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Minersphere::Index::Erebean polytopes::Iridoids

Iridoids

Published 12 January MMXXVI

The iridoids are the Erebean polytopes derived from the matrix form of the symbol [k,3,3,...], with k being the index. This generalizes the rhodoids and the tychoids to other face types beyond pentagons and heptagons. Their natural quotient order is k.

The iridoidal symmetries each have two regular polytopes, the one with the least amount of vertices being the osmoplex, having k-gonal faces, and the one with the greatest amount of vertices being known as the platinoplex, having triangular faces.

Somewhat interestingly, the group description of iridoids under natural quotient order alternate between two variants. In rank 3, it is either the direct or indirect product of PSL(2,k) with C2, with the direct product being used when k is a Pythagorean prime (p = 4n+1). In rank 4, the same thing happens but with SL(2,k)⋉PSL(2,k)⋉C2 if it is a Pythagorean prime and PSL(2,k2)⋉C2 otherwise. In rank 5, the group description simply alternates between primes, with O(5,k)⋉C2 for odd π(k) and O(5,k)×C2 otherwise.

Information that couldn't fit nicely into here (vertex layers) was put in iridoids.txt.

Natural

Rank 3

Index 11

The rank 3 index 11 cases are the 3-11-osmoplex (26-toral 60-hedron) and the 3-11-platinoplex (26-toral 220-hedron). The former has 60 hendecagonal faces as a quotient of {11,3} while the latter has 220 triangular faces as a quotient of {3,11}. The vertices of the former each have 1 opposing vertex, while the vertices of the latter each have 4 opposing vertices.

Symmetry properties
Symbol 11♯Ir3(11)
Order 1320
Petrie polygon 10
Description PSL(2,11)⋉C2
Symmetry axes
Symmetry Count
I2(11) 60
A1×A1 330
A2 220

Index 13

The rank 3 index 13 cases are the 3-13-osmoplex (50-toral 84-hedron) and the 3-13-platinoplex (50-toral 364-hedron). The vertices of the former each have 3 opposing vertices, while the vertices of the latter each have 5 opposing vertices.

Symmetry properties
Symbol 13♯Ir3(13)
Order 2184
Petrie polygon 14
Description PSL(2,13)×C2
Symmetry axes
Symmetry Count
I2(13) 84
A1×A1 546
A2 364

Index 17

The rank 3 index 17 cases are the 3-17-osmoplex (133-toral 144-hedron) and the 3-17-platinoplex (133-toral 816-hedron). The vertices of the former each have three opposing vertices, while the vertices of the latter each have four opposing vertices. The vertices of the former each have 1 opposing vertex, while the vertices of the latter each have 7 opposing vertices.

Symmetry properties
Symbol 17♯Ir3(17)
Order 4896
Petrie polygon 18
Description PSL(2,17)×C2
Symmetry axes
Symmetry Count
I2(17) 144
A1×A1 1224
A2 816

Index 19

The rank 3 index 19 cases are the 3-19-osmoplex (196-toral 180-hedron) and the 3-19-platinoplex (133-toral 1140-hedron). The vertices of the former each have three opposing vertices, while the vertices of the latter each have four opposing vertices. The vertices of the former each have 1 opposing vertex, while the vertices of the latter each have 8 opposing vertices.

Symmetry properties
Symbol 19♯Ir3(19)
Order 6840
Petrie polygon 18
Description PSL(2,19)⋉C2
Symmetry axes
Symmetry Count
I2(19) 180
A1×A1 1710
A2 1140

Index 23

The rank 3 index 23 cases are the 3-23-osmoplex (375-toral 264-hedron) and the 3-23-platinoplex (375-toral 2024-hedron). The vertices of the former each have three opposing vertices, while the vertices of the latter each have four opposing vertices. The vertices of the former each have 1 opposing vertex, while the vertices of the latter each have 8 opposing vertices.

Symmetry properties
Symbol 23♯Ir3(23)
Order 12144
Petrie polygon 24
Description PSL(2,23)⋉C2
Symmetry axes
Symmetry Count
I2(23) 264
A1×A1 3036
A2 2024

Index 29

The rank 3 index 29 cases are the 3-29-osmoplex (806-toral 420-hedron) and the 3-29-platinoplex (806-toral 4060-hedron). The vertices of the former each have three opposing vertices, while the vertices of the latter each have four opposing vertices. The vertices of the former each have 1 opposing vertex, while the vertices of the latter each have 8 opposing vertices.

Symmetry properties
Symbol 29♯Ir3(29)
Order 24360
Petrie polygon 14
Description PSL(2,29)×C2
Symmetry axes
Symmetry Count
I2(29) 420
A1×A1 6090
A2 4060

Summary

This table shows the flag counts for the symmetry k♯Irn(k), along with its Petrie polygon P and group description.

n \ k 3 5 7 11 13 17 19 23 29
3 24
P = 4
S4		 120
P = 10
A5×C2		 336
P = 8
PSL(3,2)⋉C2		 1320
P = 10
[1]		 2184
P = 14
[2]		 4896
P = 18
[2]		 6840
P = 18
[1]		 12144
P = 24
[1]		 24360
P = 14
[2]
4 120
P = 5
S5		 14400
P = 30
SL(2,5)⋉A5⋉C2		 117600
P = 24
[3]		 1771440
P = 60
[3]		 4769856
P = 84
[4]		 23970816
P = 144
[4]		 47045520
P = 180
[3]		 148035360
P = 265
[3]		 593409600
P = 28
[4]?
5 720
P = 6
S6		 9360000
P = 26
[5]		 276595200
P = 24
[6]		 25721308800
P = 122
[5]		 137037962880
P = 182
[6]		 2008994088960
P = 72
[5]		 6114035779200
P = 180
[6]		 41348052472320
P = 132
[5]		 420206392771200
P = 210
[6]?

Some trivial group descriptions: