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Regular Polytopes

This is the title of a famous book of H.S.M. Coxeter, and accordingly lots of the following can be found there too. (The below provided tables by dimensions had been assembled by D. de Winters, but his webpage ceased to exist after his death. They are thus reproduced here, in a further enriched form.)

The high degree of symmetry of regular polytopes induces lots of inter-relations onto the set of regular polytopes. There are not only dual pairings (which, according to regularity, both have to be regular polytopes!), but even the set of vertices (up to radial scalings) often are commonly used for large subsets each. Polytopes can be grouped accordingly to the dimensional size of shared sub-elements. The following systematics of Olshevsky does not only apply to regulars, but for the above reasons clearly has lots of applications here:

shared
sub-elements
name of group most convex
representant
0 army general
1 regiment colonel
2 company captain
n n-regiment n-colonel

In more detail, the army general of some regular (or even uniform) polytope always is its convex hull. Conversely, that polytope is a (full symmetrical) faceting of its army general. Alike, any polytope can be considered as an edge faceting of its regiment colonel, i.e. a faceting which respects the given edge skeleton. Thereby its vertex figure obviously belongs to the army of the vertex figure of its colonel. Note that this does not imply the other way round, that the colonel always will have a convex vertex figure (which also is known as local convexity of the polytope itself). Sure, most often it does, but there will be cases, where a coating of the edge skeleton in the sense of a convex vertex figure, demanded from one end of the edges, would ask for additional edges at the opposite end, which do not belong to the actual edge skeleton. Then the colonel would be chosen as the one with a vertex figure being as convex as possible. For instance this takes place for the 10-tet-compound (e), where there are only 2 provided edges within each potential locally coating face plane.

(Johnson extended this nomenclatura slightly, by calling 2 or more different regiments (of the same army), using the same edge length, a brigade. Further, a subset of polytopes of a brigade, which is closed under the operation of blending, is called a cohort. – Even close relatives of colonels deserve a special name, attributed by Bowers: lieutenants he calls the conjugates of colonels, which not themselves are colonels.)

Compounds too can fall into the same army as some regular polytope. This is what Coxeter once called vertex regularity.

Just as the chords of polygons are defined as its vertex-to-vertex line-segments, generally any polytope shows up similarily chords of different lengths. Those might be grouped into ones of the same size, and further the classes might be ordered (and thereby named) by increasing size. A 0-chord then would be the vertex adjoined to itself, and for uniform convex polytopes the 1-chords would be the set of edges.

In the following, compounds (esp. the polygonal ones), if not stated otherwise, are understood to be (fully) regular ones.



---- 2D ----

army
general
vertex
count
chord
number
edge
count
scaling
factor
(unscaled)
regiment colonel
{3} = x3o 3 1st 3 x {3} = x3o
{4} = x4o 4 1st 4 x {4} = x4o
{5} = x5o 5 1st 5 x {5} = x5o
2nd 5 f {5/2} = x5/2o
{6} = x6o 6 1st 6 x {6} = x6o
2nd 6 h 2-x3o-compound
(star of David)
= {6}[2{3}]{6}
{7} = x7o 7 1st 7 x {7} = x7o
2nd 7 x(7) {7/2} = x7/2o
3rd 7 x(7,3)
= 1/[1-1/x(7)]
{7/3} = x7/3o
{8} = x8o 8 1st 8 x {8} = x8o
2nd 8 x(8) 2-x4o-compound
= {8}[2{4}]{8}
3rd 8 w
= 1+q
{8/3} = x8/3o
{9} = x9o 9 1st 9 x {9} = x9o
2nd 9 x(9) {9/2} = x9/2o
3rd 9 x(9,3)
= 2+1/x(9)
3-x3o-compound
= {9}[3{3}]{9}
4th 9 x(9,4)
= 1+x(9)
{9/4} = x9/4o
{10} = x10o 10 1st 10 x {10} = x10o
2nd 10 x(10) 2-x5o-compound
= {10}[2{5}]{10}
3rd 10 x(10,3)
= f f = 1+f
{10/3} = x10/3o
4th 10 x(10,4)
= f x(10)
2-x5/2o-compound
= {10}[2{5/2}]{10}
etc.

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---- 3D ----

army
general
vertex
count
chord
number
edge
count
scaling
factor
(unscaled)
regiment colonel
(unscaled)
faceting face
face
count
(unscaled)
company captain
tet 4 1st 6 x tet x3o 4 tet = {3,3} = x3o3o
oct 6 1st 12 x oct x3o 8 oct = {3,4} = x3o4o
x4o 3
cube 8 1st 12 x cube x4o 6 cube = {4,3} = x4o3o
2nd 12 q so x3o 8 so (stella octangula)
= 2-tet-compound
= {4,3}[2{3,3}]{3,4}
ike 12 1st 30 x ike x3o 20 ike = {3,5} = x3o5o
x5o 12 gad = {5,5/2} = x5o5/2o
2nd 60 f sissid x5/2o 12 sissid = {5/2,5} = x5/2o5o
x3o 20 gike = {3,5/2} = x3o5/2o
doe 20 1st 30 x doe x5o 12 doe = {5,3} = x5o3o
2nd 60 f (sidtid) x5/2o 12
x3o 20
x4o 30 rhom = 5-cube-compound
= 2{5,3}[5{4,3}]
x5o 12
3rd 60 f q e non-regular
2-x3o-compound
20 e = 10-tet-compound
= 2{5,3}[10{3,3}]2{3,5}
4th 30 f f gissid x5/2o 12 gissid = {5/2,3} = x5/2o3o

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---- 4D ----

army
general
vertex
count
chord
number
edge
count
scaling
factor
(unscaled)
regiment
colonel
(unscaled)
faceting
face
face
count
(unscaled)
company
captain
(unscaled)
faceting
cell
cell
count
(unscaled)
3-regiment 3-colonel
pen 5 1st 10 x pen x3o 10 pen tet 5 pen = {3,3,3} = x3o3o3o
hex 8 1st 24 x hex x3o 32 hex tet 16 hex = {3,3,4} = x3o3o4o
oct 4
x4o 6
tes 16 1st 32 x tes x4o 24 tes cube 8 tes = {4,3,3} = x4o3o3o
2nd 48 q haddet x3o 64 haddet tet 32 haddet = 2-hex-compound
= {4,3,3}[2{3,3,4}]
oct 8
x4o 12
3rd 32 h
ico 24 1st 96 x ico x3o 96 ico oct 24 ico = {3,4,3} = x3o4o3o
x4o 72 gico cube 24 gico = 3-tes-compound
= 2{3,4,3}[3{4,3,3}]{3,4,3}
x6o 16
2nd 72 q stico x3o 96 stico so 24 stico = 3-hex-compound
= {3,4,3}[3{3,3,4}]2{3,4,3}
oct 12
x4o 18
3rd 96 h 2-x3o-
compound =
{6}[2{3}]{6}
16
ex 120 1st 720 x ex x3o 1200 ex tet 600 ex = {3,3,5} = x3o3o5o
ike 120 fix = {3,5,5/2} = x3o5o5/2o
x5o 720 gahi doe 120 gahi = {5,3,5/2} = x5o3o5/2o
gad 120 gohi = {5,5/2,5} = x5o5/2o5o
x10o 72
2nd 1200 f sishi x5/2o 720 sishi sissid 120 sishi = {5/2,5,3} = x5/2o5o3o
x3o 2400 dox gike 120
tet 600
oct 600 dox = 25-ico-compound
= 5-chi-compound
= 5{3,3,5}[25{3,4,3}]{3,3,5}
ike 120
x4o 1800 dac cube 600 dac = 75-tes-compound
= 25-gico-compound
= 10{3,3,5}[75{4,3,3}]5{5,3,3}
x5o 720 gaghi gad 120 gaghi = {5,5/2,3} = x5o5/2o3o
x6o 200
3rd 720 x(10) 2-x5o-
compound =
{10}[2{5}]{10}
72
4th 1800 f q [75hex] x3o 2400 [75hex] tet 1200 [75hex] = 75-hex-compound
= 25-stico-compound
= 5{3,3,5}[75{3,3,4}]10{5,3,3}
oct 300
x4o 450
5th 720 f f gishi x5/2o 720 gishi gissid 120 gishi = {5/2,3,5} = x5/2o3o5o
sissid 120 gashi = {5/2,5,5/2} = x5/2o5o5/2o
x3o 1200 gofix gike 120 gofix = {3,5/2,5} = x3o5/2o5o
tet 600 gax = {3,3,5/2} = x3o3o5/2o
x10/3o 72
6th 1200 f u 2-x3o-
compound =
{6}[2{3}]{6}
200
7th 720 f x(10) 2-x5/2o-
compound =
{10}[2{5/2}]{10}
72
hi 600 1st 1200 x hi x5o 720 hi doe 120 hi = {5,3,3} = x5o3o3o
2nd 3600 f (sidtaxhi) x5/2o 720
x3o 2400 tet 600
x4o 3600 cube 600
x5o 1440 doe 120
x10o 720
3rd 7200 f q [10ex] x3o 12000 [10ex] tet 6000 [10ex] = 10-ex-compound
= 2{5,3,3}[10{3,3,5}]
ike 1200 [10fix] = 10-fix-compound
= 2{5,3,3}[10{3,5,5/2}]2{3,3,5}
x5o 7200 [10gohi] gad 1200 [10gohi] = 10-gohi-compound
= 2{5,3,3}[10{5,5/2,5}]2{3,3,5}
doe 1200 [10gahi] = 10-gahi-compound
= 2{5,3,3}[10{5,3,5/2}]2{3,3,5}
x10o 720
4th 3600 f f (dattady) x5/2o 1440 gissid 120
x3o 1200
x4o 3600 cube 600
x5o 1440 doe 120
x6o 1200
5th 1200 f h
6th 7200 f x(10) 2-x5/2o-
compound =
{10}[2{5/2}]{10}
720
x4o 3600
7th 7200 [?] x5o 1440
8th 9600 f f q [10sishi] x5/2o 7200 [10sishi] sissid 1200 [10sishi] = 10-sishi-compound
= 2{5,3,3}[10{5/2,5,3}]2{3,3,5}
x3o 21600 [225ico] gike 1200
tet 6000
oct 5400 [225ico] = 225-ico-compound
= 9{5,3,3}[225{3,4,3}]
ike 1200
x4o 16200 [675tes] cube 5400 [675tes] = 675-tes-compound
= 18{5,3,3}[675{4,3,3}]9{3,3,5}
x5o 7200 [10gaghi] gad 1200 [10gaghi] = 10-gaghi-compound
= 2{5,3,3}[10{5,5/2,3}]2{3,3,5}
x6o 1600
9th 7200 [?] x5o 1440
10th 3600 f f f (gadtaxady) x5/2o 720 gissid 120
x3o 2400 tet 600
x10/3o 720
x4o 3600 cube 600
x5o 720
11th 7200 f q x(10) 2-x5o-
compound =
{10}[2{5}]{10}
720
12ath 7200 f f h 2-x3o-
compound =
{6}[2{3}]{6}
1200
x4o 3600
12bth 1200
13th 7200 [?] x3o 2400
14th 7200 f f x(10) 2-x5/2o-
compound =
{10}[2{5/2}]{10}
720
x4o 3600
15th 16200 f f u [675hex] x3o 21600 [675hex] tet 10800 [675hex] = 675-hex-compound
= 9{5,3,3}[675{4,3,3}]18{3,3,5}
oct 2700
x4o 4050
16th 7200 [?]
17th 7200 [?] x5/2o 1440
18ath 7200 f (2+f)
= f f sqrt(5)
[720pen] x3o 7200 [720pen] tet 3600 [720pen] = 720-pen-compound
= 6{5,3,3}[720{3,3,3}]
18bth 1200 mix x3o 1200 mix tet 600 mix = 120-pen-compound
= {5,3,3}[120{3,3,3}]{3,3,5}
19th 7200 f f f q [10gishi] x5/2o 7200 [10gishi] gissid 6000 [10gishi] = 10-gishi-compound
= 2{5,3,3}[10{5/2,3,5}]2{3,3,5}
sissid 1200 [10gashi] = 10-gashi-compound
= 2{5,3,3}[10{5/2,5,5/2}]2{3,3,5}
x3o 12000 [10gofix] gike 1200 [10gofix] = 10-gofix-compound
= 2{5,3,3}[10{3,5/2,5}]2{3,3,5}
tet 6000 [10gax] = 10-gax-compound
= 2{5,3,3}[10{3,3,5/2}]
x10/3o 720
20th 3600 [?]
21st 7200 [?] x3o 2400
22nd 9600 f f q h 2-x3o-
compound =
{6}[2{3}]{6}
1600
23rd 7200 [?] x5/2o 1440
24th 7200 [?]
25th 1200 f f f f gogishi x5/2o 720 gogishi gissid 120 gogishi = {5/2,3,3} = x5/2o3o3o
26th 3600 [?]
27th 7200 f f q x(10) 2-x5/2o-
compound =
{10}[2{5/2}]{10}
720
28th 3600 [?]
29th 1200 f f f h


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