
Date: Sun, 6 Jun 1999 19:34:03 0600 (CST)
Subject: measuring the fourth dimension
Name: Suzanne
Who is asking: Teacher
Level: Middle
Question:
Hi! The other day, two of my students asked a very interesting question: Is the fourth dimension measured with hypercubes? Their reasoning went like this: Lines are 1D and are measured with line segments, which are part of a line. Planes are 2D and are measured with squares, which are part of a plane. Space is 3D and is measured with cubes, which are part of space. So, logically, hypercubes would be used to measure the fourth dimension.
I'm in way over my head. Can anyone help me out? Many many thanks!
Suzanne
PS Joel is in Grade 6 and Brad in Grade 7.
Hi Suzanne
Your students have asked just the right question  they seem to
possess the imagination that is essential for doing mathematics. What's
more, they have hit upon the correct answer. In fact, what you call a
"hypercube" is sometimes referred to as a "measure polytope" since
hypercubes of side length 1 are used to measure the content of
"hypersolids."
Here is how the terminology works.
POLYTOPE is the general term of the sequence
point, segment, polygon, polyhedron, ... .
A line can be tiled with unit segments, a plane with unit squares,
3space with unit cubes, 4space with unit hypercubes, ...
The number of unit segments that fit into an arbitrary segment is
its length; the number of unit squares that fit into a plane figure is its
area; the number of unit cubes that fit into a solid is its volume; the
number of hypercubes that fit into a hypersolid is its content (or
hypervolume, if you prefer, but "content" is the usual terminology).
I don't know any reliable references devoted to four
dimensions at the appropriate level. Perhaps a grade 6
student might enjoy FLATLAND by Edwin A. Abbott. There is an
excerpt in THE WORLD OF MATHEMATICS, edited by James R. Newman, Volume 4,
pages 2383 to 2396, which may be more readily available than the full text.
Enough is reproduced there to give the idea and, as a bonus, it comes with
useful comments.
Cheers,
Chris

