{"id":11929,"date":"2013-04-12T21:11:48","date_gmt":"2013-04-13T02:11:48","guid":{"rendered":"http:\/\/blog.kenperlin.com\/?p=11929"},"modified":"2013-04-12T21:44:01","modified_gmt":"2013-04-13T02:44:01","slug":"11929","status":"publish","type":"post","link":"http:\/\/blog.kenperlin.com\/?p=11929","title":{"rendered":"Pyramid power proof"},"content":{"rendered":"

The other day I mentioned how if you arrange spheres into a pyramid of hexagons, you get the same number of spheres as if you’d arranged them into a cube. Today I thought it might be nice to prove it.<\/p>\n

Well, supposed we have just one sphere. We can think of this either as the tip of a pyramid or as a 1×1×1 cube. What we need to show is that every time we add a layer of hexagons to the base of the pyramid (to make a bigger pyramid), we get the same number of spheres as if we’d arranged the spheres into a cube.<\/p>\n

So first, let’s ask how many more spheres there are in the N3<\/sup> arrangement below right than in the (N-1)3<\/sup> arrangement below left:<\/p>\n

<\/p>\n\n\n
\n<\/th>\n<\/th>\n\n<\/th>\n<\/tr>\n<\/table>\n

<\/center><\/p>\n

What we need to show is that this difference is the same as the number of extra spheres needed to add an Nth layer to a pyramid of hexagons. In the image below, the new layer is highlighted in gold:<\/p>\n


\n\"pyramid-base\"<\/center><\/p>\n

Going back to the cubes, let’s color all of those extra spheres in the N3<\/sup> cube:<\/p>\n


\n<\/center><\/p>\n

Notice that we get three extra walls — each with N×(N-1) spheres — plus one extra sphere in the corner. But that’s exactly how many spheres we need to make a hexagon with N spheres on each side:<\/p>\n


\n<\/center><\/p>\n

So there you have it. We just showed that each additional hexagon layer contains exactly the difference between an (N-1)3<\/sup> cube and and an N3<\/sup> cube. So now we can be sure that a pyramid of hexagons, stacked N high, will always add up to exactly N3<\/sup> spheres!<\/p>\n","protected":false},"excerpt":{"rendered":"

The other day I mentioned how if you arrange spheres into a pyramid of hexagons, you get the same number of spheres as if you’d arranged them into a cube. Today I thought it might be nice to prove it. Well, supposed we have just one sphere. We can think of this either as the … Continue reading “Pyramid power 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":"http:\/\/blog.kenperlin.com\/index.php?rest_route=\/wp\/v2\/posts\/11929"}],"collection":[{"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=11929"}],"version-history":[{"count":23,"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=\/wp\/v2\/posts\/11929\/revisions"}],"predecessor-version":[{"id":11958,"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=\/wp\/v2\/posts\/11929\/revisions\/11958"}],"wp:attachment":[{"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11929"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11929"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11929"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}