# The Sphericon

The sphericon shown in the introduction is actually one in an infinite series of sphericon shapes. There are images of some of the other shapes in the gallery section, and the following is a description of the series and its properties.

#### Sphericon Series

Take a regular n-sided polygon, where the integer n > 2 is a multiple of 2. Label an arbitrary vertex A, and its opposing vertex B (see Figure 1 (a)). Sweep the polygon 180º around an axis AB to form a 3-dimensional solid (Figure 1 (b)).

Figure 1: Even-n sphericon construction
(a) an even-sided polygon; (b) the polygon swept around axis AB

Slice the solid along any plane that passes through A and B. Define an acis of rotation as being perpendicular to AB and passing through the centre of the solid, and rotate one half of the solid around this axis by 360k/n degrees, where 0 ≤ k < n is an integer. The series of solids that can be created in this way are known as even-n sphericons. The 'original' sphericon (that of the introduction) is an even-n sphericon where n = 4 and k = 1.

A second series, known as dual-n sphericons, can be created if A and B were positioned at opposing mid-sides, rather than opposing vertices (see Figure 2).

Figure 2: Dual-n sphericon construction
(a) an even-sided polygon; (b) the polygon swept around axis AB

A third series, known as odd-n sphericons, can be created if the integer n > 2 is not a multiple of 2. Then A is an arbitrary vertex, and B is the opposing mid-side (see Figure 3).

Figure 3: Odd-n sphericon construction
(a) an odd-sided polygon; (b) the polygon swept around axis AB

For each series, as n and k vary, a number of solids can be constructed. Each of these share some properties:

• All edges can fit on the surface of a sphere.
• The surface area is entirely made up of surfaces of the same width.
• The surfaces can be created from rectangles, sectors of circles and sectors of annuli.

#### Möbius Rings

A Möbius strip can be constructed by bending a paper strip so that the two ends join with a single half-twist. Let us define a Möbius ring as an n-sided polygon-sectioned prism that has been bent round so that the two ends join with a twist of 360k/n degrees, where -n/2kn/2 is an integer. Figure 4 shows a Möbius ring where n = 3 and k = -1.

Figure 4: Möbius ring where n = 3 and k = -1.

#### How Möbius Rings Relate to Sphericons

Note that with the ranges of k defined previously, there can exist multiple values of k that will give identical shapes for a given value of n. There are also multiple values of k that will give shapes identical other than hand. To remove these duplications, we will further restrict k for the purposes of this section. For even- and dual-n sphericons, 0 ≤ k < n/4. For odd-n sphericons and Möbius rings, 0 ≤ k < n/2.

Let us define x to be the greatest common factor of n and k. Where n is a multiple of 2, define y to be the greatest common factor of n/2 and k. Then:

Möbius Ring Odd-n Sphericon Even-n Sphericon Dual-n Sphericon
Number of Surfaces x (x + 1)/2 y y + 1
Number of Continuous Surfaces x (x - 1)/2 y y - 1
Number of Edges x (x + 1)/2 y + 1 y
Number of Continuous Edges x (x - 1)/2 y + 1 y
Table 1: The characteristics of sphericons and Möbius rings.

Therefore, whatever the value of n, when k is prime a Möbius ring will have one continuous surface and one continuous edge. When n is a multiple of 2 and k is prime, an even-n sphericon will have one continuous side and two non-continuous edges. Conversely, under the same conditions a dual-n sphericon will have a continuous edge, and two non-continuous sides.