# Poker Faces

A prismatic configuration of playing cards.

Designed and constructed in July, 2009.

A view down a 5-fold axis of symmetry.

A view down a 3-fold axis of symmetry.

A view down a 2-fold axis of symmetry.

Thirty standard playing cards are carefully slit and assembled into a configuration of six intersecting pentagonal prisms. Four of the prisms display royal flushes, the luckiest hands in poker. This sculpture is quite lucky geometrically as well: it is only possible because the aspect ratio of standard poker cards, namely $$88 \text{mm}/63 \text{mm} \approx 1.397$$, is sufficiently close to $$\frac{5-\sqrt{5}}{2} \approx 1.382$$. This number is the unique rectangle size that allows the prisms' edges to meet at 12 five-way junctures, as in this picture above. (Prove this!)

The interior of Poker Faces is a hollow Rhombic Triacontahedron. This shape can be understood as the convex hull of an icosahedron and a dodecahedron situated so that their edges bisect each other: indeed, the 12 five-way card junctures form the vertices of an icosahedron, and likewise the 20 three-way junctures form a dodecahedron, and these points are exactly the vertices of the interior rhombic triacontahedron. Each face is a "golden rhombus" whose diagonals have proportion equal to the golden ratio $$\phi = \frac{1+\sqrt{5}}{2}$$.

I made a blue Poker Faces, shown below, for the Renown Children's Hospital in Reno. It is part of a fabulous display featuring a Wonderland-esque cabinet surrounded by the Harry Potter book covers.

A blue instance of Poker Faces made for the Renown Children's Hospital.

### Instructions

If you wish to make your own Poker Faces, here are the cutting specifications. I leave assembly as a challenging puzzle.

Blueprints for cuts needed to build Poker Faces. Cut on the bold lines only; the dashed reference lines indicate the golden rhombus face. You will need 30 units, made from playing cards or heavy cardstock.