Notation for Polytopes

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

Notation for Polytopes

Postby ubersketch » Fri Jan 12, 2018 10:46 pm

NOTE: As rk said, I simply just made a more readable version of incidence matrices and Hasse diagrams.
How it works:
There is a section for each kind of element. Elements are divided by what they are connected to incidentally. That means one line segment may have two hexagons while another may have a square and a hexagon.

Lets take an example.
Dyad 1 - Square, 1 : Hexagon, 1 :
Here a line segment is connected to two kinds of faces, a square and a hexagon. But imagine within this polytope, there exists another kind of dyad bordering two hexagons. Here is how it would look.
Dyad 2 - Hexagon, 2 :
As you can see, the next to the face names, is the number of faces of that kind.
Now, let's get an example to see how this would work.
Colons divide each kind of element while slashes divide dimensions of elements.
Here's a couple of examples.

Code: Select all
Sirco

Square 1 - Square 2, 4 :
Square 2 - Square 1, 2 : Trig, 2 :
Trig - Square 2, 3 :

Dyad 1 - Square 1, 1 : Square 2, 1 :
Dyad 2 - Trig, 1 : Square 2, 1 :

Point - Square 1, 1 : Square 2, 2 : Trig, 1 : / Dyad 1, 2 : Dyad 2, 2 : /





Code: Select all
Squappy

Square - Trig, 4 :
Trig - Square, 1 : Trig, 2 :

Dyad 1 - Trig, 2 :
Dyad 2 - Square, 1 : Trig, 1 :

Point 1 - Trig, 4 :
Point 2 - Square, 1 : Trig, 2 :


I also have come up with some sort of graph which this notation is based on. Here it is.
Again, each element type is decided by what they are connected to and what they are.
Image
Last edited by ubersketch on Sat Jan 13, 2018 4:34 pm, edited 2 times in total.
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Re: Notation for Polytopes

Postby Klitzing » Sat Jan 13, 2018 10:08 am

Dear Übersketch,
it looks that you are redescovering the idea of abstract polytopes, cf. https://en.wikipedia.org/wiki/Abstract_polytope .
The diagram you were drawing is nothing but the Hasse diagram of that polyhedron.
And the "new notation" you were proclaiming is nothing but a very wordy version of the incidence matrices.
Best regards,
--- rk
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Re: Notation for Polytopes

Postby ubersketch » Sat Jan 13, 2018 4:25 pm

Klitzing wrote:Dear Übersketch,
it looks that you are redescovering the idea of abstract polytopes, cf. https://en.wikipedia.org/wiki/Abstract_polytope .
The diagram you were drawing is nothing but the Hasse diagram of that polyhedron.
And the "new notation" you were proclaiming is nothing but a very wordy version of the incidence matrices.
Best regards,
--- rk

Huh, interesting. I can't read incidence matrices well so I guess I made a readable format. As for Hasse diagrams, I also made them more human-readable.
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Re: Notation for Polytopes

Postby Klitzing » Sat Jan 13, 2018 8:12 pm

For incidence matrices you can find a more detailed explanation on my website here.
A frequent wikipedia editor wrt. polytopics also is currently considering to adopt some for wiki display, cf. his user page on "configurations".
--- rk
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