Date: 04/25/2001 at 17:30:49 From: Robert Hofmann Subject: A four-dimensional problem Why does a tesseract contain eight cubes?
Date: 04/25/2001 at 19:11:18 From: Doctor Schwa Subject: Re: A four-dimensional problem Hi Robert, There are a lot of way to think about this problem. One way is by analogy: a line segment has two endpoints a square has four line segment edges a cube has six square faces so, continuing the pattern, a tesseract has eight cubes ... (but I'm not sure what the right word is here for the 3D "sides" of a 4D object). Another way is to look at coordinates. Looking at the coordinate vertices of a cube, (0,0,0), (1,0,0), (0,1,0), (1,1,0), (0,0,1), (1,0,1), (0,1,1), (1,1,1), you can find six sets of four vertices that make a square. Similarly, using the vertices of a tesseract, (0,0,0,0) and so on (the same set of eight with an extra ,0 at the end, and eight more with a ,1 at the end) you can list the eight sets of eight vertices that make cubes. In particular: the eight that all have 0 in the first coordinate make a cube, the eight that all have 0 in the second coordinate make a cube, and so on (with 0 or 1 in each of four places; there are eight of them). Does that answer your question? If not, please write back. - Doctor Schwa, The Math Forum http://mathforum.org/dr.math/
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