Wikipedia:Reference desk/Archives/Mathematics/2022 December 28

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December 28

Why are Reeve tetrahedra skewed?

Reeve tetrahedra showing that Pick's theorem does not apply in over two dimensions

Would anyone know why John Reeve picked (1, 1, r) as the top vertex of the Reeve tetrahedron? (0, 0, r) gives the same result yet is much easier to visualise and draw. Thanks, cmɢʟee⎆τaʟκ 16:21, 28 December 2022 (UTC)

Choose $r=2$, so the four vertices are lattice points, having positions $(0,0,0),$ $(1,0,0),$ $(0,1,0)$ and $(0,0,2).$ Then, next to these four, also the lattice point $(0,0,1)$ lies on the surface of the tetrahedron.  --Lambiam 19:57, 28 December 2022 (UTC)

To clarify a bit for who aren't familiar with them, the point of the Reeve tetrahedra is to show there is no three dimensional version of Pick's theorem. If the fourth vertex is (0, 0, r) then the number of lattice points on the surface vary with r, and that invalidates the example. Reeve tetrahedra have the same number of interior lattice points (0), face lattice points (0) and edge lattice points (0), the number of vertices is the same (4), and vertices are lattice points, but the volumes are different. --RDBury (talk) 00:01, 29 December 2022 (UTC)

Very good points, thanks! I missed the extra edge points. cmɢʟee⎆τaʟκ 15:22, 31 December 2022 (UTC)

The pictures would be much clearer if you chose a vantage point so that all four vertices lie on the boundary of the projection. 100.36.106.199 (talk) 16:23, 31 December 2022 (UTC)

Also the text at the bottom is (1) in a font-size that is almost illegibly small at the size the image is displayed in the article, and (2) is largely pointless (all Reeve tetrahedra have i = 0 and b = 4, and the volume is related to r in a trivial way). I suggest removing those lines and changing the caption to read "Reeve tetrahedra for r = 1, 2, and 3" or something similar. 100.36.106.199 (talk) 16:29, 31 December 2022 (UTC)

Thanks, updated as attached. I think i and b values should be kept to show that they are constant. Cheers, cmɢʟee⎆τaʟκ 09:59, 4 January 2023 (UTC)

Thanks, this is much more legible! 100.36.106.199 (talk) 13:35, 4 January 2023 (UTC)

👍  --Lambiam 17:16, 4 January 2023 (UTC)