### The pentagonal coffee table that was mentioned in Dissections: Plane & Fancy, by Greg N. Frederickson:

Dick Ruth (at Shippensburg University, Shippensburg, Pennsylvania) asked to see some pictures of the pentagonal coffee table that I had designed and which was mentioned in my biography. Here are two photos that I made before I installed the glass top. The table consists of a pentagonal frame that holds a bronze casting of a surface consisting of identical rhombuses. It is about a meter across, and the cross braces for the legs form a pentagram. The glass top is clearly a necessity if you want to have a usable coffee table, as you can see.

The second photo gives a fairly good idea how the bronze surface is based on Penrose tiles: Viewed from above, a projection of the surface onto a plane appears to be tiled with two types of rhombuses, those with angles of 36 and 144 degrees, and those with angles of 72 and 108 degrees. The actual surface is covered with rhombuses of just one type, namely those in which the ratio of the lengths of the two diagonals is the golden ratio. The rhombuses are tilted in one of two ways, either with the short diagonal level (yielding the rhombuses whose angles are 72 and 108 degrees in the projected surface) or the long diagonal level (yielding the rhombuses whose angles are 36 and 144 degrees). The vertices of the rhombuses in the actual surface are positioned at just four different heights. The pattern, with its fivefold rotational symmetry, could be extended out as far as you like, retaining the property that just four different heights are used. I am not the first person to discover this design; I was approximately fifteen years too late for that. Exercise: Identify the first person to discover this design.

The lost wax technique was used to form the bronze casting. First I made a full scale model, cutting out over 300 pieces from a sheet of Lexan (a type of plastic). Then I glued the pieces together; to have the tilted rhombuses fit nicely together, I made beveled cuts. The man at the foundry then made a rubber mold of the model and took a wax impression from the mold. He cut the wax impression into four pieces, cast each one, and then welded them together. It would have made more sense to cast five pieces, but he didn't tell me he was going to do it in pieces. He polished the surface by blasting it with crushed walnut shells and then applied a wax coat. Don't ask how much it cost.
I talked a German baptist cabinetmaker into building the wooden frame. He was an industrious and a religious man, and was concerned not about whether the angles would be difficult to cut but rather whether the presence of the pentagram indicated that the table would be used for some sort of occult activities. I assured him that this wasn't going to be the case. He then produced the frame with superb craftsmanship. Don't ask how much it cost.
The glass top was cut out of half inch thick shatterproof glass. Getting the glass-cutter to cut a pentagonal piece that would fit in the frame was a pain. Don't ask how much it cost.
For some insight into its geometry and a related dissection, take a look at George Hart's dissection of the rhombic triacontahedron.
In 2005 I learned of a set of rhombohedra, called "Rhombo Blocks", produced by Michael Longuet-Higgins, that fit almost perfectly on top of the bronze surface. Follow the link for further details.

Back to the photo gallery for Dissections: Plane & Fancy.