Timeline for Can a regular icosahedron contain a rational point on each face?
Current License: CC BY-SA 4.0
15 events
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| Oct 22, 2023 at 14:47 | comment | added | Oscar Lanzi | In all other Platonic solids, faces are three- or fourfold rotations of each other, which allows sets of corresponding points to fit into the integer lattice (by properly positioning and scaling the polyhedron). But with the icosahedron the rotation is fivefold, which precludes fitting one set of corresponding points to the integer lattice on all faces. | |
| Aug 20, 2023 at 3:12 | comment | added | Steven Stadnicki | Curious side question: is it possible that the answer depends on the dimension? (e.g., that such an icosahedron exists in $\mathbb{R}^4$ but not $\mathbb{R}^3$) | |
| Aug 20, 2023 at 2:58 | history | edited | bof | CC BY-SA 4.0 |
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| Aug 20, 2023 at 0:13 | history | edited | Ilya Bogdanov | CC BY-SA 4.0 |
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| Aug 17, 2023 at 16:10 | comment | added | Jeremy Rouse | Here's an example that doesn't work but comes close. The icosahedron with vertices $(\pm (\phi-2), \pm (\phi-1), 0)$, $(\pm (\phi - 1), 0, \pm (\phi - 2))$, $(0, \pm (\phi-2), \pm (\phi-1))$ has the property that each point in the set $\{ (1,1,-1), (1,-1,1), (1,-1,-1), (-1,1,1), (-1,1,-1), (-1,-1,1) \}$ lies on six of the extended faces of the icosahedron (with each extended face having at least one). However, one can show that for 12 of the faces of this icosahedron, any rational point lies a distance at least $\sqrt{2}$ from the origin, while the circumradius is $\sqrt{7-4\phi} < \sqrt{2}$. | |
| Aug 17, 2023 at 10:23 | comment | added | Yaakov Baruch | I'm curious: is there a pentagon with a rational point on each edge? | |
| Aug 16, 2023 at 23:31 | comment | added | Gerry Myerson | Related: math.stackexchange.com/questions/4509194/… | |
| Aug 16, 2023 at 23:10 | history | edited | M. Winter | CC BY-SA 4.0 |
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| Aug 16, 2023 at 14:36 | comment | added | Ilya Bogdanov | @KentaSuzuki No, you cannot find a regular pentagon in the integer lattice (in any dimension). | |
| Aug 16, 2023 at 14:32 | comment | added | Kenta Suzuki | Can all of the vertices of an icosahedron be rational? | |
| Aug 16, 2023 at 10:58 | comment | added | Ilya Bogdanov | @GeraldEdgar Thanks! Fixed. | |
| Aug 16, 2023 at 10:56 | history | edited | Ilya Bogdanov | CC BY-SA 4.0 |
edited body; edited title
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| Aug 16, 2023 at 10:48 | comment | added | Gerald Edgar | Two different (mis)spellings of icosahedron in one question! Nice question anyway. | |
| Aug 16, 2023 at 10:28 | history | edited | Ilya Bogdanov |
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| Aug 16, 2023 at 10:23 | history | asked | Ilya Bogdanov | CC BY-SA 4.0 |