# Matching Rhombs on Polyrhons

2020/11/12
If you color the sides of rhombs using four different colors you get 12 distinct pieces. Unfortunately a single rhombic dodecahedron, rhon for short, cannot be covered by these pieces with matching colors at the edges.

Therefore I looked for sets of pieces with other restrictions or with all combinations of colors allowed. This way various polyrhons can be covered with these sets. The table shows how many pieces are available for a given property and a given number of colors. Click the properties to see some constructions with the yellow marked sets. Click the pictures of the constructions to turn them around as virtual objects.

Properties of Rhombs Colors
2 3 4 5 6 7 8 n
All Color Combinations
Allowed
10 45 136 325 666 1225 2080 (n^4+n^2)/2
No Same Colored Pieces 8 42 132 320 660 1218 2072 (n^4+n^2)/2-n
3:1 Coloring 4 12 24 40 60 84 112 n*(n-1)*2
All Sides Differently Colored 0 0 12 60 180 420 840 n*(n-1)(n-2)(n-3)/2

A physical object is shown above. First six dodecahedrons were printed and glued together. Then iron foil was stuck on the 60 faces, and the 60 rhombs were printed on magnetic foil. Now you can easily move the rhombs on the surface of the ring to get matching colors at the edges. It would have been better to have a snapping mechanism to fix the pieces.