Penrose found that there are two simple shapes that you can use to fill a space as large as you like and which have five-fold axes of symmetry. The shapes are built from 6 flat faces which are , that is, shapes with all sides of equal length (like a square) and which has oppopsite sides paralled (again like a square), but which does not have all its angles equal - so they are diamond shaped (rhombs, rhombuses). The Penrose tiling shown on the Flat Phi page is also made from two rhombuses and fills theplane with a fivefold symmetric pattern.
For the solid shapes, the faces are all diamonds (rhombs) but not the ones used in the Penrose tiling and pentagons and pentagrams. The surprising relationship that holds for these new rhombuses is that the ratio of the two diagonals of the diamonds (rhombuses) is Phi! <j>
So this is a different rhomb from the Penrose rhombs and we shall call it the golden rhomb.
This makes the semi-angles (half the angles inside the rhombus) have tangents of Phi and phi so the angles of the rhombus are 2x31717474..° = 2x055357435889r and 2x58282525588° = 2x1 0172219674r.
[The angles in the rhombs in the Penrose tiling are 2/5 pi and 3/5 pi (72° and 108°) in one and 1/5 pi and 4/5 pi (36° and 144°) in the other.]
The two solids are similar to a cube but the faces are golden rhombs. The first shape is made by attaching three golden rhombs at their shorter angles in the same way as three squares meet at a corner of a cube. A duplicate is made and the two fit together to make a six-sided shape like a slanted cube. This is called a prolate rhombohedron.
The other shape is made by joining three golden rhombs together in the same way but at the larger angles this time. A duplicate of this is again fitted to make a different six-sided cube-like shape. This is called an oblate rhombohedron.
The two shapes look like cubes leaning over to one side.
Take a large number of one of these shapes and you can indeed fill as large a space as you like with them. When stacking cubes or octahedra, all the shapes are aligned identically (look identical, with the same orientation). When we use a rhombohedron, some must be turned round to fit in with others. These also occur in nature, although only discovered since the 1950's and, because they are not quite as symmetrical as crystals, as called quasi-crystals.
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