Rhodoids

The rhodoids, symmetry group ♯H_n, are the Erebean polytopes derived from the matrix form of the symbol [5,3,3,...]. Thus, they describe analogs of the dodecahedron and icosahedron in the sense that they have pentagonal faces/peak figures and simplicial vertex figures/facets. Their natural quotient order is 5.

Each rhodoidal symmetry has two regular polytopes, the one with the least amount of vertices being known as the hydroplex, analogous to the icosahedron, and the one with the greatest amount of vertices being known as the cosmoplex, analogous to the dodecahedron.

They are named after the elemental name for H_n symmetry, coined by the Hi.gher.space community.

Information that couldn't fit nicely into here (incidence matrices and vertex layers) was put in rhodoids.txt.

Natural

Rank 5

The rank 5 cases are the 5-cosmoplex (650-teron) and the 5-hydroplex (78000-teron), being duals of each other with ⁵♯H₅ symmetry. The former has 650 hecatonicosachoral tera as a quotient of {5,3,3,3} while the latter has 78000 pentachoral tera as a quotient of {3,3,3,5}. It is known that the 5-hydroplex has hexagonal cycles, that is the polygon obtained by going from edge to the opposing edge on each vertex. For the hexacosichoron (4-hydroplex), this yields decagonal cycles.

Symmetry properties
Petrie polygon	Order
26	9360000
Symmetry axes
Symmetry	Count
H4	650
H3×A1	39000
H2×A2	156000
A1×A3	195000
A4	78000
Discovery
Milo Jacquet	9 December 2024

Rank 6

Symmetry properties
Petrie polygon	Order
63	29484000000
Symmetry axes
Symmetry	Count
5♯H5	3150
H4×A1	1023750
H3×A2	40950000
H2×A3	122850000
A1×A4	122850000
A5	40950000
Discovery
PlanetN9ne	2 June 2025

Rank 7

Symmetry properties
Petrie polygon	Order
120	460687500000000
Symmetry axes
Symmetry	Count
5♯H6	15625
5♯H5×A1	24609375
H4×A2	5332031250
H3×A3	159960937500
H2×A4	383906250000
A1×A5	319921875000
A6	91406250000
Discovery
PlanetN9ne	3 June 2025

Rank 8

Symmetry properties
Petrie polygon	Order
313	35760406500000000000
Symmetry axes
Symmetry	Count
5♯H7	77624
5♯H6×A1	606437500
5♯H5×A2	636759375000
H4×A3	103473398437500
H3×A4	2483361562500000
H2×A5	4966723125000000
A1×A6	3547659375000000
A7	886914843750000
Discovery
PlanetN9ne	3 June 2025

Rank 9

Symmetry properties
Petrie polygon	Order
624	13946558535000000000000000
Symmetry axes
Symmetry	Count
5♯H8	390000
5♯H7×A1	15136680000
5♯H6×A2	78836875000000
5♯H5×A3	62084039062500000
H4×A4	8070925078125000000
H3×A5	161418501562500000000
H2×A6	276717431250000000000
A1×A7	172948394531250000000
A8	38432976562500000000
Discovery
MinersHavenM43	3 June 2025

Rank 10

Symmetry properties
Petrie polygon	Order
1878	27230655539587500000000000000000
Symmetry axes
Symmetry	Count
5♯H9	1952500
5♯H8×A1	380737500000
5♯H7×A2	9851455900000000
5♯H6×A3	38482249609375000000
5♯H5×A4	24243817253906250000000
H4×A5	2626413535839843750000000
H3×A6	45024232042968750000000000
H2×A7	67536348064453125000000000
A1×A8	37520193369140625000000000
A9	7504038673828125000000000
Discovery
kapzduke	4 June 2025

Summary

Symmetry	5♯Hn	An
H2	5	5
H3	12	20
H4	120	600
5♯H5	650	78000
5♯H6	3150	40950000
5♯H7	15625	91406250000
5♯H8	77624	886914843750000
5♯H9	390000	38432976562500000000
5♯H10	1952500	7504038673828125000000000

General

This table shows the flag counts for the symmetry ^p♯H_n, along with its Petrie polygon P.

n \ p	2	3	5
3	60 P = 5	120 P = 10	120 P = 10
4	7200 P = 15	14400 P = 30	14400 P = 30
5	979200 P = 17	3443212800 P = 80	9360000 P = 26
6	2036736000 P = 65	203039372390400 P = 80	29484000000 P = 63
7	4106059776000 P = 85	P = 410	460687500000000 P = 120
8	134021791088640000 P = 65	P = 3281	35760406500000000000 P = 313
9	4408780839651901440000 P = 255		13946558535000000000000000 P = 624
10	2313728184649317875712000000 P = 1071		27230655539587500000000000000000 P = 1878
11	P = 195		P = 1562
12	P = 5355		P = 7810
13	P = 5115		P = 7810
14	P = 3277		P = 39438
15			P = 26042
16			P = 40638
17			P = 93720