Root-Two Rhombus Polypanel by HT

Description

The "Root-Two Rhombus" has the same side lengths but different internal angles to the polypanel rhombus created by Devin Montes, and this is an important difference because it allows this rhombus to play more nicely with copies of itself in three dimensions in the sense that there are many more ways they can be joined together without ending up with a gap that nothing fits into, which is a problem the other rhombus is prone to. I call this polypanel the "Root-Two Rhombus" because it has diagonals with lengths in the ratio 1:√2. It's internal angles then work out to be approximately 70.5 and 109.5 degrees. By comparison, the original rhombus has internal angles of 120 and 60 degrees. [Update: I have now also produced three other polypanel rhombuses with different internal angles, which you can find under other listings (Golden Rhombus Polypanel and Polypanel Penrose Puzzle).] One of the shapes you can make out of the Root-Two Rhombus is a rhombic dodecahedron, which is a nice symmetrical shape that is interesting mathematically because it tiles space. Twelve Root-Two Rhombuses are required to build a small rhombic dodecahedron that is about 10 cm wide. To create a rhombic dodecahedron that is twice as large, you can instead use four of these rhombuses to construct each face, requiring a total of 48 rhombuses. To create one that is three times as large, you can use nine rhombuses to construct each face, requiring a total of 108 rhombuses, and so on. The faces remain relatively…