# Beyond 3-D, Chapter 9

## David Akers

### Slicing the hypercube..

If you have a JAVA capable browser, check out my new applet.
You can now slice a hypercube from a vertex...
### Triangles on Spheres

It was stated on p. 186 that "the area of the triangular region on the
sphere is precisely the amount by which its angle sum exceeds 180
degrees." This, I assume, indicates that there is a linear relationship
between angle sum and area of the triangular region? I'm a little
confused by this. I'd like to see more of the mathematics behind this
statement.
### Taxicab Geometry

Although it wasn't mentioned in the book, I've seen this described as an
alternative geometry before. As I understand it, the basic difference
between this and planar Euclidean geometry is that the distance between
two points is defined as delta_x + delta_y, instead of sqrt (dx^2 + dy^2),
by the pythagorean theorem. Thus, a circle, defined as the locus of all
points equidistant from a center, would in fact look like a square (or a
diamond), if projected into our cartesian coordinate system. Which of the
axioms of Euclidean plane geometry hold in this system? Also: What happens
when we extend Taxicab geometry to the third or fourth dimensions?
Questions to think about..

Prof. Banchoff's response

### Links

Dan
Margalit's week 12 paper.

Michael
Matthews' week 12 paper.

Jeremy
Kahn's week 12 paper.

David Akers