Discrete Homogenous Surfaces

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This software is written by software people. The terms and definitions used here are not mathematically rigorous. If you find the ideas here interesting, write to us at bhushit at sourceforge dot net. It will be our honour to explain to you what we think and see it formalized. This software tries to address conditions that arise increment or decrement operations are performed on maximum and minimum of partial ordered discrete sets. We limit our discussions to maximum 2 partial ordered sets. The software here addresses discrete sets. As per our limited understanding of mathematics, the domain of discourse falls at the cross section of Topology, Geometry and Algebra. We have some intriguing questions: Going by this classification, we find there are many types of spheres and hemispheres (where x dimension is cylindrical but y dimension has pole like formations). We also find that if "abnormal conditions" - like "rolling over this co-ordinate", "changing the other co-ordinate only/also" etc.occur only at boundaries, there are only a few types of shapes possible - flat board, cylinders, tori, Mobius' strips, Klein's bottles, spheres, hemispheres and something like "double Mobius' strip". Are there only so many shapes? (Please see "Examples of Segments and Surfaces" link.) Why do we need only two parameters to fix location but three to ascertain the next location on certain shapes like spheres or hemispheres? How many dimension do these shapes really have? Why do only certain boundary conditions pair with the other? (For example, you can have one boundary shapes like hemispheres but never three boundary shapes.) For a game on various boards, we have various situations - for example, it is possible to checkmate a king with just a king and a rook on a conventional board but seems impossible to do so on a cylindrical board. So we need to now separate "boundary specific" strategies from "boundary independent" strategies in the game theory. Many puzzles become impossible to solve or become trivial on various boards. For example, we have proven by brute force that it is impossible to solve 8-queen puzzle on a cylindrical chess board. This may lead to interesting research. Why do we think the world around is normally flat? For example, why do we never enquire about "the neighbor" of a matrix element? Why do we never wonder about the third side of an equation? Why do we always think "in-betweenness" as a linear concept?