A new kind of polytope?

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

A new kind of polytope?

Postby Plasmath » Tue May 04, 2021 1:53 am

I've been thinking about this sort of thing for a while, and I haven't seen it anywhere else.
We all know what rays and lines are. Rays go forever in one direction, and lines extend in two.
Image
Here, the arrows will mean that the polytope extends to infinity, as this will make things easier to understand.

The question I was wondering was... what would this look like in higher dimensions?
Introducing... the polycara! I'm going to define an n-dimensional polycaron as any shape that has an infinite n-hypervolume.
So what do these look like? I've coined another name for the 2-dimensional polycara: the polycra. I've also put these into 5 general categories, each with a 'professional' and 'unprofessional' name. Without further ado, let's see what they are!

The Planaray (professional: radic prism)
Image
The main idea of this shape is the prism of a ray, or the cartesian product of a line segment and a ray. This specific planaray is what I call a right planaray, because it has two right angles. It is also technically a triangle (3 sides), but only has 2 actual angles.
We can also define many other things with the rays leaving at different angles or having a different polygon at the finite section, and that is the entire category of planarays. Here is an example of one of these irregular planarays:
Image

The Linon (professional: linear prism)
Image
This is similar to the planaray, but with a line. The specific kind of linon you see above is what I will call a regular linon, because it actually is regular. The regular linon can also be defined as the cartesian product of a line segment and a line. The other irregular kinds of linons are just alterations to the intersection of the sides, and adding more sides. Here is an example:
Image

The Angleon (professional: solid angle)
Image
For the most part, this category of shapes takes an angle and turns it into a solid shape. There are 3 basic subcategories of linons:
    Right linon: Has a right angle. This is also the Cartesian product of two rays.
    Regular linon: Has any angle between 0 and 360 degrees.
    Irregular linon: Has additional sides to a regular linon.
These are relatively easy to visualize, so I don't think I need to include examples for all 3.

The Hemiplane (professional: also hemiplane)
Image
As you may expect, the hemiplane is half of a plane. You can also have irregular hemiplanes by adding sides.

The Plane (professional: Plane)
I don't need an image for this, do I? We all know what the plane is, and it is the Cartesian product of two lines. This is the final category of polycra.

Higher-dimensional polycara
We can get all the other N-dimensional polycara categories by taking the Cartesian products of N line segments, rays, or lines (where there has to be at least 1 ray or line). Because of this, there will be 3N-1 N-dimensional polycara.
Does anybody have any similar ideas or subcategories of these polycara, or possibly some renders of polycara? I find these extremely interesting.
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Re: A new kind of polytope?

Postby mr_e_man » Wed May 12, 2021 8:52 pm

Hmm, your pictures could be a little smaller.

I think some of your figures would fit with my polytope definition, if the space includes points at infinity. But I require every vertex of a polygon to meet exactly two edges; if a ray goes to a vertex at infinity, then there must be another ray going to the same vertex, or instead an edge at infinity connecting to a second vertex at infinity.

You could use gnomonic projection of a hemisphere, with the equator mapping to points at infinity. Gnomonic projection of a great circle arc is a straight line segment. The formula is simple: a point (x,y,z) on the hemisphere (meaning x²+y²+z²=1 and z>0) maps to (x,y,z)/z = (x/z,y/z,1) on the plane. Inversely, (X,Y,1) on the plane maps to (X,Y,1)/√(X²+Y²+1) on the hemisphere. Your infinite polycara correspond to finite figures (possibly polytopes) on the closed hemisphere. ("Closed" means it includes its boundary, the equator.)
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Re: A new kind of polytope?

Postby Plasmath » Fri May 14, 2021 6:41 pm

Okay, I created the gnomonic projections on desmos by projecting the outlines of polycra and just squishing out the z position (images and stuff here) and found what these finite shapes are (I use right/regular polycra here):

Planaray: Strange truncated ellipse, with 1 point at infinity, 2 finite points, and 3 sides.
Linon: Ellipse with 2 points at infinity, no finite points, and 2 sides.
Angleon: Sort of guitar pick shape with 2 points at infinity, and 1 finite point. To connect the two points at infinity, I added a line at infinity which would normally be invisible, totaling 3 sides.
Hemiplane: Another ellipse-like shape with 2 points at infinity and no finite points. I added another line at infinity to connect the two sides, totaling 2 sides.
Plane: Didn't do this one because there are no points or lines.

This was interesting!
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Re: A new kind of polytope?

Postby ndl » Tue May 25, 2021 12:08 am

Is your linon two parallel lines that create a 2-sided polygon by connecting at infinity? That's what happens with duals of hemi-hedra (polyhedra with faces going through the center).
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Re: A new kind of polytope?

Postby Plasmath » Wed May 26, 2021 12:06 am

Yeah, the original idea came from those polytopes. You could also think or a linon as a digon in spherical space, and expanding it into normal Euclidean space (I like this definition slightly better). If we assume the second definition to be true, then the linon would be self-dual, which is interesting.
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Re: A new kind of polytope?

Postby mr_e_man » Tue Jun 08, 2021 1:04 am

Ah, we've stumbled upon the Gans disk model of the Euclidean plane!

https://math.stackexchange.com/a/1464
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