Because the hexagons all lie in a Euclidean plane, I think the distance between hexagon centres should be √3 times the edge length of the hexagons.
But that makes the *distance* between the hexagon centres:
√3 arccosh(5/4)
and the dot product
–cosh(√3 arccosh(5/4))
which is nothing you can get from Eisenstein integers.
The vertex-vertex dot product of –5/4 looks much more promising! But it’s still not obvious to me how to get the vertex coordinates.
Greg Egan /
npub12c…gd38p
2024-04-22 14:00:05
in reply to nevent1q…ncgu
Author Public Key
npub12cuzzqzv8e8y7na77a9zv47mx2v6pec60qs5h7takzmv5p0mw0fsmgd38pPublished at
2024-04-22 14:00:05Event JSON
{
"id": "21bf6c8fb5aac254471303134d61f5a3085f925aa9fd4627add062d271987b1d",
"pubkey": "563821004c3e4e4f4fbef74a2657db3299a0e71a78214bf97db0b6ca05fb73d3",
"created_at": 1713794405,
"kind": 1,
"tags": [
[
"p",
"563821004c3e4e4f4fbef74a2657db3299a0e71a78214bf97db0b6ca05fb73d3"
],
[
"p",
"9a6a1a8eefd0b53d7e0c966ab35bd904151246f03b1be98fa0d2d0eeb4940320"
],
[
"p",
"2db431c8bab241109ce32e5b373c722c601c136c4ad4eb847478105046a16dc9"
],
[
"e",
"606bd7852f9f47c8776ea69462d63cc6fee18fe90680df3ea86dfd58985590b7",
"",
"reply",
"563821004c3e4e4f4fbef74a2657db3299a0e71a78214bf97db0b6ca05fb73d3"
],
[
"e",
"f6229c8b793bfee46a674f42534461cb9760b976045fe98c6307ba61711fc2fe",
"",
"root",
"9a6a1a8eefd0b53d7e0c966ab35bd904151246f03b1be98fa0d2d0eeb4940320"
],
[
"proxy",
"https://mathstodon.xyz/@gregeganSF/112315230178959253",
"web"
],
[
"proxy",
"https://mathstodon.xyz/users/gregeganSF/statuses/112315230178959253",
"activitypub"
],
[
"L",
"pink.momostr"
],
[
"l",
"pink.momostr.activitypub:https://mathstodon.xyz/users/gregeganSF/statuses/112315230178959253",
"pink.momostr"
],
[
"-"
]
],
"content": "Because the hexagons all lie in a Euclidean plane, I think the distance between hexagon centres should be √3 times the edge length of the hexagons.\n\nBut that makes the *distance* between the hexagon centres:\n\n√3 arccosh(5/4)\n\nand the dot product\n\n–cosh(√3 arccosh(5/4))\n\nwhich is nothing you can get from Eisenstein integers.\n\nThe vertex-vertex dot product of –5/4 looks much more promising! But it’s still not obvious to me how to get the vertex coordinates.",
"sig": "9e773c49d9baeb6c477345494d45397c5e7c40d49d3bda611bc6ce8530b4d58ddffb552c772b1d1711438e229fbf78e311724426d88132579cdde85ba2902356"
}