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 the B₅ Coxeter plane

5-cube

5-orthoplex

5-demicube

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 31 uniform polytopes with B₅ symmetry. There are two regular forms, the 5-orthoplex, and 5-cube with 10 and 32 vertices respectively. The 5-demicube is added as an alternation of the 5-cube.

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 B₅ Coxeter group, and other subgroups.

Graphs

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 of these 32 polytopes can be made in the B₅, B₄, B₃, B₂, A₃, Coxeter planes. A_k has [k+1] symmetry, and B_k has [2k] symmetry.

These 32 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.

#	Graph B5 / A4 [10]	Graph B4 / D5 [8]	Graph B3 / A2 [6]	Graph B2 [4]	Graph A3 [4]	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... and Schläfli symbol Johnson and Bowers names
20						t0{4,3,3,3} 5-cube Penteract (pent)
21						t1{4,3,3,3} Rectified 5-cube Rectified 5-cube In give-dimensional geometry, a rectified 5-cube is a convex uniform 5-polytope, being a rectification of the regular 5-cube.There are 5 degrees of rectifications of a 5-polytope, the zeroth here being the 5-cube, and the 4th and last being the 5-orthoplex. Vertices of the rectified 5-cube are... Rectified penteract (rin)
22						t2{4,3,3,3} Birectified 5-cube Penteractitriacontiditeron (nit)
40						t1{3,3,3,4} Rectified 5-orthoplex Rectified triacontiditeron (rat)
39						t0{3,3,3,4} 5-orthoplex Triacontiditeron (tac)
23						t0,1{4,3,3,3} Truncated 5-cube Truncated 5-cube In five-dimensional geometry, a truncated 5-cube is a convex uniform 5-polytope, being a truncation of the regular 5-cube.There are four unique truncations of the 5-cube. Vertices of the truncated 5-cube are located as pairs on the edge of the 5-cube. Vertices of the bitruncated 5-cube are located... Truncated penteract (tan)
24						t1,2{4,3,3,3} Bitruncated 5-cube Bitruncated penteract (bittin)
25						t0,2{4,3,3,3} Cantellated 5-cube Cantellated 5-cube In six-dimensional geometry, a cantellated 5-cube is a convex uniform 5-polytope, being a cantellation of the regular 5-cube.There are 6 unique cantellation for the 5-cube, including truncations... Rhombated penteract (sirn)
26						t1,3{4,3,3,3} Bicantellated 5-cube Small birhombi-penteractitriacontiditeron (sibrant)
27						t0,3{4,3,3,3} Runcinated 5-cube Runcinated 5-cube In five-dimensional geometry, a runcinated 5-cube is a convex uniform 5-polytope that is a runcination of the regular 5-cube.... Prismated penteract (span)
28						t0,4{4,3,3,3} Stericated 5-cube Stericated 5-cube In five-dimensional geometry, a stericated 5-cube is a convex uniform 5-polytope with fourth-order truncations of the regular 5-cube.... Small celli-penteractitriacontiditeron (scant)
41						t0,1{3,3,3,4} Truncated 5-orthoplex Truncated 5-orthoplex In six-dimensional geometry, a truncated 5-orthoplex is a convex uniform 5-polytope, being a truncation of the regular 5-orthoplex.There are 4 unique truncations of the 5-orthoplex. Vertices of the truncation 5-orthoplex are located as pairs on the edge of the 5-orthoplex. Vertices of the... Truncated triacontiditeron (tot)
42						t1,2{3,3,3,4} Bitruncated 5-orthoplex Bitruncated triacontiditeron (bittit)
43						t0,2{3,3,3,4} Cantellated 5-orthoplex Cantellated 5-orthoplex In six-dimensional geometry, a cantellated 5-orthoplex is a convex uniform 5-polytope, being a cantellation of the regular 5-orthoplex.There are 6 cantellation for the 5-orthoplex, including truncations... Small rhombated triacontiditeron (sart)
44						t0,3{3,3,3,4} Runcinated 5-orthoplex Runcinated 5-orthoplex In six-dimensional geometry, a runcinated 5-orthoplex is a convex uniform 5-polytope with 3rd order truncation of the regular 5-orthoplex.There are 8 runcinations of the 5-orthoplex with permutations of truncations, and cantellations... Small prismated triacontiditeron (spat)
28						t0,4{3,3,3,4} Stericated 5-orthoplex Small celli-penteractitriacontiditeron (scant)
29						t0,1,2{4,3,3,3} Cantitruncated 5-cube Great rhombated penteract (girn)
30						t1,2,3{4,3,3,3} Bicantitruncated 5-cube Great birhombi-penteractitriacontiditeron (gibrant)
31						t0,1,3{4,3,3,3} Runcitruncated 5-cube Prismatotruncated penteract (pattin)
32						t0,2,3{4,3,3,3} Runcicantellated 5-cube Prismatorhomated penteract (prin)
33						t0,1,4{4,3,3,3} Steritruncated 5-cube Cellitruncated penteract (capt)
34						t0,2,4{4,3,3,3} Stericantellated 5-cube Cellirhombi-penteractitriacontiditeron (carnit)
35						t0,1,2,3{4,3,3,3} Runcicantitruncated 5-cube Great primated penteract (gippin)
36						t0,1,2,4{4,3,3,3} Stericantitruncated 5-cube Celligreatorhombated penteract (cogrin)
37						t0,1,3,4{4,3,3,3} Steriruncitruncated 5-cube Celliprismatotrunki-penteractitriacontiditeron (captint)
38						t0,1,2,3,4{4,3,3,3} Omnitruncated 5-cube Great celli-penteractitriacontiditeron (gacnet)
45						t0,1,2{3,3,3,4} Cantitruncated 5-orthoplex Great rhombated triacontiditeron (gart)
46						t0,1,3{3,3,3,4} Runcitruncated 5-orthoplex Prismatotruncated triacontiditeron (pattit)
47						t0,2,3{3,3,3,4} Runcicantellated 5-orthoplex Prismatorhombated triacontiditeron (pirt)
48						t0,1,4{3,3,3,4} Steritruncated 5-orthoplex Cellitruncated triacontiditeron (cappin)
49						t0,1,2,3{3,3,3,4} Runcicantitruncated 5-orthoplex Great prismatorhombated triacontiditeron (gippit)
50						t0,1,2,4{3,3,3,4} Stericantitruncated 5-orthoplex Celligreatorhombated triacontiditeron (cogart)
51						h0{4,3,3,3} 5-demicube Hemipenteract (hin)