The same decomposition works in higher dimensions. For NNN random points on a sphere, the probability they all lie in a hemisphere is also N/2N−1N / 2^{N-1}N/2N−1. The argument is identical: anchor a hemisphere at each point, observe that each event has probability 1/2N−11/2^{N-1}1/2N−1, and verify that at most one anchor can work (the complementary cap is at least a full hemisphere, so no other anchor's hemisphere can straddle it). TODO: If I figure out how to add 3d visualizations to this website, I'll cover the 3D case
Small talk may not profoundly alter your life. But its absence will profoundly alter human life as we know it. We live in a world of intense and often unnecessary division. Small talk is a tiny, free and very possibly priceless reminder of our shared humanity. If we intentionally give up talking to strangers, if we purposely decide to give in to the phone shield, the consequences will be horrible. Arguably, we are already on the verge of doing this. Let’s back up and start a conversation before it’s too late.。业内人士推荐同城约会作为进阶阅读
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arXivLabs: experimental projects with community collaborators
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