Timeline for Number of polytopes formed by connecting points on a hypercube
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Apr 24, 2020 at 22:43 | comment | added | ReverseFlowControl | @GerhardPaseman, looking at the vector representation and counting those with properties that place them on the boundaries is the only "reasonable" way I know of looking at it. A face of the hypercube has all point lying on some hyperplane at $x_i=0$ or $x_i=d$. Using higher level geometry, or fancier math, seems unnecessary. Just like in 3D $y=0$ means zx-plane, or $z=0$ means xy-plane, and so on. | |
| Apr 24, 2020 at 22:39 | comment | added | Gerhard Paseman | He is counting a different quantity, which may (or may not) bear some relation to your quantity. I suggested using Euler's formula for d=2. For higher d, one might start out with his formula to count related quantities. By the way, boundary is not clear yet in higher dimensions for me. I can see counting only lattice points on edges or on edges and faces as qualifying, but you might mean something more general. Gerhard "Not Used To High-Dimensional Combinatorics" Paseman, 2020.04.24. | |
| Apr 24, 2020 at 22:39 | comment | added | ReverseFlowControl | I think we can easily see the first case is indeed 4: 3 triangles all with one corner at the origin and then the entire square. For $n=3$, that is the right count if you realize that polytopes need not be a triangle, you could choose 4 points including the origin and get a polypote with 4 sides where an edge would be part of the boundary, there are many such cases that only counting triangles leaves out.. | |
| Apr 24, 2020 at 22:06 | comment | added | skd | Thanks. Again, apologies for possibly being silly, but it seems like the sequence γ(L(n,2)) goes 4, 120, 2036, ... It's not clear to me how this sequence relates to the question. | |
| Apr 24, 2020 at 20:57 | history | edited | ReverseFlowControl | CC BY-SA 4.0 |
Update.
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| Apr 24, 2020 at 20:48 | history | edited | ReverseFlowControl | CC BY-SA 4.0 |
Fixed notation. again.
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| Apr 24, 2020 at 20:38 | comment | added | ReverseFlowControl | Fixed. Thanks! (Also, the 15 character comment length minimum is silly.) | |
| Apr 24, 2020 at 20:37 | history | edited | ReverseFlowControl | CC BY-SA 4.0 |
Fixed minor notation issue.
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| Apr 24, 2020 at 19:56 | comment | added | skd | Thanks for the response. I'm not sure I fully understand; I don't parse the string "L(n,d) := [0,n]^d" and then the equations for L(n,d). Perhaps I've not calculated/understood your answer correctly, but it doesn't seem like any of the equations listed recover the sequence 4, 56, 340 for an n-by-n square in two dimensions. | |
| Apr 24, 2020 at 9:18 | history | edited | ReverseFlowControl | CC BY-SA 4.0 |
Fixed sum.
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| Apr 24, 2020 at 7:00 | history | answered | ReverseFlowControl | CC BY-SA 4.0 |