# Beyond 3-D, Chapter 3

## David Akers

While I found the fourth dimensional slicing techniques discussed in
Chapter 3 to be the most interesting, I was also impressed by some of the
techniques used to describe 3-dimensional slicing -- especially those
which involved symmetry. The slice which divided a 3-dimensional cube
"vertex first," creating a hexagon, reminded me of a problem from an old
math test that I took. The problem was as follows (and perhaps this could
be used as an advanced exercise..):

*
Imagine a large cube which has been partitioned into 27 equal sub-cubes.
Now slice the large cube with a plane perpendicular to its diagonal such
that the cube is divided exactly in half. How many of the 27 sub-cubes
are hit by this slice? (One can use symmetry to determine the answer)
*

A random thought which also occured to me . . consider the equation
which is used to generate the Mandelbrot set: f(z) = z^2 + a, where z and
a are complex variables. There are really FOUR variables involved here,
since both **z** and **a** contain real and imaginary components. Is
the mandelbrot set (obtained by varying **a**, and iterating **z**)
a two-dimensional slice of a four-dimensional object? Are Julia sets
(obtained by fixing **a** and then iterating) other 2-d slices of this
same object? I am a little confused.

David Akers