Map of Page

 Basics - Construction of new types
 Types of Sphericon
 When cut from apex to apex (AA)
 When cut from mid-side to mid-side (SS)
 When cut from apex to mid-side when n is odd (AS)
Hybrids - when half is AA, half SS (AASS)
 How this all applies to sphericons discussed earlier

Take half of a regular polygon (n sides) (either with a vertice (a) or the
centre of a side (b) at the top) , and sweep it around into a 3d shape (in plan
it is a circle, front elevation the polygon) (c). Then slice it down a verticle
plane (through centre) (d) and then you can rotate one half for [360 / n] or
[360 / (n - 1 ) ] or [ 360 / ( n - 2 ) ] etc until [ n - (n - 1) ] (e)

Here are two examples of a hexagon (n = 6):

(a) Half polygon (Cut from side to side (SS)),    (b) Half polygon (Cut from apex to apex (AA))

Lets take the AA half polygon (b):

(c) Sweep into 3D shape                              (d) Slice in half

(e) Rotate (in this case) 60 degrees   (f) Finished shape

This particular shape has one continuous side. Two roll around each other (as long as you turn one of them 60 degrees left, one 60 degrees right) You can watch them here.
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When n / 2 is integer (ie. n is even):

Degree of rotation                    Type
            90                             square based
            45                            octagon based
            22.5                         16agon based

When n / 3 is integer:

Degree of rotation               Type
            120                            triangle based
            60                            hexagon based
           30                            12agon based

When the polygon is AA; and when n is even; the amount of sides which are posible to make
from rotating the two halves are the factors of   [ n / 2 ].

For example: when you take a 12 agon polygon (AA), you can create a shape with 1, 2, 3, or 6 sides.

All even sided polygons can have one continuous side covering all of shape, and
so roll around each other.
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With these, n is always even. There are (n / 2) - 1 possible sphericons for each AA (this is counting sphericons with left and right hand twists as separate). If n / 4 is an integer then it has a neutral position (neither left nor right hand).

For AA sphericons, the mobius chart and sphericon chart relate as follows:  n on sphericon chart is the same as n / 2 on mobius chart.

For example, for 16 AA, on SPHERICON CHART, this is 1 2 1 4, and the 8 on the mobius chart is also 1 2 1 4. This means that, for the sphericon chart, we have 7 different solids.
4 have one continuous side only
1 has two continuous sides
1 has four continuous sides. This one is neutral, as it has the most sides.
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These are the 'dual' of their AA counterparts. As the AAs are essentially the same as SS sphericons (the only difference being that the edges are sides and visa versa) you can calculate the amount of edges on an SS sphericon by the amount of sides on it's AA counterpart and again visa versa.

For example, 16 AA rotated 90 degrees is dual to 16 SS rotated 90 degrees. i.e. the AA is 4 4 5 3, whereas the SS is 5 3 4 4. (See sphericon chart)
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These need to be cut from apex to the opposite mid-side. We will refer to these as AS.

An AS sphericon always has one non-continuous side and can have continuous sides at the same time.

If n is a prime number then there is always one non-continuous side only. The AS n sphericons on sphericon chart relate to n-mobius rings on mobius chart, but it has to be 'decoded'.

For example, for 15n AS on mobius chart, we have 1 1 3 1 5 3 1. Each number has to be divided by 2 and then rounded up to a whole number. This gives 1 1 2 1 3 2 1 on the sphericon chart. Each of these 7 numbers gives a different solid -
4 have one non-continuous side only
2 have one continuous side and one non-continuous side
1 has one non-continuous side and two continuous sides

All these can be either left or right handed, depending on which way you turn the halves. This gives another 7 solids also. We can tell all this without making them.
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These (we will call them AASS) have one half AA and one half SS of the same n, and then the halves rotated. As yet these have not been explored.
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We can therefore make an infinate amount of new sphericons, each with possibly unique properties and latices.

Sphericon: 4 sided polygon AA, rotated 90 degrees
Squircle: 4 sided polygon SS, rotated 90 degrees
Octasphericon: 8 sided polygon SS, rotated 90 degrees

(Obviously, it is also possible to rotate the 8 sided polygon with flat at top 45 degrees, or 8 sided polygon with vertice at top either 45 or 90 degrees to create 3 more different octasphericons.)

It is also possible to put either a concave or convex curve into/onto the side of any sphericon. A concave curve on the original Sphericon makes a Femisphere, a convex curve on the Spherion makes the sphere.
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For more information, contact: paul.roberts99@ntlworld.com