Assembling 10 Rhombic Hexahedrons into a Rhombic Icosahedron

Jen-chung Chuan

In this workshop with Cabri 3D we are to assemble 10 rhombic hexahedrons to form a rhombic icosahedron.

A. Construction of the rhombic icosahedron by taking the convex hull of 10 rhombi:

1) Five rhombi on top: these are the same as five faces on top of the rhombic triacontahedron, obtained by taking the convex hull of a regular dodecahedron with the icosahedron “in dual position”: 2) Five rhombi at the bottom: these are the reflections of the rhombi in 1) w.r.t. the upward translation of the center of triacontahedron by half length of the vertical edge:

3) The required rhombic icosahedron is formed by take the convex hull of the 10 rhombi: B. Construction of two basic building blocks for the rhombic icosahedron: Flat block: the "flat" rhombic hexahedron

Rounded block: the “rounded" rhombic hexahedron Each of the two blocks can be constructed as a prism with a face of rhombic tricontahedron as base.

C. Two possible "growths” associated with blocks asssembly:

Growth 1: Five rounded blocks followed by five flat blocks. 1) Starting with the rounded block, build four other congruent pieces by taking appropriate plane reflections. This completes the growth for the rounded blocks. 2) Build one flat block from the pocket formed by the rounded blocks. This "core" will serve as the only "hidden" pieces after the assembly.

3) Build four other flat blocks by taking appropriate plane reflections of the core. Growth 2: This procedure reminds us of the Greedy Algorithm. 1) Starting with the (flat) core. Build two other identical blocks by taking reflections across two faces. 2) Build up the rounded blocks each sharing a face with the core. This completes the six blocks each sharing a face with the core. 3) Build up the remaining three blocks each sharing an edge with the core.

An interesting thought: the well-known Greedy Algorithm in Computer Science now makes an appearance in the problem of “dissection of rhombic icosahedron”!