I don’t know about you, but for me the results of yesterday’s experiment in “seeing a cube like a Flatlander” were discouraging.<\/p>\n
After all, you and I have something in common: We are very familiar with three dimensional objects like cubes. All of us can understand what is going on when we see a cube rotating, and some people can even mentally rotate a cube entirely in their mind, without needing to look at a cube.<\/p>\n
Yet when I look at the representations of a rotated cube that I made in yesterday’s post, I experience a disconnect. Intellectually I know that the rotations are correct, because I can trace the path of each of the cube’s eight corners as it travels to its new position.<\/p>\n
But looking at this visual representation does not give me the automatic intuition that I get from viewing the usual perspective view of a rotating cube. My spatial intuition does not quite survive the change in visual representation — even though the new representation is indeed spatial and geometric.<\/p>\n
The conclusion I reach from this (since I am an optimist by nature) is the following: In order to use a technique like this to gain an intuition about four dimensions, we would first need to train ourselves to develop a spatial intuition for nested “outer\/inner” as a stand-in for “front\/back”.<\/p>\n
One way to do this would be to develop a game that portrays a 3D cube using “outer\/inner” to represent depth, and then challenge the player to solve a progression of puzzles that require rotating that cube. Eventually the player’s existing spatial intuition about cubes and 3D rotation would transfer over to this new representation.<\/p>\n
And then we would be ready to try to tackle the 4D hypercube. To do this, we would start with a true 3D cube, and add “outer\/inner” as a graphical stand-in for a fourth dimension.<\/p>\n
I’m not saying this multi-stage plan would work. I’m just saying that it seems like a reasonable thing to try. I’d love to get the opinions of others on this.<\/p>\n","protected":false},"excerpt":{"rendered":"
I don’t know about you, but for me the results of yesterday’s experiment in “seeing a cube like a Flatlander” were discouraging. After all, you and I have something in common: We are very familiar with three dimensional objects like cubes. All of us can understand what is going on when we see a cube … Continue reading “2 \u00d7 2 \u00d7 2 \u00d7 2, part 4”<\/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\/14053"}],"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=14053"}],"version-history":[{"count":18,"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=\/wp\/v2\/posts\/14053\/revisions"}],"predecessor-version":[{"id":14071,"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=\/wp\/v2\/posts\/14053\/revisions\/14071"}],"wp:attachment":[{"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=14053"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=14053"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/blog.kenperlin.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=14053"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}