/ max
max  rss = max  rss
\
rss  max
We have this representation:
() defines a spheric or rss product
[] defines a prismic or max product
<> defines a tegmic or sum product
unconnected nodes project in 2d as squares
connected nodes project in 2d as circles.

oo S2 circle (1,2)
o o P2 prism [1,2]
o o o P3 cube [ 1,2,3]
oooz S3 sphere (1,2,3)
oo o cylinder [(1,2),3]
ooo dome ([1,3],2]
12 < glome (1,2,3,4) 1 2
 X 
34 tesseract [1,2,3,4] > 3 4
1 4 < spherinder [(1,2,3),4 ] 1  4
 \ / 
2  3 tridome ( [1,2,3], 4] > 2 3
12 < duocylinder [(1,2),(3,4)] 14
 
34 cyclodome ( [1,2],[3,4]) > 32
12 < cubinder [(1,2),3,4] 1 2
 X 
3 4 semiglome ([1,2],3,4) > 3  4
1  2 <  spheridome (2,[1,(3,4)]) 1 2
/   \
3 4 dominder [2,(1,[3,4])] > 3 4
12 < longdome

43
all branches alike
o o o o o
P [1,2,3,4,5] peneteract
S (1,2,3,4,5) pentaglome
one branch different
oo o o o
([1,2],3,4,5) semipentaglome
[(1,2),3,4,5] cubicircle
two branches different
oo oo o
[(1,2),(3,4),5] dual cylinder
([1,2],[3,4],5) pyraglome
ooo o o
[(2,[1,3]),4,5] cubicircle
([2,(1,3)],4,5] semipentaglome
three branches different
ooo: o o
[(1,2,3),4,5] spherisquare
([1,2,3),4,5) sesquiglome
ooo oo
[(2,[1,3]),(4,5)] dormicircle
([2,(1,3)],[4,5]) siamese spheridome
ooo Ao o
[(2,[1,3,4]),5] tridominder
([2,(1,3,4)],5) glomidome
oooo o
* 1
* 2
four branches different
ooo: oo
[(1,2,3),(4,5)] sphericircle
([1,2,3],[4,5]) tricycle
oooo: o
[([1,2],[3,4]),5] cyclodominder
([(1,2),(3,4)],5) duospheridome
ooo:o o
[([(1,2),4],3),5] spheridominder
([([1,2],4),3],5) spheritridome
ooooo
* 3
* 4
oooo Ao
* 5
* 6
ooo Ao Bo
(2,[1,3,4,5]) tetradome
[2,(1,3,4,5)] glominder
five different branches
ooooo:
* 7
ooo: o Ao
(3,[(1,2),4,5]) glomosquare
[3,([1,2],4,5)] spheriglome
oooo: o
ooo: oo
* 8
* 9
ooo: o Co
* 10
wendy wrote:We pick three polygons, CO(x,y,z).
In the positive octant, the CO has four faces, the triangle, and quartersquares.
The qsquare faces have the vertex (1,1,0) to (1,0,1). This can be represented as, eg as max(x, sum(y,z)). This is a prism product of a product of x and the tegum over y and z, or say [x,<y,z>].
Marek14 wrote:[...]Then, when I was lying in my bed trying to sleep, something connected in my head and I saw a way to entangle these shapes with graph theory... and this is what resulted:
[...]Now comes the trick: there is one more graph with three nodes: O=O=O. Does this graph have a rotatope associated with it like the others?
Surprisingly, it turns out it does. First, we review its properties: It has to cast two circular shadows (when we remove the endnodes) and one square shadow (removing the middlenode). Passing it through the plane will result in two possibilities: either a square that will start from a point, grow, then shrink...
...or we start with a line. This will grow into circle (originally I thought it was through ellipse, but in fact the intermediate shapes are circles with their top and bottom cut). Then it shrinks back into line.
[...]The point here is that there are 11 different graphs on 4 nodes. 5 of them have only complete components and correspond to 5 rotatopes you have on your page. The other six have noncomplete graphs, and all of them would contain the O=O=O shape as a shadow in one or more direction (one of them, indeed, in ALL 4 directions). I think these are worth checking, in order to enlarge the zoo a bit.
Since these are obviously not true rotatopes, we will need a new name... Would "graphotopes" work? "[...]
Marek14 wrote:Here's the puzzle, though: there is only limited number of ways how to combine max and rss functions in this way, while the number of possible graphs starts to exceed them eventually. I'm not sure whether things like what I call longdome exist  but if it does, I am very interested in its equation.
wendy wrote:[...]The thing about the graphnotation, is that the allmarked graph does not give exclusively a sphere. In three dimensions, one can also get a rhombic dodecahedron, being the convex hull of a cube and an edgetangent sphere. This is the intersection of three cylinderic pipes.
So even though the sphere gives the form of o=o=o=:, it's not the only shape to do so.
quickfur wrote:Mainly, my interest was driven by trying to understand what exactly "longdome" is.
quickfur wrote:Or perhaps I'm misunderstanding what the graph nodes signify. But in either case, can we express a tricylinder in any existing notation? It's no less of an interesting shape that the other more conventional rotatopes / toratopes, having a connection with the rhombic dodecahedron, which makes one wonder if a 4D analogue might have connections with the 24cell(!).
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