Golem (Fattend Skeletons)

Today I want to look at decorations of simplicial graphs. As an example, here is a decoration of last week‘s skeleton:


The vertices of the two skeletons have been replaced by tetrahedra, oriented and scaled so that vertices of neighbors touch. This should explain what I mean by a decoration: A geometric construction that consistently modifies the graph, in order to obtain something with similar symmetries (many) and possibly other desired properties. Another simple and well known decoration is that by Voronoi cells: We replace each vertex by the set of all points that are closer to that vertex than to any other vertex. In this case, the Voronoi cells are truncated octahedra, as shown in another post. Instead of replicating it here, one can also pass to the dual tiling, which is by rhombic dodecahedra. Try viewing it cross eyed.


This is another polyhedral version of the Diamond surface of Schwarz. Like the one obtained from the truncated dodecahedra, the polyhedral approximation shares the conformal structure (by sheer symmetry).

There is another decoration that is quite remarkable:


Here, we have replaced each vertex and each edge of the original graph by an octahedron, properly scaled an oriented. We thus get two fattened skeletons that are congruent and disjoint. All faces are equilateral triangles, and all vertices have valency 6. There even is enough room between them to fit in the truncated octahedra, as one can see in the last image.


This image also shows how to position each octahedron within the cubical lattice: The central octahedron has its vertices along the edges of the lattice, dividing each edge in the ratio 1:3. That ratio ensures that the other three octahedra in the image become in fact regular. More about this next week.

Neovius surface (Algorithmic Geometry IV)

When the truncated octahedron tiles space, the diagonals of the hexagonal faces become part of a line configuration.


Following these lines we can build the bent rhombi that we encountered in Schwarz’ P-surface, but here we will focus on the more complicated bent octagons that weave around the square faces of the truncated octahedra. These serve as Plateau contours for another minimal surface, the Neovius surface, named after the Finnish mathematician Edvard Rudolf Neovius, a student of Hermann Amandus Schwarz.


One can also fill each octagon with four copies of said bent rhombus to obtain an interesting polygonal version of the Neovius surface. Here are two such filled octagons aligned. Note that we have broken a rule: The four bent rhombi that fill the octagon are not rotated about their edges to fit together, but reflected.


Rotating about the edges by 180 degrees will create larger portions of the infinite surface.


Temporarily breaking a rule can sometimes be a good thing.