I talked about a spherated cyltrianglinder margin one time, as a possible shape. No one seemed too interested in it. Another possibility could be a tiger made from a triangle torus. But, according to the name, it's probably more likely the first one. So, let's take a step back, and point out some things with the tiger, first. The tiger is the spherated margin of a duocylinder. A duocylinder is the cartesian product of two solid disks. The tiger has a bone structure of a cartesian product of two hollow circles. Inflating this bone structure gives us the tiger.

In contrast, the cyltrianglinder is the cartesian product with a solid disk and a triangle. Its margin would then be an orthogonal superimposing of the edges of a circle and triangle. In theory, the tigroid version will give us a bone structure of a hollow circle and a hollow triangle. Inflating this margin will puff up the 1-D line into a circle cross-cut. One major radius is a circle, the other is a triangular frame.

By looking at the notation for a torus, ((II)I), where ((major)minor), a triangle torus could look like ((II)'), a spherated cone, according to one definition. We could also use ((I')I) to mean a spherated hollow triangle. The minor radius is a circle, but the major is a triangular frame, only the outside edges of a I' are inflated.

Then, we have the comparison of a duocylinder (II)(II) and a tiger ((II)(II)). If the cyltrianglinder is I'(II), or using the commutative property (II)(I'), then a spherated margin of this could look like ((I')(II)) or ((II)(I')). The cuts through either major radius would be those of a hollow circle or a hollow triangle. The symmetry of the tiger dance depends on which major radius was cut through.

Lately, I've been conversing with Marek to further understand the cut algorithm. I'd like to apply it to this theoretical shape. So, to recapitulate, the tiger is:

Tiger: ((II)(II))

3D cuts

A - ((II)(I)) - 2 vertical stacked toruses

2D cuts

A1 - ((I)(Ii)) - 4 circles in vertices of square, moving out will merge both rows into 2 circles in a row, then deflates to two points and vanishes

A2 - ((II)(i)) - origin empty, moving out will make circle appear, divide into 2 concentric circles, merge into one and vanish

* 3D cuts of the tiger are both two parallel and separated torii. Slicing through either major radius always gives circular tiger symmetry. But, this thing ((I')(II)) can have a circular or triangular tiger symmetry. Triangular as in 2-simplex symmetry, circular as is 2-sphere. So, let's cut and dice this thing up, whatever it is...

* If we place a triangle I' centered at origin, height along Y, using "i" for the axis that was cut along, cuts of a triangle are:

(x

^{y})

i' - cutting along X gives 2 points at medium distance, moving +Y collapses dist to zero, moving -Y grows distance to max ending in a point-prism, i.e. line

I

^{i}- cutting along Y gives 2 points at maximum dist, moving along ±X will collapse dist to zero

I' - triangle

((I')I) - spherated hollow triangle, triangle-frame torus ( triangulus?)

((II)') - spherated cone, triangle crosscut torus

Possible Cyltrianglintigroid: ((I')(II)) oriented ((x

^{y})(zw))

3D cuts

* A - ((i')(II)) - cutting along X gives 2 parallel circle-torii at medium dist, moving +Y will collapse distance and merge, moving -Y will grow dist then a torinder appears

* B - ((I

^{i})(II)) - cutting along Y gives 2 parallel circle-torii at max distance, moving ±X will collapse dist and merge

* C - ((I')(iI)) and ((I')(Ii)) - cutting along Z or W makes 2 vertical stacked triangle-frame torii ( triangulii ) that have 2-sphere tiger symm, moving ±W or ±Z respectively will collapse dist and merge

Then, of course, the cyltrianglintigroidal pyramid would just be a tapering of this whole shape down to a point into 5D. It could look like this: ((I')(II))'

If the cyltrianglintigroid was made from the tiger-rotation of a triangle torus, then the current toratope notation cannot express it. This is from the minor radius not being required to be in the notation. The triangle torus will rotate making the two circular major radii of a tiger. But, the crosscut will be a triangle instead of a circle. Maybe it could be done, based off the spherated cone analogy, I'm not sure.

Let me know what you all think

--Philip