Császár Polyhedron

The Hungarian topologist Ákos Császár discovered the only other polyhedron, besides the tetrahedron, that doesn't have diagonals. In other polyhedra, you can drill a hole from one vertex to another through the interior of the polyhedron. In the Császár polyhedron this isn't possible because, each vertex is connected to every other vertex by an edge.

Steps

1. Score the fold lines and cut out the pattern pieces.
   
2. Fold up the first piece and glue tab A to side A.
   
3. Fold the second piece and glue tab B to side B and tab C to side C.
   
4. Glue the two pieces together by gluing tab D on the first piece to side D on the second piece.
   
5. Continue gluing tabs on the first piece to the edges on the second piece. Tab E will match side E.
   

Notes

• The dual of the Császár polyhedron is the Szilassi polyhedron.
  
• You may have heard of Euler's formula relating the number of vertices, faces, and edges of a polyhedron. v + f = e + 2. However, there is a more general equation that works for tori, too: v + f = e + 2 - 2h where h is the number of holes.
  
• If there is another polyhedron without diagonals the next higher number of vertices it can have is 12, with 6 holes through it.
  
• Ákos Császár gave an example of vertices for a Császár polyhedron in his journal article. The vertices he gave are: Vertex 1 (-3, 3, 0); Vertex 2 (-3, -3, 1); 3 (-1, -2, 3); 4 (1, 2, 3) ; 5 (3, 3, 1) ; 6 (3, -3, 0); and 7 (0, 0, 15).
  

Links

• Visit this site for an interactive 3D plot showing the vertices of the Császár polyhedron. It helps you understand how it is constructed.

References

• (Császár 1949), (Gardner, Time Travel and other Mathematical Bewilderments 1988)