Poker Faces

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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 \(\phi = \frac{1+\sqrt{5}}{2}\).