Sunday, November 20, 2016

Intervals Are Not Polytopes

One dimensional spaces are qualitatively different from spaces with 2 or more dimensions. My definition of "linear", as a space who's subspaces have the correct number of regions does not suffice to make one-dimensional spaces convertible to number lines. Here is a counter-example.


Interpret this as a list of boundaries, each of which is a pair of half-spaces, and interpret the numbers as regions. As a two-dimensional space this is a simplex with empty vertex regions. As a one-dimensional space, each subspace has m+1 regions, where m is the number of boundaries. As a result of testing my Haskell code, I discovered this counter-example. I rewrote my isLinear and superSpace functions to behave differently for one-dimensional arguments.

Stay tuned for whether there are counter-examples of 2 or more dimension. If the proofs in previous posts are correct, then there will be none. Math is scientific in the sense that we can never know for certain that a proof is correct.

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