User:Tomruen/Gosset-Elte figures

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The complete list of Gosset-Elte figures, named after Thorold Gosset and E. L. Elte, represent all the one-end-ringed single-bifurcating groups with all order 2 or 3 mirror dihedral angles.

The polytopes then fall into something like an ADE classification. Basically, the condition for k_ij to exist is that

or equality if you want to include the honeycombs. (and <1 for hyperbolic honeycombs?)

The Coxeter group [3^i,j,k] can generate up to 3 unique uniform Gosset-Elte figures with Coxeter-Dynkin diagrams with one end node ringed. By Coxeter's notation, each figure is represented by k_ij to mean the end-node on the k-length sequence is ringed.

A-family [3^n-1-i,i,0] (Rectified simplices)

Class	Simplex	Rectified	Birectified	...
A1	= 00,0
A2	= 01,0
A3	= 02,0	= 01,1
A4	= 03,0	= 02,1
A5	= 04,0	= 03,1	= 02,2
A6	= 05,0	= 04,1	= 03,2
...	...	...
An	... = 0n-1,0	... = 0n-2,1	...	...... = 0n-1-m,m

B-family [3^n-3,1,1] (demihypercubes)

Class	Demihypercubes	Orthoplexes (Regular)
B3=[31,1,0]	= 11,0
B4=[31,1,1]	= 11,1
B5=[32,1,1]	= 12,1	= 21,1
B6=[33,1,1]	= 13,1	= 31,1
...	...	...
Bn=[3n-3,1,1]	... = 1n-3,1	... = (n-3)1,1

E-family [3^n-4,2,1]

			Semiregular E-polytope
E3=[3-1,2,1]	= 2-1,1	= 12,-1	= (-1)2,1
E4=[30,2,1]	= 20,1	= 12,0	= 02,1
E5=[31,2,1]	= 21,1	= 12,1	= 12,1
E6=[32,2,1]	= 22,1	= 12,2	= 22,1
E7=[33,2,1]	= 23,1	= 13,2	= 32,1
E8=[34,2,1]	= 24,1	= 14,2	= 42,1

T-family (Euclidean honeycombs)

T7=[32,2,2]	= 22,2
T8=[33,3,1]	= 33,1	= 13,3
T9=[35,2,1]	= 25,1	= 15,2	= 52,1

Hyperbolic honeycombs ??

?8=[33,2,2]	= 32,2 (??)	= 23,2 (??)
?9=[34,2,2]	= 42,2 (??)	= 24,2 (??)
?9=[34,3,1]	= 43,1 (??)	= 34,1 (??)	= 14,3 (??)

[edit] See also

• Bimonster

[edit] References

• Gosset, Thorold (1900). "On the regular and semi-regular figures in space of n dimensions". Messenger of Mathematics 29: 43–48. 
• Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen