{"id":555,"date":"2019-06-29T22:24:26","date_gmt":"2019-06-29T22:24:26","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=555"},"modified":"2019-12-20T23:20:37","modified_gmt":"2019-12-20T23:20:37","slug":"slices-of-hanging-cubes","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/slices-of-hanging-cubes\/","title":{"rendered":"Slices of Hanging Cubes"},"content":{"rendered":"\n

Hang a cube from one of its vertices. Now, if you slice it horizontally through its center, you get a hexagon.<\/p>\n\n\n\n

What if you do this with a 4-dimensional cube, i.e., a\u00a0tesseract? The slice will yield a 3-dimensional object— what does it look like?<\/p>\n\n\n\n

Answer: you get a\u00a0octahedron!<\/p>\n\n\n\n

Presentation Suggestions:<\/strong>
Use lower dimensional analogies to help students visualize higher dimensional objects.<\/p>\n\n\n\n

The Math Behind the Fact:<\/strong>
It is not hard to see (using symmetry arguments) that the object you get must be regular. By analogy with the slice of the 3-cube, the slice of the 4-cube must cut every “face”. The number of “faces” of a 4-cube is eight. The only regular 8-sided solid is an octahedron.<\/p>\n\n\n\n

Visualizing\u00a0high dimensional\u00a0objects can be taxing, but fun!<\/p>\n\n\n\n

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

References:<\/strong>
R. Vakil, A Mathematical Mosaic<\/a>, 1996.<\/p>\n\n\n\n

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

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