Hessian polyhedron


In geometry, the Hessian polyhedron is a regular complex polyhedron 333,, in. It has 27 vertices, 72 3 edges, and 27 33 faces. It is self-dual.
Coxeter named it after Ludwig Otto Hesse for sharing the Hessian configuration or, 9 points lying by threes on twelve lines, with four lines through each point.
Its complex reflection group is 333 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, 3333, contains the Hessian polyhedron as cells and vertex figures.
It has a real representation as the 221 polytope,, 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 can be given coordinates in : for.
where.

As a Configuration

Its symmetry is given by 333 or, order 648.
The configuation matrix for 333 is:
The number of k-face elements 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.
L3k-facefkf0f1f2k-figNotes
L2f0278833L3/L2 = 27*4!/4! = 27
L1L13f137233L3/L1L1 = 27*4!/9 = 72
L233f28827L3/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.
E6
Aut
D5
D4 / A2




B6
A5
A4
A3 / D3




Related complex polyhedra

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 332, 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.
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

The elements can be seen in a configuration matrix:
M3k-facefkf0f1f2k-figNotes
L2f0548833M3/L2 = 1296/24 = 54
L1A1f1221633M3/L1A1 = 1296/6 = 216
M223f26972M3/M2 = 1296/18 = 72

Images


polyhedron
polyhedron with one face, 23 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

The rectification, doubles in symmetry as a regular complex polyhedron with 72 vertices, 216 3 edges, 54 33 faces. Its vertex figure is 32, and van oss polygon 33. It is dual to the double Hessian polyhedron.
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 33 faces
with one blue face, 33 highlighted
with one of 9 van oss polygon, 33, highlighted

Construction

The elements can be seen in two configuration matrices, a regular and quasiregular form.
M3k-facefkf0f1f2k-figNotes
f0729632M3/M2 = 1296/18 = 72-
L1A13f132162M3/L1A1 = 1296/3/2 = 216
L233f28854M3/L2 = 1296/24 = 54