A Look at the Zome Ball:
Rectangles, Triangles and Pentagons
- by Paul Hildebrandt


Why are the blue struts and holes shaped like a Golden Rectangle?

The cross-section of the blue strut is a Golden Rectangle. This means that the short edge of the rectangle is in the Divine Proportion to the long edge of the rectangle.

The Golden Rectangle (as used in Zome) is the number 2 in space. It has 2 sets of 2 parallel sides, i.e. 2x2=4 sides and 2x2=4 corners.

The Golden Rectangle has reflection symmetry in 2 directions. This means you can place a mirror on the Golden Rectangle in 2 different directions and the image will still look like a Golden Rectangle.

The Golden Rectangle also has 2-fold rotational symmetry. This means you can rotate the Golden Rectangle 180 degrees and it will always line line up with itself.

When viewed along a blue strut, many symmetrical models built with Zome will exhibit 2-fold symmetries like the Golden Rectangle. For example, a shadow of the dodecahedron model, made in such a way that the blue strut inserted in one of the balls casts the minimum shadow, has 2-fold rotational and 2-fold reflection symmetry.

Blue planes are perpendicular to blue struts. 2-dimensional models built in the blue plane will often exhibit 2-fold symmetry. The dodecahedron shadow can be built in the blue plane. This called a true 2-dimensional projection of the dodecahedron along a 2-fold axis of symmetry.


Why are the yellow struts and holes shaped like a triangle?

The cross-section of the yellow strut is an equilateral triangle. This means that all 3 edges of the triangle are the same length (and the same length as the long edge of the blue strut's cross-section) and the angle at all 3 corners is the same.

The equilateral triangle has reflection symmetry in 3 directions. This means you can place a mirror on the equilateral triangle in 3 different directions and the image will still look like an equilateral triangle.

The equilateral triangle also has 3-fold rotational symmetry. This means you can rotate the equilateral triangle 120 degrees and it will always line up with itself.

When viewed along a yellow strut, many symmetrical models built with Zome will exhibit 3-fold symmetries like the equilateral triangle. For example, a shadow of the dodecahedron model, made in such a way that a yellow strut inserted in one of the balls casts the minimum shadow, has 3-fold rotational and 3-fold reflection symmetry.

Yellow planes are perpendicular to yellow struts. 2-dimensional models built in the yellow plane will often exhibit 3-fold symmetry. The 3-fold dodecahedron shadow cannot be built in the yellow plane using Zome parts. However, a 3-fold shadow of the rhombic triacontahedron (a polyhedron with 30 faces built of all red struts) can be built. The 3-fold dodecahedron shadow is implied in this model (see dotted lines).


Why are the red struts and holes shaped like a pentagon?

The cross-section of the red strut is an equilateral pentagon. This means that all 5 edges of the pentagon are the same length (and the same length as the short edge of the blue strut's cross-section) and the angle at all 5 corners is the same.

The equilateral pentagon (as used in Zome) represents the number 5 in space. It has 5 equal sides and 5 corners.

The equilateral pentagon has reflection symmetry in 5 directions. This means you can place a mirror on the equilateral pentagon in 5 different directions and the image will still look like an equilateral pentagon.

The equilateral pentagon also has 5-fold rotational symmetry. This means you can rotate the equilateral pentagon 72 degrees and it will always line up with itself.

When viewed along a red strut, many symmetrical models built with Zome will exhibit 5-fold symmetries like the equilateral pentagon. For example, a shadow of the dodecahedron model, made in such a way that a red strut inserted in one of the balls casts the minimum shadow, has 5-fold rotational and 5-fold reflection symmetry.

Red planes are perpendicular to red struts. 2-dimensional models built in the red plane will often exhibit 5-fold symmetry. The 5-fold dodecahedron shadow cannot be built in the red plane using Zome parts. However, a 5-fold shadow of the rhombic triacontahedron (a polyhedron with 30 faces built of all red struts) can be built. The 5-fold dodecahedron shadow is implied in this model (see dotted lines).