{"id":375,"date":"2019-06-26T23:48:48","date_gmt":"2019-06-26T23:48:48","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=375"},"modified":"2020-01-03T22:34:19","modified_gmt":"2020-01-03T22:34:19","slug":"tesseract","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/tesseract\/","title":{"rendered":"Tesseract"},"content":{"rendered":"\n
\"\"<\/figure><\/div>\n\n\n\n

What does a cube look like in higher dimensions?<\/p>\n\n\n\n

Well, we can extrapolate by looking at lower dimensions. 
A 0-dimensional cube is a point, a vertex. 
A 1-dimensional “cube” is a line segment, with 2 vertices at either end. It is obtained from a 0-dimensional cube by thickening it in one dimension. 
A 2-dimensional “cube” is square, with 4 vertices, obtained by thickening up the line segment in a second dimension. 
A 3-dimensional “cube” is a cube, with 8 vertices, obtained from the square by thickening it in a third dimension.<\/p>\n\n\n\n

So, by extrapolation the 4-dimensional “cube”, also called a tesseract<\/em>or hypercube<\/em>, should have 16 vertices, and is obtained from a cube by thickening it up in a fourth dimension. Since we cannot easily visualize this, there are a number of ways we can understand this object by viewing projections, or “shadows” of it in 3-D. See Figure 1.<\/p>\n\n\n\n

Presentation Suggestions:<\/strong>
See if students can guess by extrapolation how many vertices the tesseract should have.<\/p>\n\n\n\n

The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong>
Thinking in four\u00a0dimensions\u00a0is not easy, and takes practice. However, a number of science fiction books have been written around this idea and explain it and possible applications quite well; see the reference for one notable example.<\/p>\n\n\n\n

How to Cite this Page:<\/strong> 
Su, Francis E., et al. “Tesseract.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n

References:<\/strong>
M. L’Engle, A Wrinkle in Time<\/a>.<\/p>\n\n\n\n

Fun Fact suggested by: <\/strong>
Francis Su <\/p>\n","protected":false},"excerpt":{"rendered":"

What does a cube look like in higher dimensions? Well, we can extrapolate by looking at lower dimensions. A 0-dimensional cube is a...<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[175,174,8,4],"class_list":["post-375","page","type-page","status-publish","hentry","tag-4-space","tag-four-dimensions","tag-geometry","tag-medium"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/375","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/comments?post=375"}],"version-history":[{"count":3,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/375\/revisions"}],"predecessor-version":[{"id":1677,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/375\/revisions\/1677"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=375"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=375"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}