Hessian polyhedron

Hessian polyhedron
File:Complex polyhedron 3-3-3-3-3.png Orthographic projection (triangular 3-edges outlined as black edges)
Schläfli symbol	3{3}3{3}3
Coxeter diagram	File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png
Faces	27 3{3}3 File:Complex polygon 3-3-3.svg
Edges	72 3{} File:Complex trion.png
Vertices	27
Petrie polygon	Dodecagon
van Oss polygon	12 3{4}2 File:Complex polygon 3-4-2.png
Shephard group	L3 = 3[3]3[3]3, order 648
Dual polyhedron	Self-dual
Properties	Regular

In geometry, the Hessian polyhedron is a regular complex polyhedron ₃{3}₃{3}₃, File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png, in ${\mathbb{ℂ}}^3$. It has 27 vertices, 72 ₃{} edges, and 27 ₃{3}₃ faces. It is self-dual. Coxeter named it after Ludwig Otto Hesse for sharing the Hessian configuration $\left[\begin{array}{cc}9& 4\\ 3& 12\end{array}\right]$ or (9₄12₃), 9 points lying by threes on twelve lines, with four lines through each point. Its complex reflection group is ₃[3]₃[3]₃ or File:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png, order 648, also called a Hessian group. It has 27 copies of File:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png, order 24, at each vertex. It has 24 order-3 reflections. Its Coxeter number is 12, with degrees of the fundamental invariants 3, 6, and 12, which can be seen in projective symmetry of the polytopes. The Witting polytope, ₃{3}₃{3}₃{3}₃, File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png contains the Hessian polyhedron as cells and vertex figures. It has a real representation as the 2₂₁ polytope, File:CDel nodes 10r.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, in 6-dimensional space, sharing the same 27 vertices. The 216 edges in 2₂₁ can be seen as the 72 ₃{} edges represented as 3 simple edges.

Its 27 vertices can be given coordinates in ${\mathbb{ℂ}}^3$: for (λ, μ = 0,1,2).

(0,ω^λ,−ω^μ)

(−ω^μ,0,ω^λ)

(ω^λ,−ω^μ,0)

where $\omega =\frac{-1+i\sqrt{3}}{2}$.

As a Configuration

File:Complex polyhedron 3-3-3-3-3-one-blue-face.png Hessian polyhedron with triangular 3-edges outlined as black edges, with one face outlined as blue.

File:Complex polyhedron 3-3-3-3-3-one-blue-van oss polygon.png One of 12 Van oss polygons, 3{4}2, in the Hessian polyhedron

Its symmetry is given by ₃[3]₃[3]₃ or File:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png, order 648. The configuration matrix for ₃{3}₃{3}₃ is:

$\left[\begin{array}{ccc}27& 8& 8\\ 3& 72& 3\\ 8& 8& 27\end{array}\right]$

The number of k-face elements (f-vectors) can be read down the diagonal. The number of elements of each k-face are in rows below the diagonal. The number of elements of each k-figure are in rows above the diagonal.

L3	File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png	k-face	fk	f0	f1	f2	k-fig	Notes
L2	File:CDel node x.pngFile:CDel 2.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png	( )	f0	27	8	8	3{3}3	L3/L2 = 27*4!/4! = 27
L1L1	File:CDel 3node 1.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel 3node.png	3{ }	f1	3	72	3	3{ }	L3/L1L1 = 27*4!/9 = 72
L2	File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 2.pngFile:CDel node x.png	3{3}3	f2	8	8	27	( )	L3/L2 = 27*4!/4! = 27

Images

These are 8 symmetric orthographic projections, some with overlapping vertices, shown by colors. Here the 72 triangular edges are drawn as 3-separate edges.

Coxeter plane orthographic projections

E6 [12]	Aut(E6) [18/2]	D5 [8]	D4 / A2 [6]
File:Up 2 21 t0 E6.svg (1=red,3=orange)	File:Complex polyhedron 3-3-3-3-3.png (1)	File:Up 2 21 t0 D5.svg (1,3)	File:Up 2 21 t0 D4.svg (3,9)
B6 [12/2]	A5 [6]	A4 [5]	A3 / D3 [4]
File:Up 2 21 t0 B6.svg (1,3)	File:Up 2 21 t0 A5.svg (1,3)	File:Up 2 21 t0 A4.svg (1,2)	File:Up 2 21 t0 D3.svg (1,4,7)

Related complex polyhedra

Double Hessian polyhedron
Schläfli symbol	2{4}3{3}3
Coxeter diagram	File:CDel node 1.pngFile:CDel 4.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png
Faces	72 2{4}3 File:3-generalized-2-orthoplex skew.svg
Edges	216 {} File:Complex dion.png
Vertices	54
Petrie polygon	Octadecagon
van Oss polygon	{6} File:Regular polygon 6.svg
Shephard group	M3 = 3[3]3[4]2, order 1296
Dual polyhedron	Rectified Hessian polyhedron, 3{3}3{4}2
Properties	Regular

The Hessian polyhedron can be seen as an alternation of File:CDel node 1.pngFile:CDel 4.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png, File:CDel node h.pngFile:CDel 4.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png = File:CDel label-33.pngFile:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel label3.png. This double Hessian polyhedron has 54 vertices, 216 simple edges, and 72 File:CDel node 1.pngFile:CDel 4.pngFile:CDel 3node.png faces. Its vertices represent the union of the vertices File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png and its dual File:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node 1.png. Its complex reflection group is ₃[3]₃[4]₂, or File:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 4.pngFile:CDel node.png, order 1296. It has 54 copies of File:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png, order 24, at each vertex. It has 24 order-3 reflections and 9 order-2 reflections. Its coxeter number is 18, with degrees of the fundamental invariants 6, 12, and 18 which can be seen in projective symmetry of the polytopes. Coxeter noted that the three complex polytopes File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png, File:CDel node 1.pngFile:CDel 4.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png, File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 4.pngFile:CDel node.png resemble the real tetrahedron (File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png), cube (File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png), and octahedron (File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png). The Hessian is analogous to the tetrahedron, like the cube is a double tetrahedron, and the octahedron as a rectified tetrahedron. In both sets the vertices of the first belong to two dual pairs of the second, and the vertices of the third are at the center of the edges of the second. Its real representation 54 vertices are contained by two 2₂₁ polytopes in symmetric configurations: File:CDel nodes 10r.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png and File:CDel nodes 01r.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png. Its vertices can also be seen in the dual polytope of 1₂₂.

Construction

The elements can be seen in a configuration matrix:

M3	File:CDel node 1.pngFile:CDel 4.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png	k-face	fk	f0	f1	f2	k-fig	Notes
L2	File:CDel node x.pngFile:CDel 2.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png	( )	f0	54	8	8	3{3}3	M3/L2 = 1296/24 = 54
L1A1	File:CDel node 1.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel 3node.png	{ }	f1	2	216	3	3{ }	M3/L1A1 = 1296/6 = 216
M2	File:CDel node 1.pngFile:CDel 4.pngFile:CDel 3node.pngFile:CDel 2.pngFile:CDel node x.png	2{4}3	f2	6	9	72	( )	M3/M2 = 1296/18 = 72

Images

Orthographic projections

File:Complex polyhedron 2-4-3-3-3.png File:CDel node 1.pngFile:CDel 4.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png polyhedron

File:Complex polyhedron 2-4-3-3-3 blue-edge.png File:CDel node 1.pngFile:CDel 4.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png polyhedron with one face, 2{4}3 highlighted blue

File:Complex polyhedron 2-4-3-3-3-bivertexcolor.png File:CDel node 1.pngFile:CDel 4.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png polyhedron with 54 vertices, in two 2 alternate color

File:Complex polyhedron 3-3-3-4-2-alternated.png File:CDel label-33.pngFile:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel label3.png and File:CDel label-33.pngFile:CDel nodes 01rd.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel label3.png, shown here with red and blue vertices form a regular compound File:CDel node h3.pngFile:CDel 4.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png

Rectified Hessian polyhedron

Rectified Hessian polyhedron
Schläfli symbol	3{3}3{4}2
Coxeter diagrams	File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 4.pngFile:CDel node.png File:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.png or File:CDel label3.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel label-33.png.
Faces	54 3{3}3 File:Complex polygon 3-3-3.svg
Edges	216 3{} File:Complex trion.png
Vertices	72
Petrie polygon	Octadecagon
van Oss polygon	9 3{4}3 File:Complex polygon 3-4-3.png
Shephard group	M3 = 3[3]3[4]2, order 1296 3[3]3[3]3, order 648
Dual polyhedron	Double Hessian polyhedron 2{4}3{3}3
Properties	Regular

The rectification, File:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.png doubles in symmetry as a regular complex polyhedron File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 4.pngFile:CDel node.png with 72 vertices, 216 ₃{} edges, 54 ₃{3}₃ faces. Its vertex figure is ₃{4}₂, and van oss polygon ₃{4}₃. It is dual to the double Hessian polyhedron. It has a real representation as the 1₂₂ polytope, File:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png, sharing the 72 vertices. Its 216 3-edges can be drawn as 648 simple edges, which is 72 less than 1₂₂'s 720 edges.

File:Complex polyhedron 3-3-3-4-2.png File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 4.pngFile:CDel node.png or File:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.png has 72 vertices, 216 3-edges, and 54 File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.png faces

File:Complex polyhedron 3-3-3-4-2-one-blue-face.png File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 4.pngFile:CDel node.png with one blue face, File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.png highlighted

File:Complex polyhedron 3-3-3-4-2-one-blue van oss polygon.png File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 4.pngFile:CDel node.png with one of 9 van oss polygon, File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel 3node.png, 3{4}3, highlighted

Construction

The elements can be seen in two configuration matrices, a regular and quasiregular form.

M₃ = ₃[3]₃[4]₂ symmetry

M3	File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 4.pngFile:CDel node.png	k-face	fk	f0	f1	f2	k-fig	Notes
M2	File:CDel node x.pngFile:CDel 2.pngFile:CDel 3node.pngFile:CDel 4.pngFile:CDel node.png	( )	f0	72	9	6	3{4}2	M3/M2 = 1296/18 = 72
L1A1	File:CDel 3node 1.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel node.png	3{ }	f1	3	216	2	{ }	M3/L1A1 = 1296/3/2 = 216
L2	File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 2.pngFile:CDel node x.png	3{3}3	f2	8	8	54	( )	M3/L2 = 1296/24 = 54

L₃ = ₃[3]₃[3]₃ symmetry

L3	File:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.png	k-face	fk	f0	f1	f2		k-fig	Notes
L1L1	File:CDel 3node.pngFile:CDel 2.pngFile:CDel node x.pngFile:CDel 2.pngFile:CDel 3node.png	( )	f0	72	9	3	3	3{ }×3{ }	L3/L1L1 = 648/9 = 72
L1	File:CDel node x.pngFile:CDel 2.pngFile:CDel 3node 1.pngFile:CDel 2.pngFile:CDel node x.png	3{ }	f1	3	216	1	1	{ }	L3/L1 = 648/3 = 216
L2	File:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node 1.pngFile:CDel 2.pngFile:CDel node x.png	3{3}3	f2	8	8	27	*	( )	L3/L2 = 648/24 = 27
	File:CDel node x.pngFile:CDel 2.pngFile:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.png			8	8	*	27

References

• Coxeter, H. S. M. and Moser, W. O. J.; Generators and Relations for Discrete Groups (1965), esp pp 67–80.
• Coxeter, H. S. M.; Regular Complex Polytopes, Cambridge University Press, (1974).
• Coxeter, H. S. M. and Shephard, G.C.; Portraits of a family of complex polytopes, Leonardo Vol 25, No 3/4, (1992), pp 239–244,