## Notation for Polytopes

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

### Notation for Polytopes

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
`SircoSquare 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
`SquappySquare - 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.
Last edited by ubersketch on Sat Jan 13, 2018 4:34 pm, edited 2 times in total.

ubersketch
Trionian

Posts: 126
Joined: Thu Nov 30, 2017 12:00 am

### Re: Notation for Polytopes

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
Klitzing
Pentonian

Posts: 1437
Joined: Sun Aug 19, 2012 11:16 am
Location: Heidenheim, Germany

### Re: Notation for Polytopes

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

ubersketch
Trionian

Posts: 126
Joined: Thu Nov 30, 2017 12:00 am

### Re: Notation for Polytopes

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
Klitzing
Pentonian

Posts: 1437
Joined: Sun Aug 19, 2012 11:16 am
Location: Heidenheim, Germany