Sphericon Series

In all this messing around, especially once I made the hexasphericon, I began to see a series, in a mathematical sense. The standard sphericon has a square cross section where you cut it in half after making the double-cone. The hexasphericon has a hexagonal cross section where you cut it in half.

I began to think about how what I start with determines what I end up with.

Number
of sides in
cross section
Axis of rotationRevolved ShapeFinished -con
3point-to-flat
single cone with 60-degree angles1 terminating edge, 1 terminating path
4point-to-point
double cone with 90-degree apex anglessphericon
2 terminating edges, 1 continuous path
4flat-to-flat
cylinder1 continuous edge, 2 terminating paths
5point-to-flat
original solid of revolution looks kinda like a cupcake
1 terminating edge, 1 terminating path
or
1 terminating edge, 1 terminating path
6point-to-point
"double-pencil"
hexasphericon
2 terminating edges, 1 continuous path
6flat-to-flat
7point-to-flat
1 terminating edge, 1 terminating path
or
1 terminating edge, 1 terminating path
or
2 terminating edges, 1 continuous edge, 2 continuous paths
8point-to-point
2 terminating edges, 1 continuous path
or
2 terminating edges, 1 continuous path
8flat-to-flat
double-pencil, with flattened ends
2 continuous edges, 1 continuous path + 2 terminating paths
or
1 continuous edge, 2 terminating paths

I find it very interesting to see the drastic difference it makes in the final x-con, depending solely on which axis you use to create the original revolved solid.

You might notice that for the 8-sided versions you get two variations of x-con, depending on how many degrees you rotate the halves with respect to each other. The first one shown is rotated 90 degrees, and is fairly plain. The second one shown is rotated 45 degrees, and is quite a bit more interesting visually. There is actually one more that is not shown. If you rotate the halves 135 degrees with respect to each other, you get the mirror-image (or left-handed version) of the second one shown! It "swirls" the opposite direction.
You also get two variations for the 6-sided solids. In these cases, you can rotate the halves 60 degrees or 120 degrees, and get right-handed or left-handed versions of the x-con.

It might be kind of hard to see the patterns here. However, let me try to explain what I see so far. First, let's go through the series looking at the revolved solids that are created with the axis of revolution going through the flats. All of the x-cons that are created by sawing these solids and rotating the halves have two distinct surfaces. These surfaces end in half-circles. On the other hand, all of the x-cons that are created by sawing the solids that are revolved around point-to-point axes end up having only one continuous surface!

The solids which have an odd number of sides in their cross-section are a slightly different breed. They all have only one path on their surface, but that path stops at both ends; it is not continuous like those whose with even-numbered sides whose axis is point-to-point.

Another pattern here is that there is a mathematical series created when you look at the number of ways you can rotate the halves of the solids.

Number of sidesAvailable Rotations
41
62
83

Note: in the table above, I've listed only those rotations which produce unique shapes. For example, a 6-sided section whose half is rotated 120 degrees ccw produces an object which is identical to one whose half is rotated 60 degrees cw, and vice-versa. This is due to the 180-degree symmetry of the part. Right- and left-hand versions are counted separately.

All this leads up to another yet-unanswered question: How many sides can a -con have? If I made a 200-sided revolved solid with its axis of revolution through the points, could I rotate it (in one of many, many ways) and end up with an x-con which had only one surface, which snaked around and around it?

More to come on this page sometime! Check back!


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