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词条 Hessian polyhedron
释义

  1. Coordinates

  2. As a Configuration

  3. Images

  4. Related complex polyhedra

     Construction   Images   Rectified Hessian polyhedron  Construction 

  5. References

Hessian polyhedron

Orthographic projection
(triangular 3-edges outlined as black edges)
Schläfli symbol3{3}3{3}3
Coxeter diagram3node_1|3|3node|3|3node}}
Faces27 3{3}3
Edges72 3{}
Vertices27
Petrie polygonDodecagon
van Oss polygon12 3{4}2
Shephard groupL3 = 3[3]3[3]3, order 648
Dual polyhedronSelf-dual
PropertiesRegular

In geometry, the Hessian polyhedron is a regular complex polyhedron 3{3}3{3}3, {{CDD|3node_1|3|3node|3|3node}}, 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 {{CDD|3node|3|3node|3|3node}}, order 648, also called a Hessian group. It has 27 copies of {{CDD|3node|3|3node}}, 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, {{CDD|3node_1|3|3node|3|3node|3|3node}} contains the Hessian polyhedron as cells and vertex figures.

It has a real representation as the 221 polytope, {{CDD|nodes_10r|3ab|nodes|split2|node|3|node}}, in 4-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

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

(0,ωλ,−ωμ)

(−ωμ,0,ωλ)

(ωλ,−ωμ,0)

where .

As a Configuration


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 by 3[3]3[3]3 or {{CDD|3node|3|3node|3|3node}}, order 648.[2]

The configuation 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 eac k-figure are in rows above the diagonal.

L33node_1|3|3node|3|3node k-facefkf0f1f2k-figNotes
L2 node_x|2|3node|3|3node}}( )f02788 3{3}3 L3/L2 = 27*4!/4! = 27
L1L1 3node_1|2|node_x|2|3node}} 3{ }f13723 3{ } L3/L1L1 = 27*4!/9 = 72
L2 3node_1|3|3node|2|node_x}} 3{3}3f28827( )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]

(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)

Related complex polyhedra

Double Hessian polyhedron
Schläfli symbol2{4}3{3}3
Coxeter diagramnode_1|4|3node|3|3node}}
Faces72 2{4}3
Edges216 {}
Vertices54
Petrie polygonOctadecagon
van Oss polygon{6}
Shephard groupM3 = 3[3]3[4]2, order 1296
Dual polyhedronRectified Hessian polyhedron, 3{3}3{4}2
PropertiesRegular

The Hessian polyhedron can be seen as an alternation of {{CDD|node_1|4|3node|3|3node}}, {{CDD|node_h|4|3node|3|3node}} = {{CDD|label-33|nodes_10ru|split2|node|label3}}. This double Hessian polyhedron has 54 vertices, 216 simple edges, and 72 {{CDD|node_1|4|3node}} faces. Its vertices represent the union of the vertices {{CDD|3node_1|3|3node|3|3node}} and its dual {{CDD|3node|3|3node|3|3node_1}}.

Its complex reflection group is 3[3]3[4]2, or {{CDD|3node|3|3node|4|node}}, order 1296. It has 54 copies of {{CDD|3node|3|3node}}, 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 {{CDD|3node_1|3|3node|3||3node}}, {{CDD|node_1|4|3node|3||3node}}, {{CDD|3node_1|3|3node|4|node}} resemble the real tetrahedron ({{CDD|node_1|3|node|3|node}}), cube ({{CDD|node_1|4|node|3|node}}), and octahedron ({{CDD|node_1|3|node|4|node}}). 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: {{CDD|nodes_10r|3ab|nodes|split2|node|3|node}} and {{CDD|nodes_01r|3ab|nodes|split2|node|3|node}}. Its vertices can also be seen in the dual polytope of 122.

Construction

The elements can be seen in a configuration matrix:

M3node_1|4|3node|3|3node k-facefkf0f1f2k-figNotes
L2node_x|2|3node|3|3node}}( )f0548 8 3{3}3 M3/L2 = 1296/24 = 54
L1A1 node_1|2|node_x|2|3node}} { }f122163 3{ } M3/L1A1 = 1296/6 = 216
M2 node_1|4|3node|2|node_x}} 2{4}3f269 72 ( ) M3/M2 = 1296/18 = 72

Images

Orthographic projections
{{CDD>node_1|4|3node|3|3node}} polyhedron{{CDD>node_1|4|3node|3|3node}} polyhedron with one face, 2{4}3 highlighted blue{{CDD>node_1|4|3node|3|3node}} polyhedron with 54 vertices, in two 2 alternate color {{CDD>label-33|nodes_10ru|split2|node|label3}} and {{CDD|label-33|nodes_01rd|split2|node|label3}}, shown here with red and blue vertices form a regular compound {{CDD|node_h3|4|3node|3|3node}}
{{-}}

Rectified Hessian polyhedron

Rectified Hessian polyhedron
Schläfli symbol3{3}3{4}2
Coxeter diagrams3node_1|3|3node|4|node}}
{{CDD|3node|3|3node_1|3|3node}} or {{CDD|label3|node_1|split1|nodes|label-33}}.
Faces54 3{3}3
Edges216 3{}
Vertices72
Petrie polygonOctadecagon
van Oss polygon9 3{4}3
Shephard groupM3 = 3[3]3[4]2, order 1296
3[3]3[3]3, order 648
Dual polyhedronDouble Hessian polyhedron
2{4}3{3}3
PropertiesRegular

The rectification, {{CDD|3node|3|3node_1|3|3node}} doubles in symmetry as a regular complex polyhedron {{CDD|3node_1|3|3node|4|node}} 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, {{CDD|nodes|3ab|nodes|split2|node|3|node_1}}, sharing the 72 vertices. Its 216 3-edges can be drawn as 648 simple edges, which is 72 less than 122's 720 edges.

{{CDD>3node_1|3|3node|4|node}} or {{CDD|3node|3|3node_1|3|3node}} has 72 vertices, 216 3-edges, and 54 3{4}2 faces{{CDD>3node_1|3|3node|4|node}} with one blue face, 3{3}3 highlighted{{CDD>3node_1|3|3node|4|node}} with one of 9 van oss polygon, 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
M33node_1|3|3node|4|node k-facefkf0f1f2k-figNotes
2>{{CDD|node_x|2|3node|4|node}} ( )f0729 6 3{4}2 M3/M2 = 1296/18 = 72
L1A1 3node_1|2|node_x|2|node}} 3{ }f132162 { } M3/L1A1 = 1296/3/2 = 216
L2 3node_1|3|3node|2|node_x}} 3{3}3f288 54 ( ) M3/L2 = 1296/24 = 54
L3 = 3[3]3[3]3 symmetry
L33node|3|3node_1|3|3node k-facefkf0f1f2k-figNotes
L1L1 3node|2|node_x|2|3node}} ( )f0729333{ }×3{ } L3/L1L1 = 648/9 = 72
L1 node_x|2|3node_1|2|node_x}}3{ }f1321611{ } L3/L1 = 648/3 = 216
L23node|3|3node_1|2|node_x}}3{3}3f28827*( )L3/L2 = 648/24 = 27
{{CDD|node_x|2|3node_1|3|3node}}88*27

References

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,

1 : Polytopes
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