As requested by Alex, here is an old fashioned proof that packing spheres into a cube N on a side uses the same number of spheres as arranging them into an N-high hexagonal pyramid.<\/p>\n
We already know that a 13<\/sup> cube uses just a single sphere — the same as a “pyramid” of height 1 — which is just a single sphere.<\/p>\n We just need to show that adding an Nth layer to a hexagonal pyramid uses the same number of spheres as incrementing from a cube of width N-1 to a cube of width N.<\/p>\n To get from an (N-1)3<\/sup> cube to an N3<\/sup> cube requires N3<\/sup> – (N-1)3<\/sup> more spheres. That’s N3<\/sup> – (N3<\/sup> – 3N2<\/sup> + 3N – 1) spheres, which simplifies to 3N(N-1)+1.<\/p>\n If we can show that a packed hexagon with N spheres on a side uses the same number of spheres, then we’re done. We do that as follows:<\/p>\n To go from our single sphere to a packed hexagon with 2 spheres on each side, we need to add 1 sphere for each of the hexagon’s six sides.<\/p>\n Similarly, to go from that to a packed hexagon with 3 spheres on each side, we need to add 2 spheres for each side of the hexagon.<\/p>\n So to build up to a packed hexagon with N spheres on a side, we need to add 1 + 2 + …. (N-1) spheres for each side of the hexagon.<\/p>\n The sum of the numbers from 1 through N-1 is N(N-1)\/2 (here’s the proof on Wikipedia)<\/a>.<\/p>\n Putting it all together, we have six of those sums (because the hexagon has six sides), plus the one sphere in the center.<\/p>\n Which gives us 6×( N(N-1)\/2 ) + 1, or 3N(N-1)+1.<\/p>\n As they say in latin class, QED. But I think the visual proof is a lot more elegant — and it didn’t require linking to the Wikipedia to explain picky details. \ud83d\ude42<\/p>\n","protected":false},"excerpt":{"rendered":" As requested by Alex, here is an old fashioned proof that packing spheres into a cube N on a side uses the same number of spheres as arranging them into an N-high hexagonal pyramid. We already know that a 13 cube uses just a single sphere — the same as a “pyramid” of height 1 … Continue reading “Old fashioned proof”<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/blog.kenperlin.com\/index.php?rest_route=\/wp\/v2\/posts\/11962"}],"collection":[{"href":"https:\/\/blog.kenperlin.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.kenperlin.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.kenperlin.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.kenperlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=11962"}],"version-history":[{"count":7,"href":"https:\/\/blog.kenperlin.com\/index.php?rest_route=\/wp\/v2\/posts\/11962\/revisions"}],"predecessor-version":[{"id":11969,"href":"https:\/\/blog.kenperlin.com\/index.php?rest_route=\/wp\/v2\/posts\/11962\/revisions\/11969"}],"wp:attachment":[{"href":"https:\/\/blog.kenperlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11962"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.kenperlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11962"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.kenperlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11962"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}