Hey, they did it! People have known for a long time that in 4 dimensions you can pack equal-sized balls so that each touches 24 others. In 2008 someone showed that's the best you can do. But in April this year, three mathematicians showed there's essentially just ššš š¤šš¦ you can accomplish this!
This way is easy to describe. Take the points whose coordinates are all integers, together with those whose coordinates are all integers plus 1/2. Use each of these points as the center of a ball of radius 1/2. Then each ball touches 24 others. This pattern is called the šš° š¹š®ššš¶š°š².
The new result shows that if you pack equal-sized balls in 4 dimensions so that each touches 24 others, you're šššššš to create a version of the D4 lattice - possibly rotated, translated and/or rescaled.
This sort of thing is hard to show: it took a lot of fancy math and two weeks of calculation on a standard desktop computer. I believe only other dimensions where we know a result like this are 1, 2, 8 and 24. In dimension 3 it's just false: you can pack equal-sized balls so that each touches 12 others, and this is the best you can do, but there are uncountably many different ways to do it!
https://arxiv.org/abs/2404.18794
John Carlos Baez /
npub17uā¦8pd6m
2024-10-13 16:13:27
Author Public Key
npub17u6xav5rjq4d48fpcyy6j05rz2xelp7clnl8ptvpnval9tvmectqp8pd6mPublished at
2024-10-13 16:13:27Event JSON
{
"id": "bc6b0c14c31db105d2877e84638dcb0bec27595d0788f23752d393d04a78f947",
"pubkey": "f7346eb283902ada9d21c109a93e83128d9f87d8fcfe70ad819b3bf2ad9bce16",
"created_at": 1728836007,
"kind": 1,
"tags": [
[
"proxy",
"https://mathstodon.xyz/users/johncarlosbaez/statuses/113300996557902308",
"activitypub"
]
],
"content": "Hey, they did it! People have known for a long time that in 4 dimensions you can pack equal-sized balls so that each touches 24 others. In 2008 someone showed that's the best you can do. But in April this year, three mathematicians showed there's essentially just ššš š¤šš¦ you can accomplish this! \n\nThis way is easy to describe. Take the points whose coordinates are all integers, together with those whose coordinates are all integers plus 1/2. Use each of these points as the center of a ball of radius 1/2. Then each ball touches 24 others. This pattern is called the šš° š¹š®ššš¶š°š².\n\nThe new result shows that if you pack equal-sized balls in 4 dimensions so that each touches 24 others, you're šššššš to create a version of the D4 lattice - possibly rotated, translated and/or rescaled.\n\nThis sort of thing is hard to show: it took a lot of fancy math and two weeks of calculation on a standard desktop computer. I believe only other dimensions where we know a result like this are 1, 2, 8 and 24. In dimension 3 it's just false: you can pack equal-sized balls so that each touches 12 others, and this is the best you can do, but there are uncountably many different ways to do it! \n\nhttps://arxiv.org/abs/2404.18794",
"sig": "f556e2994d2309470d77ef9250a631c08e3c6dbd55566d92fd9bb01a800d6151f4c420943f10603f2ffa682a3e7d3e0a94e2ec39e062d630857699c42e97c6bd"
}