Taxi-Cab Geometry

[SGS]

Ok.... so I have to write a paper for my math class. I am writing my paper on expanding taxi-cab geometry to a three dimensional coordinate plane. I have some ideas I am going to put into my paper and was just wondering if anyone here may have some good things that may make for some good writing material. Any ideas are appreciated. Thanks in advance.

SGS

[Rhett]

What's a three dimensional coordinate plane?

[Siegmund]

One place you might look is at the shapes of crystals, especially those in the cubic system. They are made, at the slightly-larger-than-atomic level, by stacking piles of cubes to form the other shapes their crystals take on. (Or piles of rectangular bricks, for the tetragonal and orthorhombic systems, etc etc.)

The surface of the octahedron, an uncut diamond's natural form, is the sphere of the taxicab-geometry world, i.e., the set of points which are equidistant (in x+y+z distance) from the center of the crystal.

Under a microscope, you can see the "steps" in the non-horizontal and vertical faces of these crystals - sometimes you can even see striations on the surface of the crystal (sphalerite and pyrite) that reflect the stacking of the unit cells.