Additions to

Zome Geometry

George W. Hart   and  Henri Picciotto

Ten Triangular Prisms

It is very difficult, but possible, to weave ten triangular prisms through each other to make a fascinating compound with icosahedral symmetry.  This is one of the most frustrating Zome constructions I have ever made, so don't start this unless you have some free time.  But it is very satisfying when you finally complete it.

Four Triangular Prisms

We will begin with a simpler compound of just four triangular prisms to build up some skills and intuition.  Consider a regular octahedron.  The eight triangular faces come in four opposite parallel pairs.  But a pair of opposite faces are not arranged like the two bases of a prism.  Look along a 3-fold axis to see how two opposite faces are off by 60 degrees (like the bases of an antiprism).  Equivalently, you could say they are off by 180 degrees.  However, if we superimposed another triangle over each of the octahedron's faces, and rotated each new triangle 30 degrees clockwise (as seen from outside the polyhedron), then opposite pairs of these new triangles would be arranged like bases of a prism.  We could join opposite pairs to form an intersecting arrangement of four triangular prisms.

(The result is chiral; you can get its enantiomorph by rotating all the new triangles 30 degrees counterclockwise instead.  You may also prefer to think of these rotations as 90 degrees clockwise or counterclockwise, so two sum to correct the 180 degree offset.  This makes it more straightforward to see that each edge of each new triangle is perpendicular to some edge of the original triangle.)

Four Prisms (4-fold axis)
Four Prisms (3-fold axis)
Four Prisms (2-fold axis)

The picture above shows three views of the compound of four triangular prisms.  Each prism here consists of two b3 equilateral bases and three y2 connecting edges.  The blue struts have to bend slightly, locking each prism with the other three. The structure has four separate prism components.  The Zomeballs in the four are not parallel and so can not be connected.  So you can not just build some Zome scaffolding and then remove superfluous supports.  You have to do some "weaving." Perhaps with four people this would be simpler to assemble; I have only done these solo.

The fact that the struts rub against each other means that the size of the  prism is crucial.  With other size prisms, there is too much space between the struts, and the four are free to move.  I came upon this size by trial and error.

Find the three 4-fold axes, four 3-fold axes, and six 2-fold axes.  Also notice how the 24 vertices of the result are positioned approximately as in the snub cube.  The vertices are equivalent in the sense that you can rotate the compound to bring any chosen vertex into the position of any other chosen vertex while keeping the entire compound appearing unchanged.  Thus, this is a "vertex-uniform" compound.  If it were possible to build this with squares instead of rectangles, this would be a uniform compound.

Ten Triangular Prisms

Now on to the compound of ten triangular prisms, which is also vertex uniform.  The same idea applies, but start by considering an icosahedron.  The icosahedron has ten pairs of opposite faces, and each pair has a  60 (or 180) degree offset.  Twenty new triangles placed over the icosahedron faces can be rotated 30 (or 90) degrees clockwise (or counterclockwise) to be the bases of ten prisms.
Ten Prisms (5-fold axis)
Ten Prisms (3-fold axis)
Ten Prisms (2-fold axis)

This can be built very nicely if you use ten tall prisms that have b1 equilateral bases and three y3 parallel edges. There is a slight amount of free play that allows the prisms to slide loosely in and out, roughly 1 cm.

Again, no Zome scaffolding is possible, so you just have to weave.  The images above should help.  The analysis below might help.  The assembly took a long time for me, but maybe you can find a simpler way to make it.  Perhaps it is easier with a group of ten people?  Good luck.

The 30 Yellow Struts

It may be helpful to understand the arrangement of the 30 yellow struts in this compound.  Consider the 30 edges of a dodecahedron.  Let us imagine that these edges are not connected to each other at the vertices, but somehow attached to the 2-fold axes that pass through their centers.  Imagine we can rotate each edge like a propeller on its own axis, but we require all 30 edges rotate the same amount.  Stand the dodecahedron up on a vertex, so a 3-fold axis is vertical, and around the "equator" of the dodecahedron there are six edges which are almost vertical.  (If extended, they meet to form a regular skew hexagon.)  A slight rotation of all 30 edges makes three of these six exactly vertical, parallel to the vertical 3-fold axis and each other.  By symmetry, all 30 edges become aligned (in groups of three) with the ten 3-fold axes.  The required rotation is small: the blue-yellow angle of about 21 degrees.  Extending these rotated edges in both directions gives the arrangement of the yellow struts in the compound of ten prisms.  In the model, a pentagram-like arrangement can be seen behind each 5-fold opening; each corresponds to the rotated edges of one pentagon of the underlying dodecahedron.

Related Compounds

It may or may not be useful to try the October 2000 compounds first.  The compound of six pentagonal prisms shown there is an analogous construction starting from the dodecahedron.

There is another related compound, starting with the cube and rotating the faces 45 degrees.  But it is not chiral, so unlike the above, some edges have to pass exactly through each other. This happens when an edge crosses a mirror plane at an angle, and therefor must cross its reflection.  So it can not be made by this technique.  However, a version can be made with a Zomeball at those crossing points, and is given in the book, on p. 189.