The Definitive Diagonal Cube Tutorial (Zen Magnets)

1. Overview

This video tutorial shows how to build solid diagonal cubes of any size out of Zen Magnets, with three magnet vertices. Step-by-step instructions show how to build cubes directly, layer by layer, without the need for intermediate octahedron and cuboctahedron steps, using stable three-magnet corners.

Construction is shown explicitly for cubes with edge counts
1 (2:09),
2 (4:55),
3 (8:13), and
4 (17:34).
Schematic diagrams (see 27:51 and http://imgur.com/jJpatbd) show how to build cubes with edge counts up to 12.

Below, we discuss how to build the diagonal cube in even larger sizes. In general, a cube of edge count n requires N = n(4n*n + 6n + 3) magnets.

2. Layer Numbers and Dimensions

The cube is built from single-thickness diagonal layers. These layers are horizontal when the cube is oriented with one corner directly above its opposite corner. In this orientation, n triangular layers form the top third of the cube (orange labels on the schematics), n hexagonal layers form the middle third (violet labels), and n triangular layers form the bottom third (orange labels), for a total of 3n layers.

Layer dimensions are denoted by L x M, where L and M are the edge lengths along alternating sides of a hexagon, and with L less than or equal to M by convention. If L = 1, the hexagon becomes a triangle. If L = M, you get a regular hexagon with all six sides of the same length. For L between 1 and M, you get an irregular hexagon with sides alternating between edge length L and edge length M.

The n triangular layers in the top third of the cube and the n triangular layers in the bottom third are duplicates of each other, and have dimensions:

1 x 2
1 x 4
1 x 6...
1 x 2n.

The dimensions of the hexagonal layers in the middle third depend on whether n is even (n = 2, 4, 6, ...) or odd (n = 1, 3, 5, ...). For even n, you'll need two of each of the following:

2 x 2n
4 x (2n - 2)
6 x (2n - 4)...
n x (n+2)

The two n x (n+2) layers meet at the center of the cube.

For odd n, you'll need two of each of the following for the middle third:

2 x 2n
4 x (2n - 2)
6 x (2n - 4)...
(n-1) x (n+3)

and you'll also need one (n+1) x (n+1) regular hexagon at the center of the cube.

3. Layer Construction

We now discuss how to make these layers. At the heart of each layer is one of three cores, a single magnet (1 x 1), a triangle of three magnets (1 x 2), or a triangle of six magnets (1 x 3). The difference M - L between the two sides is 0, 1, and 2 for these three cores, respectively.

For odd n, as discussed above, a single regular hexagon forms the central layer of the cube. This hexagon has a 1 x 1 core. The next layer up (or down) has a 1 x 2 core, then the next layer has a 1 x 3 core, and the core sequence repeats, 1 x 1, 1 x 2, 1 x 3, etc. until you reach the top (or bottom) triangle of the entire cube, which is a 1 x 2 core.

For even n, as discussed above, two hexagons meet at the center of the cube. Each of these has a 1 x 3 core. Starting from the upper (or lower) of these two and moving up (or down), the cores needed for successive layers are 1 x 1, 1 x 2, 1 x 3, 1 x 1, 1 x 2, 1 x 3, etc. until you reach the top (or bottom) triangle of the cube, which again is a 1 x 2 core.

Adding a complete ring (shades of green in the schematics) to an L x M layer converts it into an (L+1) x (M+1) layer. Thus, the difference M - L between the two sides doesn't change when you add a complete ring.

Adding three chains each of length L - 1 along sides of length L of an L x M layer (shown in shades of blue or white in the schematics) converts it into an (L-1) x (M+2) layer, subtracting 1 magnet from the sides of length L and adding 2 magnets to the sides of length M, and increasing the difference M - L by 3.

4. General Layer Construction Procedure

In general, an L x M layer requires (L+M-1)(L+M)/2 + (L-1)(M-1) magnets. To build an L x M layer (with L less than or equal to M), first divide the edge length difference M - L by 3. The quotient Q is the number of chains you need to add (shades of blue and white) to each of the three short sides. The remainder R determines the core that you need to use to build the layer; remainders R of 0, 1, and 2 imply 1 x 1, 1 x 2, and 1 x 3 cores, respectively. The number of complete rings that you need to add around the core (shades of green) is C = L + Q - 1.

For example, a 1 x 14 triangle has L = 1, M = 14, and M - L = 14 - 1 = 13. Dividing 13 by 3 gives a quotient Q = 4, remainder R = 1, and number of complete rings C = 1 + 4 - 1 = 4. Thus, to build the 1 x 14 triangle, start with a 1 x 2 core, wind 4 complete rings around it (shades of green), and add 4 chains (three blue, one white) to each of the three short sides, as shown in the schematic for edge length 7.