{"id":13971,"date":"2013-12-09T23:50:50","date_gmt":"2013-12-10T04:50:50","guid":{"rendered":"http:\/\/blog.kenperlin.com\/?p=13971"},"modified":"2013-12-10T22:06:33","modified_gmt":"2013-12-11T03:06:33","slug":"2-%c3%97-2-%c3%97-2-%c3%97-2-part-3","status":"publish","type":"post","link":"http:\/\/blog.kenperlin.com\/?p=13971","title":{"rendered":"2 \u00d7 2 \u00d7 2 \u00d7 2, part 3"},"content":{"rendered":"

Continuing our discussion from yesterday, if I were a Flatland creature trying to understand a cube, it’s not good enough merely to look at it. I need to do things with it.<\/p>\n

Before going on, it is important to clarify something: In this discussion I sometimes use color. The color itself is not important. Color just provides a way to label things, to make the pictures easier to talk about. The only thing that is really important is the geometry: Where things are. That’s where all of the insight will come from.<\/p>\n

Our Flatland creature, after staring at her picture of the 2 × 2 × 2 cube, decides she would like to rotate it and see what happens. Being a rather clever Flatlandian, she knows that a cube can be rotated in at least three different ways: (1) about its Left\/Right dimension; (2) about its Up\/Down dimension, and (3) about its Front\/Back dimension.<\/p>\n

To make it easier to see what happens when the cube rotates, we’re going to add a little more color for labeling. Specifically, we will make the colors of the four “bottom” cubes more vivid:<\/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

Now let’s see what happens when our Flatland friend rotates this cube 90o<\/sup> in each of the three ways:
\n
\n
\n<\/p>\n

about
horizontal<\/br>axis<\/TH>
\n		<\/TD>
\n		about
vertical
axis<\/TH>
\n		<\/TD>
\n		about
front\/back
axis<\/TH>
\n<\/TR><\/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

<\/TD><\/p>\n

<\/TD><\/p>\n

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

<\/TD><\/p>\n

<\/TD><\/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

<\/TD><\/p>\n

<\/TR><\/p>\n

<\/TABLE><\/p>\n

What does the result tell us, if anything, about what rotation in three dimensions might seem like to a Flatland creature? More tomorrow. <\/p>\n","protected":false},"excerpt":{"rendered":"

Continuing our discussion from yesterday, if I were a Flatland creature trying to understand a cube, it’s not good enough merely to look at it. I need to do things with it. Before going on, it is important to clarify something: In this discussion I sometimes use color. The color itself is not important. Color … Continue reading “2 \u00d7 2 \u00d7 2 \u00d7 2, part 3”<\/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\/13971"}],"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=13971"}],"version-history":[{"count":81,"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=\/wp\/v2\/posts\/13971\/revisions"}],"predecessor-version":[{"id":14052,"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=\/wp\/v2\/posts\/13971\/revisions\/14052"}],"wp:attachment":[{"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=13971"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=13971"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=13971"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}