### Compound of 5 Cubes and an Icosahedron

index

This object is:

• 5 cubes of equal size plus an icosahedron with slightly shorter sides.
• A stellation of a truncated icosahedron.
• 12 pentagrams arranged into a dodecahedron plus an icosahedron.

Construction

This construction is made out of garden bamboo sticks, the longest of which are 1½ meters. There are 12x5 long sticks, and 30 shorter sticks, making a total of 90. The 60 long sticks which make up the 5cube compound are separated into 6 loops of 10 which are twisted into pairs of pentagrams, the joints being made of elephant/gaffer tape. The remaining 30 sticks of the icosahedron are made into 3 modules each consisting of 10 sticks, again joined using strong tape. These various parts are held together with garden wire, which is used to securely tie the vertices together, and optionally, the intersections. The whole construction ( after removal of the wire twists ) can be folded into a bundle of sticks that can be thrown in a corner or in the back of your car, to be rebuilt elsewhere.

The rigidity of the icosahedron portion of the structure is normal. The rigidity of the 5 cubes or 12 pentagrams compound follows from the fact that each strut is actually part of a big equilateral triangle. See if you can spot one of these triangles in the animation on the right. Also see if you can spot the 5 cubes and icosahedron flashing by, and some of the pentagrams.

In addition to the obvious patterns in this construction there are a variety of other interesting patterns, pentagons, golden triangles, etcetera..

 Segment Lengths The ratio of the lengths of the long and short sticks is interesting. Suppose the length of the long sticks we can call 1. Suppose g is the small golden  ratio (root(5)+1)/2. The long struts are struts of pentagrams, and by noticing the construction contains two sizes of equilateral triangles, we have.. long strut length = 1 = g + g² = 2g² + g³ short strut length = 2g³ + g² = g + g³ which is nice.

 Here are three stars from the 5 cube dodecahedron in yellow, and a triangle from the icosahedron in pink. You can see how the three yellow stars join in a rigid yellow triangle, and 3 sides of a square.

 This is one of the modules that make up the icosahedron. It is made of ten sticks joined with tape. Three of these tied at right angles to each other make a stable icosahedron.
 Here is a close up of one of the joints of the construction. Messy isn't it.