{"id":13930,"date":"2013-12-08T22:12:58","date_gmt":"2013-12-09T03:12:58","guid":{"rendered":"http:\/\/blog.kenperlin.com\/?p=13930"},"modified":"2013-12-08T22:12:58","modified_gmt":"2013-12-09T03:12:58","slug":"2-%c3%97-2-%c3%97-2-%c3%97-2-part-2","status":"publish","type":"post","link":"http:\/\/blog.kenperlin.com\/?p=13930","title":{"rendered":"2 \u00d7 2 \u00d7 2 \u00d7 2, part 2"},"content":{"rendered":"

One of the problems in describing things in four dimensions is that they are hard to see. If I used the classic view, where Z and W are both in perspective, it would probably just be confusing.<\/p>\n

So let’s ask ourselves: If we lived in a 2D Flatland, and we wanted to do puzzles on a 2 × 2 × 2 cube, how would we visualize the cube?<\/p>\n

To be clear, we are talking about very smart Flatland people here, who know perfectly well what a cube is. The understand it has six faces, twelve edges and eight corners. They just have trouble visualizing it, since the world they grew up with — and therefore all their intuitions — is two dimensional.<\/p>\n

If I were such a 2D individual, and I ran out of places to put that pesky third dimension — the one that doesn’t fit in my world — I might try nesting things one inside the other.<\/p>\n

If you take that approach, then an intrepid Flatlander might visual a cube by doing something like this:<\/p>\n

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

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

Our Flatland friend has found a way to position all eight subcubes of the 2 × 2 × 2 cube, by nesting one thing inside another.<\/p>\n

In the above picture, a Back<\/b> cube is represented by a pink inner square and a Front<\/b> cube is represented by a blue outer square.<\/p>\n

Maybe we three dimensional folks can use a similar trick to allow us to think about 2 × 2 × 2 × 2 hypercube puzzles. More tomorrow.<\/p>\n","protected":false},"excerpt":{"rendered":"

One of the problems in describing things in four dimensions is that they are hard to see. If I used the classic view, where Z and W are both in perspective, it would probably just be confusing. So let’s ask ourselves: If we lived in a 2D Flatland, and we wanted to do puzzles on … Continue reading “2 \u00d7 2 \u00d7 2 \u00d7 2, part 2”<\/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\/13930"}],"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=13930"}],"version-history":[{"count":40,"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=\/wp\/v2\/posts\/13930\/revisions"}],"predecessor-version":[{"id":13970,"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=\/wp\/v2\/posts\/13930\/revisions\/13970"}],"wp:attachment":[{"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=13930"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=13930"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=13930"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}