Hessian polyhedron

Orthographic projection
(triangular 3-edges outlined as black edges)
Schläfli symbol 3{3}3{3}3
Coxeter diagram
Faces 27 3{3}3
Edges 72 3{}
Vertices 27
Petrie polygon Dodecagon
van Oss polygon 12 3{4}2
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, , in . 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 or (94123), 9 points lying by threes on twelve lines, with four lines through each point.[1]

Its complex reflection group is 3[3]3[3]3 or , order 648, also called a Hessian group. It has 27 copies of , 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, contains the Hessian polyhedron as cells and vertex figures.

It has a real representation as the 221 polytope, , 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.

Coordinates

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Its 27 vertices can be given coordinates in  : for (λ, μ = 0,1,2).

(0,ωλ,−ωμ)
(−ωμ,0,ωλ)
λ,−ωμ,0)

where  .

As a Configuration

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Hessian polyhedron with triangular 3-edges outlined as black edges, with one face outlined as blue.
 
One of 12 Van oss polygons, 3{4}2, in the Hessian polyhedron

Its symmetry is given by 3[3]3[3]3 or      , order 648.[2]

The configuration matrix for 3{3}3{3}3 is:[3]

 

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       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

Images

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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]
 
(1=red,3=orange)
 
(1)
 
(1,3)
 
(3,9)
B6
[12/2]
A5
[6]
A4
[5]
A3 / D3
[4]
 
(1,3)
 
(1,3)
 
(1,2)
 
(1,4,7)
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Double Hessian polyhedron
Schläfli symbol 2{4}3{3}3
Coxeter diagram      
Faces 72 2{4}3  
Edges 216 {}  
Vertices 54
Petrie polygon Octadecagon
van Oss polygon {6}  
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      ,       =      . This double Hessian polyhedron has 54 vertices, 216 simple edges, and 72     faces. Its vertices represent the union of the vertices       and its dual      .

Its complex reflection group is 3[3]3[4]2, or      , order 1296. It has 54 copies of    , 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      ,      ,       resemble the real tetrahedron (     ), cube (     ), and octahedron (     ). 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.[4]

Its real representation 54 vertices are contained by two 221 polytopes in symmetric configurations:         and        . Its vertices can also be seen in the dual polytope of 122.

Construction

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The elements can be seen in a configuration matrix:

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 = 1296/6 = 216
M2       2{4}3 f2 6 9 72 ( ) M3/M2 = 1296/18 = 72

Images

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Orthographic projections
 
      polyhedron
 
      polyhedron with one face, 2{4}3 highlighted blue
 
      polyhedron with 54 vertices, in two 2 alternate color
 
      and      , shown here with red and blue vertices form a regular compound      

Rectified Hessian polyhedron

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Rectified Hessian polyhedron
Schläfli symbol 3{3}3{4}2
Coxeter diagrams      
      or      .
Faces 54 3{3}3  
Edges 216 3{}  
Vertices 72
Petrie polygon Octadecagon
van Oss polygon 9 3{4}3  
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,       doubles in symmetry as a regular complex polyhedron       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.[5]

It has a real representation as the 122 polytope,        , sharing the 72 vertices. Its 216 3-edges can be drawn as 648 simple edges, which is 72 less than 122's 720 edges.

 
      or       has 72 vertices, 216 3-edges, and 54     faces
 
      with one blue face,     highlighted
 
      with one of 9 van oss polygon,    , 3{4}3, highlighted

Construction

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

M3 = 3[3]3[4]2 symmetry
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
L3 = 3[3]3[3]3 symmetry
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

References

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  1. ^ Coxeter, Complex Regular polytopes, p.123
  2. ^ Coxeter Regular Convex Polytopes, 12.5 The Witting polytope
  3. ^ Coxeter, Complex Regular polytopes, p.132
  4. ^ Coxeter, Complex Regular Polytopes, p.127
  5. ^ Coxeter, H. S. M., Regular Complex Polytopes, second edition, Cambridge University Press, (1991). p.30 and p.47
  • 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,