Associated Topics || Dr. Math Home || Search Dr. Math

### Tesseract

```
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,
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).

- Doctor Schwa, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Higher-Dimensional Geometry
High School Higher-Dimensional Geometry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search