Holyhedron!

A Holyhedron is a polyhedron in which every face has a "hole". Until recently it was unknown whether a finite holyhedron existed. Jade Vinson was the first to discover one, with 78,585,627 faces.

A $20 Holyhedron

Here is a holyhedron with 492 faces. John Conway has offered a prize of $10000 divided by the number of faces, so this one should be worth approximately $20.3252.

This holyhedron consists of nine layers. Each image in the first row shows a single one of these layers, while the corresponding image in the second row shows all layers up to and including that one. The final image in the lower right corner is the full holyhedron. Click on any of the small images to view a larger version of it.

12 faces 12*6 = 72 faces 20*4 = 80 faces 30*4 = 120 faces 12*6 = 72 faces 20*4 = 80 faces 4*4 = 16 faces 6*4 = 24 faces 4*4 = 16 faces
mystery1_1layer_100x100.png mystery2_1layer_100x100.png mystery3_1layer_100x100.png mystery4_1layer_100x100.png mystery5_1layer_100x100.png mystery6_1layer_100x100.png mystery7_1layer_100x100.png mystery8_1layer_100x100.png mystery9_1layer_100x100.png
mystery1_100x100.png mystery2_100x100.png mystery3_100x100.png mystery4_100x100.png mystery5_100x100.png mystery6_100x100.png mystery7_100x100.png mystery8_100x100.png mystery9_100x100.png
Here is a cutaway view of the entire holyhedron (click on it for a larger image).
mysteryCutaway_100x100.png

The excellent program JavaView was used to model the holyhedron and to render these images.

The following graph shows the "piercing relation" among the layers; it is a schematic of which parts pierce which other parts of the holyhedron, and may be helpful in understanding what is going on in the pictures above. The nine nodes in the graph represent the nine layers of the holyhedron, colored the same as in the above images. The label on each graph edge is the number of piercings that the "from" layer makes into the "to" layer. The two graph edges leading nowhere from the High Spikes nodes represent the eight vertices on the convex hull of the holyhedron, which don't pierce anything.
PiercingGraph.png

Postscript version (suitable for printing)


Here are some vrml models of it. The "exploding" ones may or may not work in any particular vrml viewer; sorry, that's life.
Here are some other references to holyhedra on the web:


?!
Last Modified: Fri May 23 23:15:29 PDT 2003
Don Hatch
hatch@plunk.org
Valid HTML 4.01!
Back to Don Hatch's home page.