John Carlos Baez on Nostr: It's wonderful that you can take a 3-sphere - the surface of a ball in 4d space - and ...
It's wonderful that you can take a 3-sphere - the surface of a ball in 4d space - and cut it in half with a torus. In other words, you can take two doughnuts, glue them together along their boundaries in a certain way, and get a 3-sphere.
It's more wonderful that you can take a 3-sphere and chop it into 600 regular tetrahedra in a symmetrical way. This is called the 600-cell.
But what really surprised me is that when you combine these ideas, you can get your torus to cut through 100 of the regular tetrahedra, which form a 10 × 10 arrangement as below!
Since you've got a torus, this picture wraps around: you have to imagine the left edge attached to the right, and the top to the bottom.
I learned this from a Wikipedia article that "contains an excessive amount of intricate detail":
https://en.wikipedia.org/wiki/600-cell#Union_of_two_toriSomeone has spent a huge amount of time visualizing the 600-cell, and explains it in ways that I have trouble following. I fear some of this is "original work", but I hope it never gets lost.
Published at
2026-08-18 13:27:11 UTCEvent JSON
{
"id": "78446b524ec4eb386c1fc71d48e2483589610cf62274f4c2e908a7537d4e6ceb",
"pubkey": "f7346eb283902ada9d21c109a93e83128d9f87d8fcfe70ad819b3bf2ad9bce16",
"created_at": 1787059631,
"kind": 1,
"tags": [
[
"imeta",
"url https://media.mathstodon.xyz/media_attachments/files/117/116/717/065/165/437/original/98da673a7a3f0b1e.jpg",
"m image/jpeg",
"dim 532x525",
"blurhash UEL;aH%M~p%Mxua}fQa}_3ayWBayxua}fQa}"
],
[
"proxy",
"https://mathstodon.xyz/users/johncarlosbaez/statuses/117116740014448265",
"activitypub"
],
[
"client",
"Mostr",
"31990:6be38f8c63df7dbf84db7ec4a6e6fbbd8d19dca3b980efad18585c46f04b26f9:mostr",
"wss://relay.ditto.pub"
]
],
"content": "It's wonderful that you can take a 3-sphere - the surface of a ball in 4d space - and cut it in half with a torus. In other words, you can take two doughnuts, glue them together along their boundaries in a certain way, and get a 3-sphere.\n\nIt's more wonderful that you can take a 3-sphere and chop it into 600 regular tetrahedra in a symmetrical way. This is called the 600-cell.\n\nBut what really surprised me is that when you combine these ideas, you can get your torus to cut through 100 of the regular tetrahedra, which form a 10 × 10 arrangement as below!\n\nSince you've got a torus, this picture wraps around: you have to imagine the left edge attached to the right, and the top to the bottom.\n\nI learned this from a Wikipedia article that \"contains an excessive amount of intricate detail\":\n\nhttps://en.wikipedia.org/wiki/600-cell#Union_of_two_tori\n\nSomeone has spent a huge amount of time visualizing the 600-cell, and explains it in ways that I have trouble following. I fear some of this is \"original work\", but I hope it never gets lost.\n\nhttps://media.mathstodon.xyz/media_attachments/files/117/116/717/065/165/437/original/98da673a7a3f0b1e.jpg",
"sig": "a790499fecd95149d0eb22d116cce69baf77020e8df8c57800640e983af19930851c389e5d9368ae96685d7993594e11046e273655fe46c8b3a1354828aaf233"
}
