List of D5 polytopes
Encyclopedia
5-demicube |
5-orthoplex |
In 5-dimensional geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, there are 32 uniform polytopes with D5 symmetry, 8 are unique, and 24 are shared with the B5 symmetry. There are two special forms, the 5-orthoplex, and 5-demicube with 10 and 16 vertices respectively.
They can be visualized as symmetric orthographic projection
Orthographic projection
Orthographic projection is a means of representing a three-dimensional object in two dimensions. It is a form of parallel projection, where all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface...
s in Coxeter planes of the D6 Coxeter group, and other subgroups.
Graphs
Symmetric orthographic projectionOrthographic projection
Orthographic projection is a means of representing a three-dimensional object in two dimensions. It is a form of parallel projection, where all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface...
s of these 8 polytopes can be made in the D5, D4, D3, A3, Coxeter planes. Ak has [k+1] symmetry, Dk has [2(k-1)] symmetry. The B5 plane is included, with only half the [10] symmetry displayed.
These 8 polytopes are each shown in these 5 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.
| # | Coxeter plane projections | Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... Schläfli symbol symbols Johnson and Bowers names |
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|---|---|---|---|---|---|---|
| [10/2] | [8] | [6] | [4] | [4] | ||
| B5 | D5 | D4 | D3 | A3 | ||
| 1 | (121) 5-demicube Hemipenteract (hin) |
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| 2 | t0,1(121) Truncated 5-demicube Truncated hemipenteract (thin) |
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| 3 | t0,2(121) Cantellated 5-demicube Cantellated 5-demicube In six-dimensional geometry, a cantellated 5-demicube is a convex uniform 5-polytope, being a cantellation of the uniform 5-demicube.There are 2 unique cantellation for the 5-demicube including a truncation.- Cantellated 5-demicube:... Small rhombated hemipenteract (sirhin) |
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| 4 | t0,3(121) Runcinated 5-demicube Runcinated 5-demicube In five-dimensional geometry, a runcinated 5-demicube is a convex uniform 5-polytope with a runcination operation, a 3rd order truncations the uniform 5-demicube.... Small prismated hemipenteract (siphin) |
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| 5 | t0,1,2(121) Cantitruncated 5-demicube Great rhombated hemipenteract (girhin) |
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| 6 | t0,1,3(121) Runcitruncated 5-demicube Prismatotruncated hemipenteract (pithin) |
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| 7 | t0,2,3(121) Runcicantellated 5-demicube Prismatorhombated hemipenteract (pirhin) |
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| 8 | t0,1,2,3(121) Runcicantitruncated 5-demicube Great prismated hemipenteract (giphin) |