Surface Tension in Taxicab Geometry

Taxicab geometry is a wonderful world, where common truths of normal (Euclidean) geometry don’t hold. For example, The length of the diagonal of a square is twice the length of its side (not square root of 2). Another example: the figure whose every point is equidistant from its center is not a circle; it is a diamond shape.
Now it would be interesting to see how soap bubbles will behave in Taxicab geometry. It is easier to examine the continuous geometry version, where the size of zig-zags is infinitely small, because the tools of calculus can be used.

1. What is the equilibrium shape of a single bubble? That is, which shape with a given area has the minimum parameter?
2. What is the equilibrium shape of two equal-area bubbles? When the two bubbles share a common boundary, the total boundary length shrinks.
3. What if the two bubbles have different areas?

The corresponding questions for discrete Taxicab geometry (all lines lie on a square lattice) are somewhat harder, but once some intuition has been developed for the continuous case, one can take the continuous solutions as a starting point and then look for variations.

A Question That Should Not Be Asked In Interviews

Given a board that is n x n squares, where n is a power of 2, and that has one square removed from an arbitrary position, give an algorithm to completely cover the board with L shaped tiles made from three adjacent unit squares. An 8×8 board is shown in the figure; one removed square is shown in black; one L shaped green tile is placed on the board. Remember that the tiles can be placed in any orientation.

 

Famous Places Mumbles #16