【Hacker News搬运】n球之间的n球

hackernews

Title: An n-ball Between n-balls

n球之间的n球

Text:

Url: https://www.arnaldur.be/writing/about/an-n-ball-between-n-balls

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Post by: Hugsun

Comments:

Sharlin: A good way to conceptualize what’s going on is not the idea that balls become "spiky" in high dimensions – like the article says, balls are always perfectly symmetrical by definition. But it’s the box becoming spiky, "caltrop-shaped", its vertices reaching farther and farther out from the origin as the square root of dimension, while the centers of its sides remain at exactly +-1. And the 2^N surrounding balls are also getting farther from the origin, while their radius remains 1/2. Now it should be quite easy to imagine how the center ball gets more and more room until it grows out of the spiky box.

Sharlin: 将正在发生的事情概念化的一个好方法不是球变成&quot；的想法；尖刺”；在高维度上，正如文章所说，球从定义上讲总是完全对称的。但这是盒子变得尖刺，”；“菱角形”；，其顶点距离原点越来越远，作为维度的平方根，而其边的中心保持在+-1。2^N周围的球也离原点越来越远，而它们的半径保持为1；2.现在应该很容易想象中心球如何获得越来越多的空间，直到它从尖刺的盒子里长出来。

steventhedev: This is a really good demonstration of the curse of dimensionality[0][0]: <a href="https://en.m.wikipedia.org/wiki/Curse_of_dimensionality" rel="nofollow">https://en.m.wikipedia.org/wiki/Curse_of_dimensionality</a>

steventhedev: 这是对维度灾难的一个很好的演示[0][0]：<a href=“https://en.m.wikipedia.org:wiki:curse_of_dimensional”rel=“nofollow”>https://&#x2F；en.m.wikipedia.org；维基；Curse_of_dimensional</a>

drdeca: Why did I imagine that this would be about two shapes that are merely topologically n-balls, each having part of their boundary be incident with one of the two hemi(n-1)-spheres of the boundary of an n-ball (and otherwise not intersecting it)? (So like, in 3D, if you took some ball and two lumps of clay of different colors, and smooshed each piece of clay over half of the surface of the ball, with each of the two lumps of clay remaining topologically a 3-ball.)I don’t know that there would even be anything interesting to say about that.

drdeca: 为什么我想象这将是关于两个形状，它们在拓扑上只是n球，每个形状的边界的一部分都与n球边界的两个半球（n-1）中的一个相交（否则不相交）？（所以，在3D中，如果你拿一些球和两块不同颜色的粘土，把每块粘土在球表面的一半上抹平，每块粘土在拓扑上都是一个3球。）我不知道这会有什么有趣的地方。

ColinWright: For other HN discussions of this phenomenon you can see some previous submissions of another article on it.That article doesn't have the nice animations, but it is from 14 years ago ...<a href="https://news.ycombinator.com/item?id=12998899">https://news.ycombinator.com/item?id=12998899</a><a href="https://news.ycombinator.com/item?id=3995615">https://news.ycombinator.com/item?id=3995615</a>And from October 29, 2010:<a href="https://news.ycombinator.com/item?id=1846682">https://news.ycombinator.com/item?id=1846682</a>

ColinWright: 对于HN关于这一现象的其他讨论，您可以看到之前提交的另一篇关于这一问题的文章。这篇文章没有；我没有好看的动画，但它是14年前的 <a href=“https://news.ycombinator.com/item？id=12998899”>https://news.ymbinator.com&#x2F；news.ecombinator.com；项目？id=12998899</a><a href=“https:&#x2F；news.ycombinator.com&#x2F；item？id=3995615”>https:&quot&#x2F；news.ecombinator.com；项目？id=3995615</a>从2010年10月29日起：<a href=“https:&#x2F；news.ycombinator.com&#x2M；item？id=1846682”>https:&#x2F；news.ecombinator.com；项目？id=1846682</a>

Imustaskforhelp: Can I just say how my mind is utterly blown by the animations

Imustaskforhelp: 我能说我的头脑被这些动画完全震撼了吗