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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 

  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

Does that answer your question?  If not, please write back.

- Doctor Schwa, The Math Forum   
Associated Topics:
College Higher-Dimensional Geometry
High School Higher-Dimensional Geometry

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