# Beyond 3-D, Chapter 9

## David Akers

### Slicing the hypercube..

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### 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?