User:Tomruen/complex polytopes

Regular and uniform complex polytopes

Complex Polygons (C2)

	5{}+{}
=
=
	=

The complex reflection group is _p[q]_r, order $g=8/q\cdot (1/p+2/q+1/r-1)^{-2}$ has, configuration matrix: ${\begin{bmatrix}{\begin{matrix}g/r&r\\p&g/p\end{matrix}}\end{bmatrix}}$

= (Order 2p² and p²) - Related to p-p duoprisms

Regular G(p,1,2) k-face fk f0 f1 k-fig Notes A1 ( ) f0 p2 2 { } G(p,1,2)/A1 = 2p2/2 = p2 p[ ] p{ } f1 p 2p ( ) G(p,1,2)/p[ ] = 2p2/p = 2p

Quasiregular p[]p[] k-face fk f0 f1 k-fig Notes ( ) f0 p2 1 1 { } p[]p[] = p2 p[] p{ } f1 p p * ( ) p[]p[]/p[] = p p[] p{ } p * p p[]p[]/p[] = p

(order pq) - related to p-q duoprism

Quasiregular

p[]q[]		k-face	fk	f0	f1		k-fig	Notes
		( )	f0	pq	1	1	{ }	p[]q[] = pq
p[]		p{ }	f1	p	q	*	( )	p[]q[]/p[] = q
q[]		q{ }		q	*	p		p[]q[]/q[] = p

= (order 18 and 9) - related to 3-3 duoprism

Regular M2 k-face fk f0 f1 k-fig Notes A1 ( ) f0 9 2 { } M2/A1 = 18/2 = 9 L1 3{ } f1 3 6 ( ) M2/L1 = 18/3 = 6

Quasiregular L2 1 k-face fk f0 f1 k-fig Notes ( ) f0 9 1 1 { } L2 1 = 9 L1 3{ } f1 3 3 * ( ) L2 1/L1 = 9/3 = 3 3 * 3

(order 6) - related to triangular prism

Quasiregular

L1A1		k-face	fk	f0	f1		k-fig	Notes
		( )	f0	6	1	1	{ }	L1A1 = 6
L1		3{ }	f1	3	2	*	( )	L1A1/L1 = 6/3 = 2
A1		{ }		2	*	3		L1A1/A1 = 6/2 = 3

(Order 18) - related 3-3 duopyramid

Regular

M2		k-face	fk	f0	f1	k-fig	Notes
L1		( )	f0	6	3	3{ }	M2/L1 = 18/3 = 6
A1		{ }	f1	2	9	( )	M2/A1 = 18/2 = 9

(Order 18)

Quasiregular

M2		k-face	fk	f0	f1		k-fig	Notes
		( )	f0	18	1	1	{ }	M2 = 18
A1		{ }	f1	2	9	*	( )	M2/A1 = 18/2 = 9
L1		3{ }		3	*	6		M2/L1 = 18/3 = 6

Möbius–Kantor polygon = , (order 24)

Regular

L2		k-face	fk	f0	f1	k-fig	Notes
L1		( )	f0	8	3	3{ }	L2/L1 = 4!/3 = 8
		3{ }	f1	3	8	( )

= (order 48 and 24)

Regular G6 k-face fk f0 f1 k-fig Notes A1 ( ) f0 24 2 { } G6/A1 = 48/2 = 24 L1 3{ } f1 3 16 ( ) G6/L1 = 48/3 = 16

Quasiregular L2 k-face fk f0 f1 k-fig Notes ( ) f0 24 1 1 { } L2 = 24 L1 3{ } f1 3 8 * ( ) L2/L1 = 24/3 = 8 3 * 8

Complex polyhedra (C3)

There are 9 unique regular and uniform complex polyhedra from 14 Wythoff constructions (ringed patterns) in the L₃ and M₃ Shephard groups. These polyhedra can be seen a complex analogues of tetrahedral symmetry and octahedral symmetry of the regular tetrahedron, cube, and octahedron.

Type	L3 = , order 648		M3 = , order 1296
Regular	=	(27,72,27)		(54,216,72)	=	(72,216,54)
Truncation	=	(27,72+216,27+27)		(648,216+432,72+72)	=	(648,216+432,72+72)
Quasiregular	=	(27,216,54+54)	=		(216,432,54+72)
Cantellation	=	(216,216+216,27+27+72)			(216,216+432,54+72)
Cantitruncation	=	(648,216+216+216,27+27+72)			(1296,432+432+648,54+54+216)

Hessian polyhedron

= - analogous to real tetrahedron

Regular

L3		k-face	fk	f0	f1	f2	k-fig	Notes
L2		( )	f0	27	8	8	3{3}3	L3/L2 = 27*4!/4! = 27
L1L1		3{ }	f1	3	72	3	3{ }	L3/L1L1 = 27*4!/9 = 72
L2		3{3}3	f2	8	8	27	( )	L3/L2 = 27*4!/4! = 27

Rectified Hessian polyhedron

= - analogous to real octahedron

Regular M3 k-face fk f0 f1 f2 k-fig Notes M2 ( ) f0 72 9 6 3{4}2 M3/M2 = 1296/18 = 72 L1A1 3{ } f1 3 216 2 { } M3/L1A1 = 1296/3/2 = 216 L2 3{3}3 f2 8 8 54 ( ) M3/L2 = 1296/24 = 54

Quasiregular L3 k-face fk f0 f1 f2 k-fig Notes L1L1 ( ) f0 72 9 3 3 3{ }×3{ } L3/L1L1 = 648/9 = 72 L1 3{ } f1 3 216 1 1 { } L3/L1 = 648/3 = 216 L2 3{3}3 f2 8 8 27 * ( ) L3/L2 = 648/24 = 27 8 8 * 27

Truncated Hessian polyhedron

= - analogous to real truncated tetrahedron

Truncated

L3		k-face	fk	f0	f1		f2		k-fig	Notes
L1		( )	f0	27	1	3	3	3		L3/L1 = 648/24 = 27
L1L1		3{ }	f1	3	72	*	3	0		L3/L1L1 = 648/3/3 = 72
L1				3	*	216	1	2		L3/L1 = 648/3 = 216
L2		t(3{3}3)	f2	24	8	8	27	*	( )	L3/L2 = 648/24 = 27
		3{3}3		8	0	8	*	27

Cantellated Hessian polyhedron

= - analogous to real cuboctahedron

Cantellated L3 k-face fk f0 f1 f2 k-fig Notes L1 ( ) f0 216 1 3 3 3 0 3{ }×{ } L3/L1 = 648/3 = 216 3{ } f1 3 216 * 2 0 0 { } 3 * 216 1 1 0 L2 3{3}3 f2 8 8 0 27 * * ( ) L3/L2 = 648/24 = 27 L1L1 3{ }×3{ } 9 3 3 * 72 * L3/L1L1 = 648/9 = 72 L2 3{3}3 8 0 8 * * 27 L3/L2 = 648/24 = 27

Rectified M3 k-face fk f0 f1 f2 k-fig Notes L1A1 ( ) f0 216 6 3 2 3{ }×{ } M3/L1A1 = 1296/6 = 216 L1 3{ } f1 3 432 1 1 { } M3/L1 = 1296/3 = 432 L2 3{3}3 f2 8 8 54 * ( ) M3/L2 = 1296/24 = 54 M2 3{4}2 9 6 * 72 M3/M2 = 1296/18 = 72

Cantitruncated Hessian polyhedron

= - analogous to real truncated octahedron

Truncated M3 k-face fk f0 f1 f2 k-fig Notes A1 ( ) f0 648 ? ? ? ? M3/L1 = 1296/2 = 648 L1A1 3{ } f1 3 216 * ? ? M3/L1A1 = 1296/3/2 = 216 L1 3 * 432 ? ? M3/L1 = 1296/3 = 432 L2 t(3{3}3) f2 24 8 8 54 * ( ) M3/L2 = 1296/24 = 54 M2 3{4}2 9 0 6 * 72 M3/M2 = 1296/48 = 27

Cantitruncated L3 k-face fk f0 f1 f2 k-fig Notes ( ) f0 648 ? ? ? ? ? ? L3 = 648 L1 3{ } f1 3 216 * * ? ? ? L3/L1 = 648/3 = 216 3 * 216 * ? ? ? 3 * * 216 ? ? ? L2 3{3}3 f2 24 8 8 0 27 * * ( ) L3/L2 = 648/24 = 27 L1L1 3{ }×3{ } 9 3 0 3 * 72 * L3/L1/L1 = 648/3/3 = 72 L2 3{3}3 24 0 8 8 * * 27 L3/L2 = 648/24 = 27

Double Hessian polyhedron

Double Hessian polyhedron - analogous to real cube

Regular

M3		k-face	fk	f0	f1	f2	k-fig	Notes
L2		( )	f0	54	8	8	3{3}3	M3/L2 = 1296/24 = 54
L1A1		{ }	f1	2	216	3	3{ }	M3/L1A1 = 1296/3/2 = 216
M2		2{4}3	f2	6	9	72	( )	M3/M2 = 1296/18 = 72

Truncated double Hessian polyhedron

- analogous to real truncated cube

Truncated

M3		k-face	fk	f0	f1		f2		k-fig	Notes
L1		( )	f0	648	?	?	?	?		M3/L1 = 1296/3 = 432
L1A1		{ }	f1	2	216	*	?	?		M3/L1A1 = 1296/6 = 216
L1		3{ }		3	*	432	?	?		M3/L1 = 1296/3 = 432
M2		t(3{4}2)	f2	24	8	8	72	*	( )	M3/M2 = 1296/18 = 72
L2		3{3}3		8	0	8	*	72		M3/L2 = 1296/24 = 54

Cantellated double Hessian polyhedron

- analogous to real rhombicuboctahedron

Cantellated

M3		k-face	fk	f0	f1		f2			k-fig	Notes
L1		( )	f0	216	1	3	3	3	0		M3/L1 = 1296/3 = 216
A1		{ }	f1	3	648	*	2	0	0	{ }	M3/A1 = 1296/2 = 648
L1		3{ }		3	*	216	1	1	0		M3/L1 = 1296/3 = 216
M2		3{4}2	f2	9	6	0	72	*	*	( )	M3/M2 =1296/18 = 72
L1A1		3{ }×{ }		6	3	2	*	216	*		M3/L1A1 = 1296/6 = 216
L2		3{3}3		8	0	8	*	*	54		M3/L2 = 1296/24 = 54

Cantitruncated double Hessian polyhedron

- analogous to real truncated cuboctahedron

Cantitruncated

M3		k-face	fk	f0	f1			f2			k-fig	Notes
		( )	f0	1296	?	?	?	?	?	?		M3 = 1296
L1		3{ }	f1	3	432	*	*	?	?	?		M3/L1 = 1296/3 = 432
				3	*	432	*	?	?	?
A1		{ }		3	*	*	648	?	?	?		M3/A1 = 1296/2 = 648
L2		t(3{3}3)	f2	24	8	8	0	54	*	*	( )	M3/L2 = 1296/24 = 54
L1A1		3{ }×{ }		6	3	0	2	*	216	*		M3/L1/A1 = 1296/6 = 216
M2		t(3{4}2)		18	0	9	6	*	*	27		M3/M2 = 1296/48 = 27

Witting polytope (C4)

Witting polytope - - Real representation 4₂₁ polytope

L4		k-face	fk	f0	f1	f2	f3	k-fig	Notes
L3		( )	f0	240	27	72	27	3{3}3{3}3	L4/L3 = 216*6!/27/4! = 240
L3L1		3{ }	f1	3	2160	8	8	3{3}3	L4/L3L1 = 216*6!/4!/3 = 2160
		3{3}3	f2	8	8	2160	3	3{ }
L3		3{3}3{3}3	f3	27	72	27	240	( )	L4/L3 = 216*6!/27/4! = 240

- Honeycomb of Witting polytope: L5 is order 155520N - Real representation 5₂₁ honeycomb

L5		k-face	fk	f0	f1	f2	f3	f4	k-figure	Notes
L4		( )	f0	N	240	2160	2160	240	3{3}3{3}3{3}3	L5/L4 = N
L3L1		3{ }	f1	3	80N	27	72	27	3{3}3{3}3	L5/L3L1 = NL4/L3L1 = 80N
L2L2		3{3}3	f2	8	8	270N	8	8	3{3}3	L5/L2L2 = NL4/L2L2 = 270N
L3L1		3{3}3{3}3	f3	27	72	27	80N	3	3{ }	L5/L3L1 = NL4/L3L1 = 80N
L4		3{3}3{3}3{3}3	f4	240	2160	2160	240	N	( )	L5/L4 = NL4/L4 = N