Hessian polyhedronOrthographic projection(triangular 3-edges outlined as black edges)Schläfli symbol3{3}3{3}33(24)3(24)3Coxeter diagramFaces27 3{3}3 Edges72 3{} Vertices27Petrie polygonDodecagonvan Oss polygon12 3{4}2 Shephard groupL3 = 333, order 648Dual polyhedronSelf-dualPropertiesRegular 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.Coxeter, Complex Regular polytopes, p.123
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, 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 Its 27 vertices can 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 is given by 333 or , order 648.Coxeter Regular Convex Polytopes, 12.5 The Witting polytope
The configuration matrix for 3{3}3{3}3 is:Coxeter, Complex Regular polytopes, p.132
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-facefkf0f1f2k-figNotesL2 ( ) f02788 3{3}3 L3/L2 = 27*4!/4! = 27L1L1 3{ } f13723 3{ } L3/L1L1 = 27*4!/9 = 72L2 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 projectionsE6Aut(E6)[18/2]D5D4 / A2(1=red,3=orange)(1)(1,3)(3,9)B6[12/2]A5A4A3 / D3(1,3)(1,3)(1,2)(1,4,7)
Related complex polyhedra Double Hessian polyhedronSchläfli symbol2{4}3{3}32(18)3(24)3Coxeter diagramFaces72 2{4}3 Edges216 {} Vertices54Petrie polygonOctadecagonvan Oss polygon{6} Shephard groupM3 = 332, order 1296Dual polyhedronRectified Hessian polyhedron, 3{3}3{4}2PropertiesRegular 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.Coxeter, Complex Regular Polytopes, p.127
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: M3 k-facefkf0f1f2k-figNotesL2( ) f0548 8 3{3}3 M3/L2 = 1296/24 = 54L1A1 { }f122163 3{ } M3/L1A1 = 1296/6 = 216M2 2{4}3f269 72 ( ) M3/M2 = 1296/18 = 72
Images
+ 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 Rectified Hessian polyhedronSchläfli symbol3{3}3{4}23(24)3(18)2Coxeter diagrams or . Faces54 3{3}3 Edges216 3{} Vertices72Petrie polygonOctadecagonvan Oss polygon9 3{4}3 Shephard groupM3 = 332, order 1296333, order 648Dual polyhedronDouble Hessian polyhedron2{4}3{3}3PropertiesRegular 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.Coxeter, H. S. M., Regular Complex Polytopes, second edition, Cambridge University Press, (1991). p.30 and p.47
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 The elements can be seen in two configuration matrices, a regular and quasiregular form. + M3 = 332 symmetryM3 k-facefkf0f1f2k-figNotesM2 ( ) f0729 6 3{4}2 M3/M2 = 1296/18 = 72L1A1 3{ }f132162 { } M3/L1A1 = 1296/3/2 = 216L2 3{3}3f288 54 ( ) M3/L2 = 1296/24 = 54
+ L3 = 333 symmetryL3 k-facefkf0f1f2k-figNotesL1L1 ( ) f0729333{ }×3{ } L3/L1L1 = 648/9 = 72L1 3{ }f1321611{ } L3/L1 = 648/3 = 216L2 3{3}3f28827*( ) L3/L2 = 648/24 = 2788*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,