Apologies if I fail to use correct terminology below. I am an artist trying to resolve a geometric problem.
I want to create a torus with specific poloidal and toroidal radius that has a surface that divides into 162 diamond tiles of equal area, (which would mean the shape of each tile would stretch/contract depending on location). I have tried mapping a specific square image (attached) composed from 162 diamond tiles of equal area onto a torus using a 3d graphics program. - You can see the 162 derives from a 9x9 formation (doubled to count the tiles between).-
Let us say the 5th horizontal row of diamond tiles down from the top reading (left to right) 6, +5, +4, +3... lie exactly along the outer equator of the torus, and the horizontal row of diamond tiles five complete rows up from the bottom reading (left to right) -8, -9, -1, -2.... lie exactly on the inside equator line.
The program deals with the problem of wrapping the map around the torus surface by allowing it to stretch and contract horizontally only. Therefore the tiles which are projected on or near the inner equator of the torus are contracted, and take up a smaller area compared to those on the outer equator which are expanded and take up a larger area. This helps visualisation but does not solve my problem.
My question is this:
a) Can a torus be created that will divide into a grid with the same 9x9 structure of 162 diamond tiles of equal area?
b) If so how would I go about creating a square or rectangular map for this torus, that when stretched around the torus (which would be of specific proportions) in a 3d program would give the same impression of all the tiles being of equal area (but not shape of course)?
Thanks,
Tom