List of D7 polytopes
Encyclopedia
7-demicube |
7-orthoplex |
In 7-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 128 uniform polytopes with D7 symmetry, 32 are unique, and 96 are shared with the B7 symmetry. There are two regular forms, the 7-orthoplex, and 7-demicube with 14 and 64 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 32 polytopes can be made in the D7, D6, D5, D4, D3, A5, A3, Coxeter planes. Ak has [k+1] symmetry, Dk has [2(k-1)] symmetry. B7 is also included although only half of its [14] symmetry exists in these polytopes.
These 32 polytopes are each shown in these 8 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.
| # | Coxeter plane graphs | 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 |
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| B7 [14/2] |
D7 [12] |
D6 [10] |
D5 [8] |
D4 [6] |
D3 [4] |
A5 [6] |
A3 [4] |
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| 1 | 7-demicube Demihepteract (Hesa) |
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| 2 | Truncated 7-demicube Truncated 7-demicube In seven-dimensional geometry, a truncated 7-demicube is a uniform 7-polytope, being a truncation of the 7-demicube.- Cartesian coordinates :The Cartesian coordinates for the 1344 vertices of a truncated 7-demicube centered at the origin and edge length 6√2 are coordinate permutations:with an odd... Truncated demihepteract (Thesa) |
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| 3 | Cantellated 7-demicube Cantellated 7-demicube In seven-dimensional geometry, a cantellated 7-demicube is a convex uniform 7-polytope, being a cantellation of the uniform 7-demicube. There are 2 unique cantellation for the 7-demicube including a truncation.- Cantellated 7-demicube:... Small rhombated demihepteract (Sirhesa) |
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| 4 | Runcinated 7-demicube Small prismated demihepteract (Sphosa) |
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| 5 | Stericated 7-demicube Small cellated demihepteract (Sochesa) |
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| 6 | Pentellated 7-demicube Small terated demihepteract (Suthesa) |
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| 7 | Cantitruncated 7-demicube Great rhombated demihepteract (Girhesa) |
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| 8 | Runcitruncated 7-demicube Prismatotruncated demihepteract (Pothesa) |
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| 9 | Runcicantellated 7-demicube Prismatorhomated demihepteract (Prohesa) |
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| 10 | Steritruncated 7-demicube Cellitruncated demihepteract (Cothesa) |
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| 11 | Stericantellated 7-demicube Cellirhombated demihepteract (Crohesa) |
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| 12 | Steriruncinated 7-demicube Celliprismated demihepteract (Caphesa) |
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| 13 | Pentitruncated 7-demicube Teritruncated demihepteract (Tuthesa) |
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| 14 | Penticantellated 7-demicube Terirhombated demihepteract (Turhesa) |
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| 15 | Pentiruncinated 7-demicube Teriprismated demihepteract (Tuphesa) |
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| 16 | Pentistericated 7-demicube Tericellated demihepteract (Tuchesa) |
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| 17 | Runcicantitruncated 7-demicube Great prismated demihepteract (Gephosa) |
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| 18 | Stericantitruncated 7-demicube Celligreatorhombated demihepteract (Cagrohesa) |
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| 19 | Steriruncitruncated 7-demicube Celliprismatotruncated demihepteract (Capthesa) |
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| 20 | Steriruncicantellated 7-demicube Celliprismatorhombated demihepteract (Coprahesa) |
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| 21 | Penticantitruncated 7-demicube Terigreatorhombated demihepteract (Tugrohesa) |
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| 22 | Pentiruncitruncated 7-demicube Teriprismatotruncated demihepteract (Tupthesa) |
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| 23 | Pentiruncicantellated 7-demicube Teriprismatorhombated demihepteract (Tuprohesa) |
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| 24 | Pentisteritruncated 7-demicube Tericellitruncated demihepteract (Tucothesa) |
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| 25 | Pentistericantellated 7-demicube Tericellirhombated demihepteract (Tucrohesa) |
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| 26 | Pentisteriruncinated 7-demicube Tericelliprismated demihepteract (Tucophesa) |
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| 27 | Steriruncicantitruncated 7-demicube Great cellated demihepteract (Gochesa) |
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| 28 | Pentiruncicantitruncated 7-demicube Terigreatoprimated demihepteract (Tugphesa) |
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| 29 | Pentistericantitruncated 7-demicube Tericelligreatorhombated demihepteract (Tucagrohesa) |
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| 30 | Pentisteriruncitruncated 7-demicube Tericelliprismatotruncated demihepteract (Tucpathesa) |
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| 31 | Pentisteriruncicantellated 7-demicube Tericellprismatorhombated demihepteract (Tucprohesa) |
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| 32 | Pentisteriruncicantitruncated 7-demicube Great terated demihepteract (Guthesa) |