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.
of sides in
|Axis of rotation||Revolved Shape||Finished -con|
|3||point-to-flat||single cone with 60-degree angles|
|4||point-to-point||double cone with 90-degree apex angles||sphericon|
|5||point-to-flat||original solid of revolution looks kinda like a cupcake|
|8||flat-to-flat||double-pencil, with flattened ends|
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 sides||Available Rotations|
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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