3}
| |
| |- align=center
| |
| ![[Coxeter-Dynkin diagram|Coxeter diagram]]
| |
| |{{CDD|node_1|4|node|4|node}}
| |
| |{{CDD|node|6|node|3|node_1}}
| |
| |{{CDD|node_1|6|node|3|node}}
| |
| |- align=center
| |
| !Image
| |
| |[[File:Uniform tiling 44-t0.png|100px]]
| |
| |[[File:Uniform tiling 63-t2.png|100px]]
| |
| |[[File:Uniform tiling 63-t0.png|100px]]
| |
| |- align=center
| |
| ![[Symmetry group|Symmetry]]
| |
| |*442<BR>(p4m)
| |
| |colspan=2|*632<BR>(p6m)
| |
| |}
| |
| | |
| There is one degenerate regular tiling, {∞,2}, made from two [[apeirogon]]s, each filling half the plane. This tiling is related to a 2-faced [[dihedron]], {p,2}, on the sphere.
| |
| | |
| ===== Euclidean star-tilings =====
| |
| | |
| There are no regular plane tilings of [[star polygon]]s. There are many enumerations that fit in the plane (1/''p'' + 1/''q'' = 1/2), like {8/3,8}, {10/3,5}, {5/2,10}, {12/5,12}, etc., but none repeat periodically.
| |
| | |
| ==== Hyperbolic tilings ====
| |
| | |
| Tessellations of [[Lobachevski plane|hyperbolic 2-space]] can be called ''[[hyperbolic tiling]]s''. There are infinitely many regular tilings in H<sup>2</sup>. As stated above, every positive integer pair {''p'',''q''} such that 1/''p'' + 1/''q'' < 1/2 gives a hyperbolic tiling. In fact, for the general [[Schwarz triangle]] (''p'', ''q'', ''r'') the same holds true for 1/''p'' + 1/''q'' + 1/''r'' < 1.
| |
| | |
| There are a number of different ways to display the hyperbolic plane, including the [[Poincaré disc model]] which maps the plane into a circle, as shown below. It should be recognized that all of the polygon faces in the tilings below are equal-sized and only appear to get smaller near the edges due to the projection applied, very similar to the effect of a camera [[fisheye lens]].
| |
| | |
| A sampling:
| |
| {{Regular hyperbolic tiling table}}
| |
| | |
| ===== Hyperbolic star-tilings =====
| |
| There are 2 infinite forms of hyperbolic tilings whose [[face (geometry)|faces]] or [[vertex figure]]s are star polygons: {''m''/2, ''m''} and their duals {''m'', ''m''/2} with ''m'' = 7, 9, 11, .... The {''m''/2, ''m''} tilings are [[stellation]]s of the {''m'', 3} tilings while the {''m'', ''m''/2} dual tilings are [[faceting]]s of the {3, ''m''} tilings and [[greatening (geometry)|greatening]]s of the {''m'', 3} tilings.
| |
| | |
| The patterns {''m''/2, ''m''} and {''m'', ''m''/2} continue for odd ''m'' < 7 as [[polyhedron|polyhedra]]: when ''m'' = 5, we obtain the [[small stellated dodecahedron]] and [[great dodecahedron]], and when ''m'' = 3, we obtain the [[tetrahedron]]. If ''m'' is even, depending on how we choose to define {''m''/2}, we can either obtain degenerate double covers of other tilings or [[polyhedral compound|compound]] tilings.
| |
| | |
| {| class="wikitable" style="text-align:center;"
| |
| |-
| |
| ! Name
| |
| ! [[Schläfli symbol|Schläfli]]
| |
| ! [[Coxeter-Dynkin diagram|Coxeter diagram]]
| |
| ! Image
| |
| ! Face type<br />{p}
| |
| ! [[Vertex figure]]<br />{q}
| |
| ! [[Density (polytope)|Density]]
| |
| ! [[Symmetry group|Symmetry]]
| |
| ! Dual
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| |[[Order-7 heptagrammic tiling]]
| |
| |{7/2,7}
| |
| |{{CDD|node_1|7|rat|d2|node|7|node}}
| |
| |[[Image:Hyperbolic tiling 7-2 7.png|60px]]
| |
| |{7/2}<br />[[Image:Star polygon 7-2.svg|30px]]
| |
| |{7}<br />[[Image:Heptagon.svg|30px]]
| |
| | 3
| |
| | *732
| |
| | Heptagrammic-order heptagonal tiling
| |
| |- BGCOLOR="#e0e0ff" align=center
| |
| |[[Heptagrammic-order heptagonal tiling]]
| |
| |{7,7/2}
| |
| |{{CDD|node_1|7|node|7|rat|d2|node}}
| |
| |[[Image:Hyperbolic tiling 7 7-2.png|60px]]
| |
| |{7}<br />[[Image:Heptagon.svg|30px]]
| |
| |{7/2}<br />[[Image:Star polygon 7-2.svg|30px]]
| |
| | 3
| |
| | *732
| |
| | Order-7 heptagrammic tiling
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| | [[Order-9 enneagrammic tiling]]
| |
| |{9/2,9}
| |
| |{{CDD|node_1|9|rat|d2|node|9|node}}
| |
| |[[Image:Hyperbolic tiling 9-2 9.png|60px]]
| |
| |{9/2}<br />[[Image:Star polygon 9-2.svg|30px]]
| |
| |{9}<br />[[Image:Nonagon.svg|30px]]
| |
| | 3
| |
| | *932
| |
| | Enneagrammic-order enneagonal tiling
| |
| |- BGCOLOR="#e0e0ff" align=center
| |
| | [[Enneagrammic-order enneagonal tiling]]
| |
| |{9,9/2}
| |
| |{{CDD|node_1|9|node|9|rat|d2|node}}
| |
| |[[Image:Hyperbolic tiling 9 9-2.png|60px]]
| |
| |{9}<br />[[Image:Nonagon.svg|30px]]
| |
| |{9/2}<br />[[Image:Star polygon 9-2.svg|30px]]
| |
| | 3
| |
| | *932
| |
| | Order-9 enneagrammic tiling
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| | Order-''p'' ''p''-grammic tiling
| |
| |{''p''/2,''p''}
| |
| |{{CDD|node_1|p|rat|d2|node|p|node}}
| |
| |
| |
| |{''p''/2}
| |
| |{''p''}
| |
| | 3
| |
| | *''p''32
| |
| | ''p''-grammic-order ''p''-gonal tiling
| |
| |- BGCOLOR="#e0e0ff" align=center
| |
| | ''p''-grammic-order ''p''-gonal tiling
| |
| |{''p'',''p''/2}
| |
| |{{CDD|node_1|p|node|p|rat|d2|node}}
| |
| |
| |
| |{''p''}
| |
| |{''p''/2}
| |
| | 3
| |
| | *''p''32
| |
| | Order-''p'' ''p''-grammic tiling
| |
| |}
| |
| | |
| == Four-dimensional regular polytopes ==
| |
| | |
| Regular 4-polytopes (called [[polychora]]) with [[Schläfli symbol]] <math>\{p,q,r\}</math> have cells of type <math>\{p,q\}</math>, faces of type <math>\{p\}</math>, edge figures
| |
| <math>\{r\}</math>, and vertex figures <math>\{q,r\}</math>.
| |
| * A [[vertex figure]] (of a polychoron) is a polyhedron, seen by the arrangement of neighboring vertices around a given vertex. For regular polychora, this vertex figure is a regular polyhedron.
| |
| * An [[Vertex figure#Edge figure|edge figure]]'' is a polygon, seen by the arrangement of faces around an edge. For regular polychora, this edge figure will always be a regular polygon.
| |
| | |
| The existence of a regular polychoron <math>\{p,q,r\}</math> is constrained by the existence of the regular polyhedra <math>\{p,q\}, \{q,r\}</math>.
| |
| | |
| Each will exist in a space dependent upon this expression:
| |
| : <math>\sin \left ( \frac{\pi}{p} \right ) \sin \left(\frac{\pi}{r}\right) - \cos\left(\frac{\pi}{q}\right)</math>
| |
| :: <math>> 0</math> : Hyperspherical 3-space honeycomb or 4-space polychoron
| |
| :: <math>= 0</math> : Euclidean 3-space honeycomb
| |
| :: <math>< 0</math> : Hyperbolic 3-space honeycomb
| |
| | |
| These constraints allow for 21 forms: 6 are convex, 10 are nonconvex, ''one'' is a Euclidean 3-space honeycomb, and 4 are hyperbolic honeycombs.
| |
| | |
| The [[Euler characteristic]] <math>\chi</math> for polychora is
| |
| <math>\chi = V+F-E-C</math>
| |
| and is zero for all forms.
| |
| | |
| ===Convex===
| |
| | |
| The 6 [[convex regular polychoron|convex regular polychora]] are shown in the table below. All these polychora have an [[Euler characteristic]] (χ) of 0.
| |
| | |
| {| class="wikitable"
| |
| ! Name<br />
| |
| ! [[Schläfli symbol|Schläfli]]<br />{p,q,r}
| |
| ! [[Coxeter-Dynkin diagram|Coxeter]]<br />{{CDD|node|p|node|q|node|r|node}}
| |
| ! [[Cell (geometry)|Cells]]<br /> {p,q}
| |
| ! [[Face (geometry)|Faces]]<br /> {p}
| |
| ! [[Edge (geometry)|Edges]]<br /> {r}
| |
| ! [[Vertex (geometry)|Vertices]]<br /> {q,r}
| |
| ! Dual<br /> {r,q,p}
| |
| |- BGCOLOR="#e0e0e0" align=center
| |
| | [[5-cell]]<br />([[Simplex|4-simplex]])<br />(Pentachoron)
| |
| | {3,3,3}
| |
| |{{CDD|node_1|3|node|3|node|3|node}}
| |
| | 5<br />{3,3}
| |
| | 10<br />{3}
| |
| | 10<br /> {3}
| |
| | 5<br /> {3,3}
| |
| | (self)
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| | [[8-cell]]<br />([[Hypercube|4-cube]])<br />(Tesseract)
| |
| | {4,3,3}
| |
| |{{CDD|node_1|4|node|3|node|3|node}}
| |
| | 8<br /> {4,3}
| |
| | 24<br /> {4}
| |
| | 32<br /> {3}
| |
| | 16<br /> {3,3}
| |
| | 16-cell
| |
| |- BGCOLOR="#e0e0ff" align=center
| |
| | [[16-cell]]<br />([[Cross-polytope|4-orthoplex]])
| |
| | {3,3,4}
| |
| |{{CDD|node_1|3|node|3|node|4|node}}
| |
| | 16<br /> {3,3}
| |
| | 32<br /> {3}
| |
| | 24<br /> {4}
| |
| | 8<br /> {3,4}
| |
| | Tesseract
| |
| |- BGCOLOR="#e0e0e0" align=center
| |
| | [[24-cell]]
| |
| | {3,4,3}
| |
| |{{CDD|node_1|3|node|4|node|3|node}}
| |
| | 24<br /> {3,4}
| |
| | 96<br /> {3}
| |
| | 96<br /> {3}
| |
| | 24<br /> {4,3}
| |
| | (self)
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| | [[120-cell]]
| |
| | {5,3,3}
| |
| |{{CDD|node_1|5|node|3|node|3|node}}
| |
| | 120<br /> {5,3}
| |
| | 720<br /> {5}
| |
| | 1200<br /> {3}
| |
| | 600<br /> {3,3}
| |
| | 600-cell
| |
| |- BGCOLOR="#e0e0ff" align=center
| |
| | [[600-cell]]
| |
| | {3,3,5}
| |
| |{{CDD|node_1|3|node|3|node|5|node}}
| |
| | 600<br /> {3,3}
| |
| | 1200<br /> {3}
| |
| | 720<br /> {5}
| |
| | 120<br /> {3,5}
| |
| | 120-cell
| |
| |}
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| ! [[5-cell]] || [[8-cell]] || [[16-cell]] || [[24-cell]] || [[120-cell]] || [[600-cell]]
| |
| |-
| |
| ! {3,3,3} || {4,3,3} || {3,3,4} || {3,4,3} || {5,3,3} || {3,3,5}
| |
| |-
| |
| !colspan=6|Wireframe ([[Petrie polygon]]) skew [[orthographic projection]]s
| |
| |-
| |
| | [[File:Complete graph K5.svg|105px]]
| |
| | [[File:4-cube graph.svg|105px]]
| |
| | [[File:4-orthoplex.svg|105px]]
| |
| | [[File:24-cell graph F4.svg|105px]]
| |
| | [[File:Cell120Petrie.svg|105px]]
| |
| | [[File:Cell600Petrie.svg|105px]]
| |
| |-
| |
| !colspan=6|Solid [[orthographic projection]]s
| |
| |-
| |
| | [[Image:Tetrahedron.png|105px]]<BR>[[Tetrahedron|tetrahedral<BR>envelope]]<BR>(cell/vertex-centered)
| |
| | [[Image:Hexahedron.png|105px]]<BR>[[Cube|cubic envelope]]<BR>(cell-centered)
| |
| | [[File:16-cell ortho cell-centered.png|105px]]<BR>[[Cube|Cubic<BR>envelope]]<BR>(cell-centered)
| |
| | [[Image:Ortho solid 24-cell.png|105px]]<BR>[[Cuboctahedron|cuboctahedral<BR>envelope]]<BR>(cell-centered)
| |
| | [[Image:Ortho solid 120-cell.png|105px]]<BR>[[Truncated rhombic triacontahedron|truncated rhombic<BR>triacontahedron<BR>envelope]]<BR> (cell-centered)
| |
| | [[Image:Ortho solid 600-cell.png|105px]]<BR>[[Pentakis icosidodecahedron|Pentakis<BR>icosidodecahedral]]<BR>envelope<BR>(vertex-centered)
| |
| |-
| |
| !colspan=6|Wireframe [[Schlegel diagram]]s ([[Perspective projection]])
| |
| |-
| |
| | [[Image:Schlegel wireframe 5-cell.png|105px]]<BR>(Cell-centered)
| |
| | [[Image:Schlegel wireframe 8-cell.png|105px]]<BR>(Cell-centered)
| |
| | [[Image:Schlegel wireframe 16-cell.png|105px]]<BR>(Cell-centered)
| |
| | [[Image:Schlegel wireframe 24-cell.png|105px]]<BR>(Cell-centered)
| |
| | [[Image:Schlegel wireframe 120-cell.png|105px]]<BR>(Cell-centered)
| |
| | [[Image:Schlegel wireframe 600-cell vertex-centered.png|105px]]<BR>(Vertex-centered)
| |
| |-
| |
| !colspan=6|Wireframe [[stereographic projection]]s ([[3-sphere|Hyperspherical]])
| |
| |-
| |
| | [[Image:Stereographic polytope 5cell.png|105px]]
| |
| | [[Image:Stereographic polytope 8cell.png|105px]]
| |
| | [[Image:Stereographic polytope 16cell.png|105px]]
| |
| | [[Image:Stereographic polytope 24cell.png|105px]]
| |
| | [[Image:Stereographic polytope 120cell.png|105px]]
| |
| | [[Image:Stereographic polytope 600cell.png|105px]]
| |
| |}
| |
| | |
| ==== Degenerate (spherical) ====
| |
| | |
| [[Ditope|Dichora]] and [[hosotope|hosochora]] exist as regular tessellations of the [[3-sphere]].
| |
| | |
| Regular '''dichora''' (2 facets) include: {3,3,2}, {3,4,2}, {4,3,2}, {5,3,2}, {3,5,2}, {p,2,2}, and their '''hosochora''' [[Dual polytope|duals]] (2 vertices): {2,3,3}, {2,4,3}, {2,3,4}, {2,3,5}, {2,5,3}, {2,2,p}. Polychora of the form {2,p,2} are both dichora and hosochora.
| |
| | |
| ===Non-convex===
| |
| There are ten [[star polytope|regular star polychora]], which can be called [[Schläfli–Hess polychoron]]. Their vertices are based on the convex [[120-cell]] ''{5,3,3}'' and [[600-cell]] ''{3,3,5}''.
| |
| | |
| [[Ludwig Schläfli]] found four of them and skipped the last six because he would not allow forms that failed the [[Euler characteristic]] on cells or vertex figures (for zero-hole tori: F+V−E=2). [[Edmund Hess]] (1843–1903) completed the full list of ten in his German book ''Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder'' (1883)[http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=ABN8623.0001.001].
| |
| | |
| There are 4 unique [[edge arrangement]]s and 7 unique [[face arrangement]]s from these 10 nonconvex polychora, shown as [[orthogonal projection]]s:
| |
| | |
| {| class="wikitable"
| |
| ! Name<BR>
| |
| ! Wireframe
| |
| ! Solid
| |
| ! [[Schläfli symbol|Schläfli]]<BR>{p, q, r}<BR>[[Coxeter–Dynkin diagram|Coxeter–Dynkin]]
| |
| ! Cells<BR>{p, q}
| |
| ! Faces<BR>{p}
| |
| ! Edges<BR>{r}
| |
| ! Vertices<BR>{q, r}
| |
| ![[density (polytope)|Density]]
| |
| ! [[Euler characteristic|χ]]
| |
| ![[Coxeter group|Symmetry group]]
| |
| ! Dual<BR>{r, q,p}
| |
| |- align=center BGCOLOR="#e0e0ff"
| |
| | [[Icosahedral 120-cell]]
| |
| | [[Image:Schläfli-Hess polychoron-wireframe-3.png|75px]]
| |
| | [[Image:ortho solid 007-uniform polychoron 35p-t0.png|75px]]
| |
| | {3,5,5/2}<BR>{{CDD|node_1|3|node|5|node|5|rat|d2|node}}
| |
| | 120<BR>[[Icosahedron|{3,5}]]<BR>[[Image:Icosahedron.png|25px]]
| |
| | 1200<BR>[[Triangle|{3}]]<BR>[[Image:Triangle.Equilateral.svg|25px]]
| |
| | 720<BR>[[Pentagram|{5/2}]]<BR>[[Image:Pentagram.svg|25px]]
| |
| | 120<BR>[[Great dodecahedron|{5,5/2}]]<BR>[[Image:Great dodecahedron.png|25px]]
| |
| | 4
| |
| | 480
| |
| | ''H''<sub>4</sub>
| |
| | Small stellated 120-cell
| |
| |- align=center BGCOLOR="#ffe0e0"
| |
| | [[Small stellated 120-cell]]
| |
| | [[Image:Schläfli-Hess polychoron-wireframe-2.png|75px]]
| |
| | [[Image:ortho solid 010-uniform polychoron p53-t0.png|75px]]
| |
| | {5/2,5,3}<BR>{{CDD|node|3|node|5|node|5|rat|d2|node_1}}
| |
| | 120<BR>[[Small stellated dodecahedron|{5/2,5}]]<BR>[[Image:Small stellated dodecahedron.png|25px]]
| |
| | 720<BR>[[Pentagram|{5/2}]]<BR>[[Image:Pentagram.svg|25px]]
| |
| | 1200<BR>[[Triangle|{3}]]<BR>[[Image:Triangle.Equilateral.svg|25px]]
| |
| | 120<BR>[[Dodecahedron|{5,3}]]<BR>[[Image:Dodecahedron.png|25px]]
| |
| | 4
| |
| | −480
| |
| | ''H''<sub>4</sub>
| |
| | Icosahedral 120-cell
| |
| |- align=center BGCOLOR="#e0ffe0"
| |
| | [[Great 120-cell]]
| |
| | [[Image:Schläfli-Hess polychoron-wireframe-3.png|75px]]
| |
| | [[Image:ortho solid 008-uniform polychoron 5p5-t0.png|75px]]
| |
| | {5,5/2,5}<BR>{{CDD|node_1|5|node|5|rat|d2|node|5|node}}
| |
| | 120<BR>[[Great dodecahedron|{5,5/2}]]<BR>[[Image:Great dodecahedron.png|25px]]
| |
| | 720<BR>[[Pentagon|{5}]]<BR>[[Image:Pentagon.svg|25px]]
| |
| | 720<BR>[[Pentagon|{5}]]<BR>[[Image:Pentagon.svg|25px]]
| |
| | 120<BR>[[Small stellated dodecahedron|{5/2,5}]]<BR>[[Image:Small stellated dodecahedron.png|25px]]
| |
| | 6
| |
| | 0
| |
| | ''H''<sub>4</sub>
| |
| | Self-dual
| |
| |- align=center BGCOLOR="#e0e0ff"
| |
| | [[Grand 120-cell]]
| |
| | [[Image:Schläfli-Hess polychoron-wireframe-3.png|75px]]
| |
| | [[Image:ortho solid 009-uniform polychoron 53p-t0.png|75px]]
| |
| | {5,3,5/2}<BR>{{CDD|node_1|5|node|3|node|5|rat|d2|node}}
| |
| | 120<BR>[[Dodecahedron|{5,3}]]<BR>[[Image:Dodecahedron.png|25px]]
| |
| | 720<BR>[[Pentagon|{5}]]<BR>[[Image:Pentagon.svg|25px]]
| |
| | 720<BR>[[Pentagram|{5/2}]]<BR>[[Image:Pentagram.svg|25px]]
| |
| | 120<BR>[[Great icosahedron|{3,5/2}]]<BR>[[Image:Great icosahedron.png|25px]]
| |
| | 20
| |
| | 0
| |
| | ''H''<sub>4</sub>
| |
| | Great stellated 120-cell
| |
| |- align=center BGCOLOR="#ffe0e0"
| |
| | [[Great stellated 120-cell]]
| |
| | [[Image:Schläfli-Hess polychoron-wireframe-4.png|75px]]
| |
| | [[Image:ortho solid 012-uniform polychoron p35-t0.png|75px]]
| |
| | {5/2,3,5}<BR>{{CDD|node|5|node|3|node|5|rat|d2|node_1}}
| |
| | 120<BR>[[Great stellated dodecahedron|{5/2,3}]]<BR>[[Image:Great stellated dodecahedron.png|25px]]
| |
| | 720<BR>[[Pentagram|{5/2}]]<BR>[[Image:Pentagram.svg|25px]]
| |
| | 720<BR>[[Pentagon|{5}]]<BR>[[Image:Pentagon.svg|25px]]
| |
| | 120<BR>[[Icosahedron|{3,5}]]<BR>[[Image:Icosahedron.png|25px]]
| |
| | 20
| |
| | 0
| |
| | ''H''<sub>4</sub>
| |
| | Grand 120-cell
| |
| |- align=center BGCOLOR="#e0ffe0"
| |
| | [[Grand stellated 120-cell]]
| |
| | [[Image:Schläfli-Hess polychoron-wireframe-4.png|75px]]
| |
| | [[Image:ortho solid 013-uniform polychoron p5p-t0.png|75px]]
| |
| | {5/2,5,5/2}<BR>{{CDD|node_1|5|rat|d2|node|5|node|5|rat|d2|node}}
| |
| | 120<BR>[[Small stellated dodecahedron|{5/2,5}]]<BR>[[Image:Small stellated dodecahedron.png|25px]]
| |
| | 720<BR>[[Pentagram|{5/2}]]<BR>[[Image:Pentagram.svg|25px]]
| |
| | 720<BR>[[Pentagram|{5/2}]]<BR>[[Image:Pentagram.svg|25px]]
| |
| | 120<BR>[[Great dodecahedron|{5,5/2}]]<BR>[[Image:Great dodecahedron.png|25px]]
| |
| | 66
| |
| | 0
| |
| | ''H''<sub>4</sub>
| |
| | Self-dual
| |
| |- align=center BGCOLOR="#e0e0ff"
| |
| | [[Great grand 120-cell]]
| |
| | [[Image:Schläfli-Hess polychoron-wireframe-2.png|75px]]
| |
| | [[Image:ortho solid 011-uniform polychoron 53p-t0.png|75px]]
| |
| | {5,5/2,3}<BR>{{CDD|node_1|5|node|5|rat|d2|node|3|node}}
| |
| | 120<BR>[[Great dodecahedron|{5,5/2}]]<BR>[[Image:Great dodecahedron.png|25px]]
| |
| | 720<BR>[[Pentagon|{5}]]<BR>[[Image:Pentagon.svg|25px]]
| |
| | 1200<BR>[[Triangle|{3}]]<BR>[[Image:Triangle.Equilateral.svg|25px]]
| |
| | 120<BR>[[Great stellated dodecahedron|{5/2,3}]]<BR>[[Image:Great stellated dodecahedron.png|25px]]
| |
| | 76
| |
| | −480
| |
| | ''H''<sub>4</sub>
| |
| | Great icosahedral 120-cell
| |
| |- align=center BGCOLOR="#ffe0e0"
| |
| | [[Great icosahedral 120-cell]]
| |
| | [[Image:Schläfli-Hess polychoron-wireframe-4.png|75px]]
| |
| | [[Image:ortho solid 014-uniform polychoron 3p5-t0.png|75px]]
| |
| | {3,5/2,5}<BR>{{CDD|node|5|node|5|rat|d2|node|3|node_1}}
| |
| | 120<BR>[[Great icosahedron|{3,5/2}]]<BR>[[Image:Great icosahedron.png|25px]]
| |
| | 1200<BR>[[Triangle|{3}]]<BR>[[Image:Triangle.Equilateral.svg|25px]]
| |
| | 720<BR>[[Pentagon|{5}]]<BR>[[Image:Pentagon.svg|25px]]
| |
| | 120<BR>[[Small stellated dodecahedron|{5/2,5}]]<BR>[[Image:Small stellated dodecahedron.png|25px]]
| |
| | 76
| |
| | 480
| |
| | ''H''<sub>4</sub>
| |
| | Great grand 120-cell
| |
| |- align=center BGCOLOR="#e0e0ff"
| |
| | [[Grand 600-cell]]
| |
| | [[Image:Schläfli-Hess polychoron-wireframe-4.png|75px]]
| |
| | [[Image:ortho solid 015-uniform polychoron 33p-t0.png|75px]]
| |
| | {3,3,5/2}<BR>{{CDD|node_1|3|node|3|node|5|rat|d2|node}}
| |
| | 600<BR>[[Tetrahedron|{3,3}]]<BR>[[Image:Tetrahedron.png|25px]]
| |
| | 1200<BR>[[Triangle|{3}]]<BR>[[Image:Triangle.Equilateral.svg|25px]]
| |
| | 720<BR>[[Pentagram|{5/2}]]<BR>[[Image:Pentagram.svg|25px]]
| |
| | 120<BR>[[Great icosahedron|{3,5/2}]]<BR>[[Image:Great icosahedron.png|25px]]
| |
| | 191
| |
| | 0
| |
| | ''H''<sub>4</sub>
| |
| | Great grand stellated 120-cell
| |
| |- align=center BGCOLOR="#ffe0e0"
| |
| | [[Great grand stellated 120-cell]]
| |
| | [[Image:Schläfli-Hess polychoron-wireframe-1.png|75px]]
| |
| | [[Image:ortho solid 016-uniform polychoron p33-t0.png|75px]]
| |
| | {5/2,3,3}<BR>{{CDD|node|3|node|3|node|5|rat|d2|node_1}}
| |
| | 120<BR>[[Great stellated dodecahedron|{5/2,3}]]<BR>[[Image:Great stellated dodecahedron.png|25px]]
| |
| | 720<BR>[[Pentagram|{5/2}]]<BR>[[Image:Pentagram.svg|25px]]
| |
| | 1200<BR>[[Triangle|{3}]]<BR>[[Image:Triangle.Equilateral.svg|25px]]
| |
| | 600<BR>[[Tetrahedron|{3,3}]]<BR>[[Image:Tetrahedron.png|25px]]
| |
| | 191
| |
| | 0
| |
| | ''H''<sub>4</sub>
| |
| | Grand 600-cell
| |
| |}
| |
| There are 4 ''failed'' potential nonconvex regular polychora permutations: {3,5/2,3}, {4,3,5/2}, {5/2,3,4}, {5/2,3,5/2}. Their cells and vertex figures exist, but they do not cover a hypersphere with a finite number of repetitions.
| |
| | |
| ===Tessellations of Euclidean 3-space===
| |
| [[Image:Cubic honeycomb.png|150px|thumb|Edge framework of cubic honeycomb, {4,3,4}]]
| |
| | |
| There is only one regular tessellation of 3-space (''[[honeycomb (geometry)|honeycombs]]''):
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| !Name
| |
| ![[Schläfli symbol|Schläfli<BR>symbol]]<BR>{p,q,r}
| |
| ![[Coxeter-Dynkin diagram|Coxeter]]<BR>{{CDD|node|p|node|q|node|r|node}}
| |
| !Cell<BR>type<BR>{p,q}
| |
| !Face<BR>type<BR>{p}
| |
| !Edge<BR>figure<BR>{r}
| |
| ![[Vertex figure|Vertex<BR>figure]]<BR>{q,r}
| |
| ![[Euler characteristic|χ]]
| |
| ![[Dual polytope|Dual]]
| |
| |- BGCOLOR="#e0e0e0" align=center
| |
| |[[Cubic honeycomb]]||{4,3,4}||{{CDD|node_1|4|node|3|node|4|node}}||{4,3}||{4}||{4}||{3,4}||0||Self-dual
| |
| |}
| |
| | |
| ===Degenerate tessellations of Euclidean 3-space===
| |
| [[File:Order-4 square hosohedral honeycomb-sphere.png|thumb|Regular {2,4,4} honeycomb, seen projected into a sphere.]]
| |
| There are six degenerate regular tessellations, pairs based on the three regular Euclidean tilings. Their cells and vertex figures are all regular [[hosohedron]]s {2,n}, [[dihedron]]s, {n,2}, and Euclidean tilings. These degenerate regular tilings are constructionally related to prismatic uniform honeycombs by truncation operations.
| |
| {| class=wikitable
| |
| |-
| |
| ![[Schläfli symbol|Schläfli]]<BR>{p,q,r}
| |
| ![[Coxeter-Dynkin diagram|Coxeter<BR>diagram]]
| |
| !Cell<BR>type<BR>{p,q}
| |
| !Face<BR>type<BR>{p}
| |
| !Edge<BR>figure<BR>{r}
| |
| ![[Vertex figure|Vertex<BR>figure]]<BR>{q,r}
| |
| |- align=center
| |
| |[[order-4 square hosohedral honeycomb|{2,4,4}]]||{{CDD|node_1|2|node|4|node|4|node}}||{2,4}||{2}||{4}||{4,4}
| |
| |- align=center
| |
| |[[Order-6 triangular hosohedral honeycomb|{2,3,6}]]||{{CDD|node_1|2|node|3|node|6|node}}||{2,3}||{2}||{6}||{3,6}
| |
| |- align=center
| |
| |[[hexagonal hosohedral honeycomb|{2,6,3}]]||{{CDD|node_1|2|node|6|node|3|node}}||{2,6}||{2}||{3}||{6,3}
| |
| |- align=center
| |
| |[[Order-2 square tiling|{4,4,2}]]||{{CDD|node_1|4|node|4|node|2|node}}||{4,4}||{4}||{2}||{4,2}
| |
| |- align=center
| |
| |[[Order-2 triangular tiling|{3,6,2}]]||{{CDD|node_1|3|node|6|node|2|node}}||{3,6}||{3}||{2}||{6,2}
| |
| |- align=center
| |
| |[[Order-2 hexagonal tiling|{6,3,2}]]||{{CDD|node_1|6|node|3|node|2|node}}||{6,3}||{6}||{2}||{3,2}
| |
| |}
| |
| | |
| ===Tessellations of hyperbolic 3-space===
| |
| {| align=right
| |
| |
| |
| {| class="wikitable"
| |
| |+ 4 compact regular honeycombs
| |
| |- bgcolor="#d0e0ff" align=center
| |
| |[[File:H3 534 CC center.png|150px]]<BR>[[Order-4 dodecahedral honeycomb|{5,3,4}]]
| |
| |[[File:H3 535 CC center.png|150px]]<BR>[[Order-5 dodecahedral honeycomb|{5,3,5}]]
| |
| |- bgcolor="#d0e0ff" align=center
| |
| |[[File:H3_435_CC_center.png|150px]]<BR>[[Order-5 cubic honeycomb|{4,3,5}]]
| |
| |[[File:H3_353_CC_center.png|150px]]<BR>[[Order-3 icosahedral honeycomb|{3,5,3}]]
| |
| |}
| |
| |-
| |
| |
| |
| {| class="wikitable"
| |
| |+ 4 of 11 paracompact regular honeycombs
| |
| |- bgcolor="#e0d0ff" align=center
| |
| |[[File:H3_344_CC_center.png|150px]]<BR>[[Order-4 octahedral honeycomb|{3,4,4}]]
| |
| |[[File:H3_363_FC_boundary.png|150px]]<BR>[[Triangular tiling honeycomb|{3,6,3}]]
| |
| |- bgcolor="#e0d0ff" align=center
| |
| |[[File:H3_443_FC_boundary.png|150px]]<BR>[[Square tiling honeycomb|{4,4,3}]]
| |
| |[[File:H3 444 FC boundary.png|150px]]<BR>[[Order-4_square_tiling_honeycomb|{4,4,4}]]
| |
| |}
| |
| |}
| |
| Tessellations of [[Hyperbolic space|hyperbolic 3-space]] can be called ''[[honeycomb (geometry)|hyperbolic honeycombs]]''. There are 15 hyperbolic honeycombs in H<sup>3</sup>, 4 compact and 11 paracompact.
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| !Name
| |
| ![[Schläfli symbol|Schläfli]]<BR>Symbol<BR>{p,q,r}
| |
| ![[Coxeter-Dynkin diagram|Coxeter]]<BR>{{CDD|node|p|node|q|node|r|node}}
| |
| !Cell<BR>type<BR>{p,q}
| |
| !Face<BR>type<BR>{p}
| |
| !Edge<BR>figure<BR>{r}
| |
| ![[Vertex figure|Vertex<BR>figure]]<BR>{q,r}
| |
| ![[Euler characteristic|χ]]
| |
| ![[Dual polytope|Dual]]
| |
| |- BGCOLOR="#e0e0e0" align=center
| |
| |[[Icosahedral honeycomb]]||{3,5,3}||{{CDD|node_1|3|node|5|node|3|node}}||[[icosahedron|{3,5}]]||{3}||{3}||[[dodecahedron|{5,3}]]||0||Self-dual
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| |[[Order-5 cubic honeycomb]]||{4,3,5}||{{CDD|node_1|4|node|3|node|5|node}}||[[cube|{4,3}]]||{4}||{5}||[[icosahedron|{3,5}]]||0||{5,3,4}
| |
| |- BGCOLOR="#e0e0ff" align=center
| |
| |[[Order-4 dodecahedral honeycomb]]||{5,3,4}||{{CDD|node_1|5|node|3|node|4|node}}||[[Dodecahedron|{5,3}]]||{5}||{4}||[[octahedron|{3,4}]]||0||{4,3,5}
| |
| |- BGCOLOR="#e0e0e0" align=center
| |
| |[[Order-5 dodecahedral honeycomb]]||{5,3,5}||{{CDD|node_1|5|node|3|node|5|node}}||[[Dodecahedron|{5,3}]]||{5}||{5}||[[icosahedron|{3,5}]]||0||Self-dual
| |
| |}
| |
| | |
| There are also 11 paracompact H<sup>3</sup> honeycombs (those with infinite (Euclidean) cells and/or vertex figures): {3,3,6}, {6,3,3}, {3,4,4}, {4,4,3}, {3,6,3}, {4,3,6}, {6,3,4}, {4,4,4}, {5,3,6}, {6,3,5}, and {6,3,6}.
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| !Name
| |
| ![[Schläfli symbol|Schläfli]]<BR>Symbol<BR>{p,q,r}
| |
| ![[Coxeter-Dynkin diagram|Coxeter]]<BR>{{CDD|node|p|node|q|node|r|node}}
| |
| !Cell<BR>type<BR>{p,q}
| |
| !Face<BR>type<BR>{p}
| |
| !Edge<BR>figure<BR>{r}
| |
| ![[Vertex figure|Vertex<BR>figure]]<BR>{q,r}
| |
| ![[Euler characteristic|χ]]
| |
| ![[Dual polytope|Dual]]
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| |[[Order-6 tetrahedral honeycomb]]||{3,3,6}||{{CDD|node_1|3|node|3|node|6|node}}||[[tetrahedron|{3,3}]]||{3}||{6}||[[triangular tiling|{3,6}]]||0||{6,3,3}
| |
| |- BGCOLOR="#e0e0ff" align=center
| |
| |[[Hexagonal tiling honeycomb]]||{6,3,3}||{{CDD|node_1|6|node|3|node|3|node}}||[[hexagonal tiling|{6,3}]]||{6}||{3}||[[tetrahedron|{3,3}]]||0||{3,3,6}
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| |[[Order-4 octahedral honeycomb]]||{3,4,4}||{{CDD|node_1|3|node|4|node|4|node}}||[[octahedron|{3,4}]]||{3}||{4}||[[square tiling|{4,4}]]||0||{4,4,3}
| |
| |- BGCOLOR="#e0e0ff" align=center
| |
| |[[Square tiling honeycomb]]||{4,4,3}||{{CDD|node_1|4|node|4|node|3|node}}||[[square tiling|{4,4}]]||{4}||{3}||[[cube|{4,3}]]||0||{3,3,4}
| |
| |- BGCOLOR="#e0e0e0" align=center
| |
| |[[Triangular tiling honeycomb]]||{3,6,3}||{{CDD|node_1|3|node|6|node|3|node}}||[[triangular tiling|{3,6}]]||{3}||{3}||[[hexagonal tiling|{6,3}]]||0||Self-dual
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| |[[Order-6 cubic honeycomb]]||{4,3,6}||{{CDD|node_1|4|node|3|node|6|node}}||[[Cube|{4,3}]]||{4}||{4}||[[octahedron|{3,4}]]||0||{6,3,4}
| |
| |- BGCOLOR="#e0e0ff" align=center
| |
| |[[Order-4 hexagonal tiling honeycomb]]||{6,3,4}||{{CDD|node_1|6|node|3|node|4|node}}||[[hexagonal tiling|{6,3}]]||{6}||{4}||[[Octahedron|{3,4}]]||0||{4,3,6}
| |
| |- BGCOLOR="#e0e0e0" align=center
| |
| |[[Order-4 square tiling honeycomb]]||{4,4,4}||{{CDD|node_1|4|node|4|node|4|node}}||[[square tiling|{4,4}]]||{4}||{4}||[[square tiling|{4,4}]]||0||{4,4,4}
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| |[[Order-6 dodecahedral honeycomb]]||{5,3,6}||{{CDD|node_1|5|node|3|node|6|node}}||[[dodecahedron|{5,3}]]||{5}||{5}||[[icosahedron|{3,5}]]||0||{6,3,5}
| |
| |- BGCOLOR="#e0e0ff" align=center
| |
| |[[Order-5 hexagonal tiling honeycomb]]||{6,3,5}||{{CDD|node_1|6|node|3|node|5|node}}||[[hexagonal tiling|{6,3}]]||{6}||{5}||[[icosahedron|{3,5}]]||0||{5,3,6}
| |
| |- BGCOLOR="#e0e0e0" align=center
| |
| |[[Order-6 hexagonal tiling honeycomb]]||{6,3,6}||{{CDD|node_1|6|node|3|node|6|node}}||[[hexagonal tiling|{6,3}]]||{6}||{6}||[[triangular tiling|{3,6}]]||0||Self-dual
| |
| |}
| |
| | |
| A subset of unform polychora and honeycombs are contained in the form {p,3,q} for values 3 to 6. Lorentzian honeycombs exist for p or q greater than 6, and a couple a drawn below as tilings on the ideal half-space plane.
| |
| {{Regular_hyperbolic_honeycomb_table}}
| |
| | |
| == Five-dimensional regular polytopes and higher ==
| |
| In [[Five-dimensional space|five dimensions]], a regular polytope can be named as
| |
| <math>\{p,q,r,s\}</math> where <math>\{p,q,r\}</math> is the hypercell (or ''[[teron (geometry)|teron]]'') type, <math>\{p,q\}</math> is the cell type, <math>\{p\}</math> is the face type, and <math>\{s\}</math> is the face figure, <math>\{r,s\}</math> is the edge figure, and <math>\{q,r,s\}</math> is the vertex figure.
| |
| | |
| A '''5-polytope''' has been called a [[polyteron]], and if infinite (i.e. a [[Honeycomb (geometry)|honeycomb]]) a [[tetracomb]].
| |
| | |
| : A [[vertex figure]] (of a 5-polytope) is a polychoron, seen by the arrangement of neighboring vertices to each vertex.
| |
| : An [[edge figure]] (of a 5-polytope) is a polyhedron, seen by the arrangement of faces around each edge.
| |
| : A [[face figure]] (of a 5-polytope) is a polygon, seen by the arrangement of cells around each face.
| |
| | |
| A regular 5-polytope <math>\{p,q,r,s\}</math> exists only if <math>\{p,q,r\}</math> and <math>\{q,r,s\}</math> are regular polychora.
| |
| | |
| The space it fits in is based on the expression:
| |
| : <math>\frac{\cos^2\left(\frac{\pi}{q}\right)}{\sin^2\left(\frac{\pi}{p}\right)} + \frac{\cos^2\left(\frac{\pi}{r}\right)}{\sin^2\left(\frac{\pi}{s}\right)}</math>
| |
| :: <math>< 1</math> : Spherical 4-space tessellation or 5-space polytope
| |
| :: <math>= 1</math> : Euclidean 4-space tessellation
| |
| :: <math>> 1</math> : hyperbolic 4-space tessellation
| |
| | |
| Enumeration of these constraints produce ''3'' convex polytopes, ''zero'' nonconvex polytopes, ''3'' 4-space tessellations, and ''5'' hyperbolic 4-space tessellations. There are no non-convex regular polytopes in five dimensions or higher.
| |
| | |
| Higher-dimensional polytopes have sometimes received names. 6-polytopes have sometimes been called ''polypeta'', 7-polytopes ''polyexa'', 8-polytopes ''polyzetta'', and 9-polytopes ''polyyotta''.<!--please do not add "polyxennon" for 10D; see http://www.steelpillow.com/polyhedra/ditela.html and [[Talk:Polytope#Proposed_demotion_of_.22polyxennon.22]]-->
| |
| | |
| ===Convex===
| |
| In dimensions 5 and higher, there are only three kinds of convex regular polytopes.<ref>{{Harv|Coxeter|1973|loc=Table I: Regular polytopes, (iii) The three regular polytopes in ''n'' dimensions (n>=5), pp. 294–295}}</ref>
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| !Name
| |
| ![[Schläfli symbol|Schläfli]]<BR>Symbol<BR>{p<sub>1</sub>,...,p<sub><var>n</var>−1</sub>}
| |
| ![[Coxeter-Dynkin diagram|Coxeter]]
| |
| !<var>k</var>-faces
| |
| !Facet<BR>type
| |
| ![[Vertex figure|Vertex<BR>figure]]
| |
| ![[Dual polytope|Dual]]
| |
| |- BGCOLOR="#e0e0e0" align=center
| |
| |[[Simplex|''n''-simplex]]||{3<sup><var>n</var>−1</sup>}||{{CDD|node_1|3|node|3}}...{{CDD|3|node|3|node}}||[[binomial coefficient|<math>{{n+1} \choose {k+1}}</math>]]||{3<sup><var>n</var>−2</sup>}||{3<sup><var>n</var>−2</sup>}||Self-dual
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| |[[Hypercube|<var>n</var>-cube]]||{4,3<sup><var>n</var>−2</sup>}||{{CDD|node_1|4|node|3}}...{{CDD|3|node|3|node}}||<math>2^{n-k}{n \choose k}</math>||{4,3<sup><var>n</var>−3</sup>}||{3<sup><var>n</var>−2</sup>}||<var>n</var>-orthoplex
| |
| |- BGCOLOR="#e0e0ff" align=center
| |
| |[[Cross-polytope|<var>n</var>-orthoplex]]||{3<sup><var>n</var>−2</sup>,4}||{{CDD|node_1|3|node|3}}...{{CDD|3|node|4|node}}||<math>2^{k+1}{n \choose {k+1}}</math>||{3<sup><var>n</var>−2</sup>}||{3<sup><var>n</var>−3</sup>,4}||<var>n</var>-cube
| |
| |}
| |
| | |
| ====5 dimensions====
| |
| {| class="wikitable"
| |
| |-
| |
| !Name
| |
| ![[Schläfli symbol|Schläfli]]<BR>Symbol<BR>{p,q,r,s}<BR>[[Coxeter-Dynkin diagram|Coxeter]]
| |
| !Facets<BR>{p,q,r}
| |
| !Cells<BR>{p,q}
| |
| !Faces<BR>{p}
| |
| !Edges
| |
| !Vertices
| |
| !Face<BR>figure<BR>{s}
| |
| !Edge<BR>figure<BR>{r,s}
| |
| ![[Vertex figure|Vertex<BR>figure]]<BR>{q,r,s}
| |
| |- BGCOLOR="#e0e0e0" align=center
| |
| |[[5-simplex]]
| |
| |{3,3,3,3}<BR>{{CDD|node_1|3|node|3|node|3|node|3|node}}||6<BR>{3,3,3}||15<BR>{3,3}||20<BR>{3}||15||6||{3}||{3,3}||{3,3,3}
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| |[[5-cube]]
| |
| |{4,3,3,3}<BR>{{CDD|node_1|4|node|3|node|3|node|3|node}}||10<BR>{4,3,3}||40<BR>{4,3}||80<BR>{4}||80||32||{3}||{3,3}||{3,3,3}
| |
| |- BGCOLOR="#e0e0ff" align=center
| |
| |[[5-orthoplex]]
| |
| |{3,3,3,4}<BR>{{CDD|node_1|3|node|3|node|3|node|4|node}}||32<BR>{3,3,3}||80<BR>{3,3}||80<BR>{3}||40||10||{4}||{3,4}||{3,3,4}
| |
| |}
| |
| | |
| {| class=wikitable
| |
| |- align=center valign=top
| |
| |[[Image:5-simplex t0.svg|120px]]<BR>[[5-simplex]]
| |
| |[[Image:5-cube graph.svg|120px]]<BR>[[5-cube]]
| |
| |[[File:5-orthoplex.svg|120px]]<BR>[[5-orthoplex]]
| |
| |}
| |
| | |
| ====6 dimensions====
| |
| {| class=wikitable
| |
| !Name!![[Schläfli symbol|Schläfli<BR>symbol]]!!Vertices!!Edges!!Faces!!Cells!!4-faces||5-faces!!χ
| |
| |- BGCOLOR="#e0e0e0" align=center
| |
| |[[6-simplex]]||{3,3,3,3,3}||7||21||35||35||21||7||0
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| |[[6-cube]]||{4,3,3,3,3}||64||192||240||160||60||12||0
| |
| |- BGCOLOR="#e0e0ff" align=center
| |
| |[[6-orthoplex]]||{3,3,3,3,4}||12||60||160||240||192||64||0
| |
| |}
| |
| | |
| {| class=wikitable
| |
| |- align=center valign=top
| |
| |[[Image:6-simplex t0.svg|120px]]<BR>[[6-simplex]]
| |
| |[[Image:6-cube graph.svg|120px]]<BR>[[6-cube]]
| |
| |[[File:6-orthoplex.svg|120px]]<BR>[[6-orthoplex]]
| |
| |}
| |
| | |
| ====7 dimensions====
| |
| {| class=wikitable
| |
| !Name!![[Schläfli symbol|Schläfli<BR>symbol]]!!Vertices!!Edges!!Faces!!Cells!!4-faces||5-faces!!6-faces!!χ
| |
| |- BGCOLOR="#e0e0e0" align=center
| |
| |[[7-simplex]]||{3,3,3,3,3,3}||8||28||56||70||56||28||8||2
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| |[[7-cube]]||{4,3,3,3,3,3}||128||448||672||560||280||84||14||2
| |
| |- BGCOLOR="#e0e0ff" align=center
| |
| |[[7-orthoplex]]||{3,3,3,3,3,4}||14||84||280||560||672||448||128||2
| |
| |}
| |
| | |
| {| class=wikitable
| |
| |- align=center valign=top
| |
| |[[Image:7-simplex t0.svg|120px]]<BR>[[7-simplex]]
| |
| |[[Image:7-cube graph.svg|120px]]<BR>[[7-cube]]
| |
| |[[File:7-orthoplex.svg|120px]]<BR>[[7-orthoplex]]
| |
| |}
| |
| | |
| ====8 dimensions====
| |
| {| class=wikitable
| |
| !Name!![[Schläfli symbol|Schläfli<BR>symbol]]!!Vertices!!Edges!!Faces!!Cells!!4-faces||5-faces!!6-faces!!7-faces!!χ
| |
| |- BGCOLOR="#e0e0e0" align=center
| |
| |[[8-simplex]]||{3,3,3,3,3,3,3}||9||36||84||126||126||84||36||9||0
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| |[[8-cube]]||{4,3,3,3,3,3,3}||256||1024||1792||1792||1120||448||112||16||0
| |
| |- BGCOLOR="#e0e0ff" align=center
| |
| |[[8-orthoplex]]||{3,3,3,3,3,3,4}||16||112||448||1120||1792||1792||1024||256||0
| |
| |}
| |
| | |
| {| class=wikitable
| |
| |- align=center valign=top
| |
| |[[Image:8-simplex t0.svg|120px]]<BR>[[8-simplex]]
| |
| |[[Image:8-cube.svg|120px]]<BR>[[8-cube]]
| |
| |[[File:8-orthoplex.svg|120px]]<BR>[[8-orthoplex]]
| |
| |}
| |
| | |
| ====9 dimensions====
| |
| {| class=wikitable
| |
| !Name!![[Schläfli symbol|Schläfli<BR>symbol]]!!Vertices!!Edges!!Faces!!Cells!!4-faces||5-faces!!6-faces!!7-faces!!8-faces!!χ
| |
| |- BGCOLOR="#e0e0e0" align=center
| |
| |[[9-simplex]]||{3<sup>8</sup>}||10||45||120||210||252||210||120||45||10||2
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| |[[9-cube]]||{4,3<sup>7</sup>}||512||2304||4608||5376||4032||2016||672||144||18||2
| |
| |- BGCOLOR="#e0e0ff" align=center
| |
| |[[9-orthoplex]]||{3<sup>7</sup>,4}||18||144||672||2016||4032||5376||4608||2304||512||2
| |
| |}
| |
| | |
| {| class=wikitable
| |
| |- align=center valign=top
| |
| |[[Image:9-simplex t0.svg|120px]]<BR>[[9-simplex]]
| |
| |[[Image:9-cube.svg|120px]]<BR>[[9-cube]]
| |
| |[[File:9-orthoplex.svg|120px]]<BR>[[9-orthoplex]]
| |
| |}
| |
| | |
| ====10 dimensions====
| |
| {| class=wikitable
| |
| !Name!![[Schläfli symbol|Schläfli<BR>symbol]]!!Vertices!!Edges!!Faces!!Cells!!4-faces||5-faces!!6-faces!!7-faces!!8-faces!!9-faces!!χ
| |
| |- BGCOLOR="#e0e0e0" align=center
| |
| |[[10-simplex]]||{3<sup>9</sup>}||11||55||165||330||462||462||330||165||55||11||0
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| |[[10-cube]]||{4,3<sup>8</sup>}||1024||5120||11520||15360||13440||8064||3360||960||180||20||0
| |
| |- BGCOLOR="#e0e0ff" align=center
| |
| |[[10-orthoplex]]||{3<sup>8</sup>,4}||20||180||960||3360||8064||13440||15360||11520||5120||1024||0
| |
| |}
| |
| | |
| {| class=wikitable
| |
| |- align=center valign=top
| |
| |[[Image:10-simplex t0.svg|120px]]<BR>[[10-simplex]]
| |
| |[[File:10-cube.svg|120px]]<BR>[[10-cube]]
| |
| |[[File:10-orthoplex.svg|120px]]<BR>[[10-orthoplex]]
| |
| |}
| |
| | |
| ...
| |
| | |
| === Non-convex ===
| |
| | |
| There are no non-convex regular polytopes in five dimensions or higher.
| |
| | |
| ===Tessellations of Euclidean space===
| |
| ====Tessellations of Euclidean 4-space====
| |
| There are three kinds of infinite regular tessellations ([[Honeycomb (geometry)|honeycombs]]) that can tessellate Euclidean four-dimensional space:
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| !Name
| |
| ![[Schläfli symbol|Schläfli]]<BR>Symbol<BR>{p,q,r,s}
| |
| !Facet<BR>type<BR>{p,q,r}
| |
| !Cell<BR>type<BR>{p,q}
| |
| !Face<BR>type<BR>{p}
| |
| !Face<BR>figure<BR>{s}
| |
| !Edge<BR>figure<BR>{r,s}
| |
| ![[Vertex figure|Vertex<BR>figure]]<BR>{q,r,s}
| |
| ![[Dual polytope|Dual]]
| |
| |- BGCOLOR="#e0e0e0" align=center
| |
| |[[Tesseractic honeycomb]]||{4,3,3,4}||{4,3,3}||{4,3}||{4}||{4}||{3,4}||{3,3,4}||Self-dual
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| |[[16-cell honeycomb]]||{3,3,4,3}||{3,3,4}||{3,3}||{3}||{3}||{4,3}||{3,4,3}||{3,4,3,3}
| |
| |- BGCOLOR="#e0e0ff" align=center
| |
| |[[24-cell honeycomb]]||{3,4,3,3}||{3,4,3}||{3,4}||{3}||{3}||{3,3}||{4,3,3}||{3,3,4,3}
| |
| |}
| |
| | |
| {| class="wikitable"
| |
| |[[Image:Tesseractic tetracomb.png|160px]]<BR>Projected portion of {4,3,3,4}<BR>(Tesseractic honeycomb)
| |
| |[[Image:Demitesseractic tetra hc.png|160px]]<BR>Projected portion of {3,3,4,3}<BR>(16-cell honeycomb)
| |
| |[[Image:Icositetrachoronic tetracomb.png|160px]]<BR>Projected portion of {3,4,3,3}<BR>(24-cell honeycomb)
| |
| |}
| |
| | |
| ====Tessellations of Euclidean 5-space and higher====
| |
| The [[hypercubic honeycomb]] is the only family of regular honeycomb that can tessellate each dimension, five or higher, formed by [[hypercube]] facets, four around every [[Ridge (geometry)|ridge]].
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| !Name
| |
| ![[Schläfli symbol|Schläfli]]<BR>{''p''<sub>1</sub>, ''p''<sub>2</sub>, ..., ''p''<sub>''n''−1</sub>}
| |
| !Facet<BR>type
| |
| ![[Vertex figure|Vertex<BR>figure]]
| |
| ![[Dual polytope|Dual]]
| |
| |- BGCOLOR="#e0e0e0" align=center
| |
| |[[Square tiling]]||{4,4}||{4}||{4}||Self-dual
| |
| |- BGCOLOR="#e0e0e0" align=center
| |
| |[[Cubic honeycomb]]||{4,3,4}||{4,3}||{3,4}||Self-dual
| |
| |- BGCOLOR="#e0e0e0" align=center
| |
| |[[Tesseractic honeycomb]]||{4,3<sup>2</sup>,4}||{4,3<sup>2</sup>}||{3<sup>2</sup>,4}||Self-dual
| |
| |- BGCOLOR="#e0e0e0" align=center
| |
| |[[5-cube honeycomb]]||{4,3<sup>3</sup>,4}||{4,3<sup>3</sup>}||{3<sup>3</sup>,4}||Self-dual
| |
| |- BGCOLOR="#e0e0e0" align=center
| |
| |[[6-cube honeycomb]]||{4,3<sup>4</sup>,4}||{4,3<sup>4</sup>}||{3<sup>4</sup>,4}||Self-dual
| |
| |- BGCOLOR="#e0e0e0" align=center
| |
| |[[7-cube honeycomb]]||{4,3<sup>5</sup>,4}||{4,3<sup>5</sup>}||{3<sup>5</sup>,4}||Self-dual
| |
| |- BGCOLOR="#e0e0e0" align=center
| |
| |[[8-cube honeycomb]]||{4,3<sup>6</sup>,4}||{4,3<sup>6</sup>}||{3<sup>6</sup>,4}||Self-dual
| |
| |- BGCOLOR="#e0e0e0" align=center
| |
| |n-[[hypercubic honeycomb]]||{4,3<sup>n−2</sup>,4}||{4,3<sup>n−2</sup>}||{3<sup>n−2</sup>,4}||Self-dual
| |
| |}
| |
| | |
| ===Tessellations of hyperbolic space===
| |
| ====Tessellations of hyperbolic 4-space====
| |
| There are seven convex regular [[Honeycomb (geometry)|honeycombs]] and four star-honeycombs in H<sup>4</sup> space.<ref>Coxeter, The Beauty of Geometry: Twelve Essays, Chapter 10: Regular honeycombs in hyperbolic space, Summary tables IV p. 213</ref> Five convex ones are compact, and two are paracompact.
| |
| | |
| Five compact regular honeycombs in H<sup>4</sup>:
| |
| {| class="wikitable"
| |
| |-
| |
| !Name
| |
| ![[Schläfli symbol|Schläfli]]<BR>Symbol<BR>{p,q,r,s}
| |
| !Facet<BR>type<BR>{p,q,r}
| |
| !Cell<BR>type<BR>{p,q}
| |
| !Face<BR>type<BR>{p}
| |
| !Face<BR>figure<BR>{s}
| |
| !Edge<BR>figure<BR>{r,s}
| |
| ![[Vertex figure|Vertex<BR>figure]]<BR>{q,r,s}
| |
| ![[Dual polytope|Dual]]
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| |[[Order-5 5-cell honeycomb]]||{3,3,3,5}||[[5-cell|{3,3,3}]]||[[tetrahedron|{3,3}]]||{3}||{5}||[[icosahedron|{3,5}]]||[[600-cell|{3,3,5}]]||{5,3,3,3}
| |
| |- BGCOLOR="#e0e0ff" align=center
| |
| |[[120-cell honeycomb]]||{5,3,3,3}||[[120-cell|{5,3,3}]]||[[dodecahedron|{5,3}]]||{5}||{3}||[[tetrahedron|{3,3}]]||[[5-cell|{3,3,3}]]||{3,3,3,5}
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| |[[Order-5 tesseractic honeycomb]]||{4,3,3,5}||[[tesseract|{4,3,3}]]||[[cube|{4,3}]]||{4}||{5}||[[icosahedron|{3,5}]]||[[600-cell|{3,3,5}]]||{5,3,3,4}
| |
| |- BGCOLOR="#e0e0ff" align=center
| |
| |[[Order-4 120-cell honeycomb]]||{5,3,3,4}||[[120-cell|{5,3,3}]]||[[dodecahedron|{5,3}]]||{5}||{4}||[[octahedron|{3,4}]]||[[16-cell|{3,3,4}]]||{4,3,3,5}
| |
| |- BGCOLOR="#e0e0e0" align=center
| |
| |[[Order-5 600-cell honeycomb]]||{5,3,3,5}||[[600-cell|{5,3,3}]]||[[dodecahedron|{5,3}]]||{5}||{5}||[[icosahedron|{3,5}]]||[[600-cell|{3,3,5}]]||Self-dual
| |
| |}
| |
| | |
| The two paracompact regular H<sup>4</sup> honeycombs are: {3,4,3,4}, {4,3,4,3}.
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| !Name
| |
| ![[Schläfli symbol|Schläfli]]<BR>Symbol<BR>{p,q,r,s}
| |
| !Facet<BR>type<BR>{p,q,r}
| |
| !Cell<BR>type<BR>{p,q}
| |
| !Face<BR>type<BR>{p}
| |
| !Face<BR>figure<BR>{s}
| |
| !Edge<BR>figure<BR>{r,s}
| |
| ![[Vertex figure|Vertex<BR>figure]]<BR>{q,r,s}
| |
| ![[Dual polytope|Dual]]
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| |[[Order-4 24-cell honeycomb]]||{3,4,3,4}||[[24-cell|{3,4,3}]]||[[octahedron|{3,4}]]||{3}||{4}||[[octahedron|{3,4}]]||[[cubic honeycomb|{4,3,4}]]||{4,3,4,3}
| |
| |- BGCOLOR="#e0e0ff" align=center
| |
| |[[Cubic honeycomb honeycomb]]||{4,3,4,3}||[[cubic honeycomb|{4,3,4}]]||[[cube|{4,3}]]||{4}||{3}||[[cube|{4,3}]]||[[24-cell|{3,4,3}]]||{3,4,3,4}
| |
| |}
| |
| | |
| There are four regular star-honeycombs in H<sup>4</sup> space:
| |
| {| class="wikitable"
| |
| |-
| |
| !Name
| |
| ![[Schläfli symbol|Schläfli]]<BR>Symbol<BR>{p,q,r,s}
| |
| !Facet<BR>type<BR>{p,q,r}
| |
| !Cell<BR>type<BR>{p,q}
| |
| !Face<BR>type<BR>{p}
| |
| !Face<BR>figure<BR>{s}
| |
| !Edge<BR>figure<BR>{r,s}
| |
| ![[Vertex figure|Vertex<BR>figure]]<BR>{q,r,s}
| |
| ![[Dual polytope|Dual]]
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| |[[Small stellated 120-cell honeycomb]]||{5/2,5,3,3}||[[Small stellated 120-cell|{5/2,5,3}]]||{5/2,5}||{5}||{5}||{3,3}||[[120-cell|{5,3,3}]]||{3,3,5,5/2}
| |
| |- BGCOLOR="#e0e0ff" align=center
| |
| |[[Pentagrammic-order 600-cell honeycomb]]||{3,3,5,5/2}||[[120-cell|{3,3,5}]]||[[tetrahedron|{3,3}]]||{3}||{5/2}||{5,5/2}||{3,5,5/2}||{5/2,5,3,3}
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| |[[Order-5 icosahedral 120-cell honeycomb]]||{3,5,5/2,5}||{3,5,5/2}||[[icosahedron|{3,5}]]||{3}||{5}||{5/2,5}||{5,5/2,5}||{5,5/2,5,3}
| |
| |- BGCOLOR="#e0e0ff" align=center
| |
| |[[Great 120-cell honeycomb]]||{5,5/2,5,3}||{5,5/2,5}||{5,5/2}||{5}||{3}||[[dodecahedron|{5,3}]]||{5/2,5,3}||{3,5,5/2,5}
| |
| |}
| |
| | |
| ====Tessellations of hyperbolic 5-space====
| |
| | |
| There are 5 regular honeycombs in H<sup>5</sup>, all paracompact, which include infinite (Euclidean) facets or vertex figures: {3,4,3,3,3}, {3,3,4,3,3}, {3,3,3,4,3}, {3,4,3,3,4}, and {4,3,3,4,3}.
| |
| | |
| There are no compact regular tessellations of hyperbolic space of dimension 5 or higher and no paracompact regular tessellations in hyperbolic space of dimension 6 or higher.
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| !Name
| |
| ![[Schläfli symbol|Schläfli]]<BR>Symbol<BR>{p,q,r,s,t}
| |
| !Facet<BR>type<BR>{p,q,r,s}
| |
| !4-face<BR>type<BR>{p,q,r}
| |
| !Cell<BR>type<BR>{p,q}
| |
| !Face<BR>type<BR>{p}
| |
| !Cell<BR>figure<BR>{t}
| |
| !Face<BR>figure<BR>{s,t}
| |
| !Edge<BR>figure<BR>{r,s,t}
| |
| ![[Vertex figure|Vertex<BR>figure]]<BR>{q,r,s,t}
| |
| ![[Dual polytope|Dual]]
| |
| |- BGCOLOR="#ffe0e0" align=center
| |
| |[[5-orthoplex honeycomb]]||{3,3,3,4,3}||[[5-orthoplex|{3,3,3,4}]]||[[5-cell|{3,3,3}]]||[[tetrahedron|{3,3}]]||{3}||{3}||[[cube|{4,3}]]||[[24-cell|{3,4,3}]]||[[16-cell honeycomb|{3,3,4,3}]]||{3,4,3,3,3}
| |
| | |
| |- BGCOLOR="#e0e0ff" align=center
| |
| |[[24-cell honeycomb honeycomb]]||{3,4,3,3,3}||[[24-cell honeycomb|{3,4,3,3}]]||[[24-cell|{3,4,3}]]||[[octahedron|{3,4}]]||{3}||{3}||[[tetrahedron|{3,3}]]||[[5-cell|{3,3,3}]]||[[5-simplex|{3,3,3,3}]]||{3,3,3,4,3}
| |
| | |
| |- BGCOLOR="#e0ffe0" align=center
| |
| |[[16-cell honeycomb honeycomb]]||{3,3,4,3,3}||[[16-cell honeycomb|{3,3,4,3}]]||[[16-cell|{3,3,4}]]||[[tetrahedron|{3,3}]]||{3}||{3}||[[tetrahedron|{3,3}]]||[[tesseract|{4,3,3}]]||[[16-cell honeycomb|{3,4,3,3}]]||self-dual
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| |- BGCOLOR="#ffe0e0" align=center
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| |[[Order-4 24-cell honeycomb honeycomb]]||{3,4,3,3,4}||[[16-cell honeycomb|{3,4,3,3}]]||[[24-cell|{3,4,3}]]||[[octahedron|{3,4}]]||{3}||{4}||[[octahedron|{3,4}]]||[[tesseract|{3,3,4}]]||[[tesseractic honeycomb|{4,3,3,4}]]||{4,3,3,4,3}
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| |[[Tesseractic honeycomb honeycomb]]||{4,3,3,4,3}||[[16-cell honeycomb|{4,3,3,4}]]||[[24-cell|{4,3,3}]]||[[octahedron|{4,3}]]||{4}||{3}||[[cube|{4,3}]]||[[24-cell|{3,4,3}]]||[[16-cell honeycomb|{3,3,4,3}]]||{3,4,3,3,4}
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| |}
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| ====Tessellations of hyperbolic 6-space and higher====
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| Even allowing for infinite (Euclidean) facets and/or vertex figures, there are no regular tessellations of hyperbolic space of dimension 6 or higher.
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| == Apeirotopes ==
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| An '''apeirotope''' is, like any other polytope, an unbounded hyper-surface. The difference is that whereas a polytope's hyper-surface curls back on itself to close round a finite volume of hyperspace, an apeirotope does not curl back.
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| Some people regard apeirotopes as just a special kind of polytope, while others regard them as rather different things.
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| === Two dimensions ===
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| A regular '''[[apeirogon]]''' is a regular division of an infinitely long line into equal segments, joined by vertices. It has regular embeddings in the plane, and in higher-dimensional spaces. In two dimensions it can form a straight line or a zig-zag. In three dimensions, it traces out a helical spiral. The zig-zag and spiral forms are said to be '''skew'''.
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| === Three dimensions ===
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| An '''[[apeirohedron]]''' is an infinite polyhedral surface. Like an apeirogon, it can be flat or skew. A flat apeirohedron is just a tiling of the plane. A skew apeirohedron is an intricate honeycomb-like structure which divides space into two regions.
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| There are thirty regular apeirohedra in Euclidean space.<ref>{{Harv |McMullen |Schulte |2002 |loc=Section 7E}}</ref> These include the tessellations of type ''{4,4}'', ''{6,3}'', ''{3,6}'' above, as well as (in the plane) polytopes of type: ''{∞,3}'', ''{∞,4}'', ''{∞,6}'' and in 3-dimensional space, blends of these with either an apeirogon or a line segment, and the "pure" 3-dimensional apeirohedra (12 in number)
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| See also [[regular skew polyhedron]].
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| === Four and higher dimensions ===
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| The apeirochora have not been completely classified.
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| == Abstract polytopes ==
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| The [[abstract polytope]]s arose out of an attempt to study polytopes apart from the geometrical space they are embedded in. They include the tessellations of spherical, euclidean and hyperbolic space, tessellations of other [[manifold]]s, and many other objects that do not have a well-defined topology, but instead may be characterised by their "local" topology. There are infinitely many in every dimension. See [http://www.abstract-polytopes.com/atlas/ this atlas] for a sample. Some notable examples of abstract polytopes that do not appear elsewhere in this list are the [[11-cell]] and the [[57-cell]].
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| == See also ==
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| * [[Polygon]]
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| ** [[Regular polygon]]
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| ** [[Star polygon]]
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| * [[Polyhedron]]
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| ** [[Regular polyhedron]] (5 regular [[Platonic solid]]s and 4 [[Kepler–Poinsot solid]]s)
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| *** [[Uniform polyhedron]]
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| * [[Polychoron]]
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| ** [[Convex regular 4-polytope]] (6 regular polychora)
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| *** [[Uniform polychoron]]
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| ** [[Schläfli–Hess polychoron]] (10 regular star polychora)
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| * [[Tessellation]]
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| ** [[Tilings of regular polygons]]
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| ** [[Convex uniform honeycomb]]
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| * [[Regular polytope]]
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| ** [[Uniform polytope]]
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| == References ==
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| {{reflist}}
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| {{refbegin}}
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| *[[H.S.M. Coxeter|Coxeter]], ''[[Regular Polytopes (book)|Regular Polytopes]]'', 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
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| *[[H.S.M. Coxeter|Coxeter]], ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999 ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, pp. 212–213) [http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf] [[PDF]]
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| * [[Duncan MacLaren Young Sommerville|D. M. Y. Sommerville]], ''An Introduction to the Geometry of '''n''' Dimensions.'' New York, E. P. Dutton, 1930. 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes
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| {{refend}}
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| == External links ==
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| *[http://www.math.utah.edu/~alfeld/math/polyhedra/polyhedra.html The Platonic Solids]
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| *[http://www.georgehart.com/virtual-polyhedra/kepler-poinsot-info.html Kepler-Poinsot Polyhedra]
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| *[http://www.weimholt.com/andrew/polytope.shtml Regular 4d Polytope Foldouts]
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| *[http://web.archive.org/web/20070207021813/members.aol.com/Polycell/glossary.html Multidimensional Glossary (Look up '''Hexacosichoron''' and '''Hecatonicosachoron''')]
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| *[http://www.stat.berkeley.edu/~evans/shapiro/TesseractApplet.html Polytope Viewer]
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| *[http://presh.com/hovinga/ Polytopes and optimal packing of p points in n dimensional spheres]
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| *[http://www.abstract-polytopes.com/atlas/ An atlas of small regular polytopes]
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| {{Polytopes}}
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| {{Honeycombs}}
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| [[Category:Mathematics-related lists|Regular polytopes]]
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| [[Category:Polytopes| ]]
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| [[Category:Multi-dimensional geometry]]
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