Curved Stellations and Visualization

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

Curved Stellations and Visualization

Postby ubersketch » Wed Dec 20, 2017 12:22 am

So, I have created a new method of representing stellations I call the Reuleaux triangle method and the circle method. Note that the Reuleaux triangle method is not mathematically accurate and is used for practicality purposes and the circle method is more mathematically accurate.

Monoga and diga will be considered hosotopes as they are 2d equivalents to them. Hosotopes are under a group called microtopes, spherical polytopes that collapse into what appears to be a lower dimensional polytope after being made to have no curvature.

Reuleaux triangle method:
First create a Reuleaux polygon (draw vertices and edges on a circle, create points halfway from the band of the circle and the faces of the hexagon, create a curve going through each point, each ending on the original vertices of the hexagon.) On the vertices, augment an equilateral triangle. Finally, turn two of the edges into Reuleaux triangle edges and remove the edge touching the vertices.

Circle method:
To be posted. Currently your only hope is connect curves of constant widths or find it yourself.

These methods can be generalized to higher dimensions.

Diga

{1^1} dig - This appears to be a compound of 2 mogs (monoga)

Triga

{1^1}trig - This appears to be an actual trigram, not just a compound and I call it the triquetra!
{2^1]trig - This appears to just be a compound of 3 mogs.
[3^1}trig - Another trigram. From here it's just more trigrams and monoga.

Tetraga

{1^1}square - This is a compound of 2 digs. This shape's higher dimensional equivalent, the tetrahemihexacron, is very interesting as I will tell about in another topic.
{2^1}square - A tetragram.
{3^1}square - A compound of 4 mogs. From here it becomes less interesting.

Pentaga

{1^1}peg - A polygram that is actually possible in euclidean geometry. Hooray!
{2^1}peg - Another pentagram. From here, its just more pentagrams.

Hexaga

{1^1}hig - Compound of 2 trigs. Possible in euclidean geometry.
{2^1}hig - A compound of 3 digs.
{3^1}hig - Compound of 2 triquetras.
{4^1}hig - An actual hexagram.
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