Proof and the Art of Mathematics: Examples and Extensions by Joel David Hamkins

Author:Joel David Hamkins [Hamkins, Joel David]
Language: eng
Format: epub
Tags: Game Theory, Mathematics, Proof; discrete mathematics; graph theory; infinity; induction; number theory; proofs without words; theory of games; order theory; real analysis, General, Number Theory, Geometry
ISBN: 9780262542203
Google: mSAXEAAAQBAJ
Publisher: MIT Press
Published: 2021-02-23T00:25:38.324552+00:00

Theorem.  The figures that can be formed with lattices in the infinite 2 × 1 regular brick lattice are exactly the same as those in the 1 × 1 integer lattice, because these are in fact the same lattice. Consequently, the only regular polygons to be found in the brick lattice are squares.

Proof.  Consider the 2 × 1 regular brick tiling, where by regular I mean that on each subsequent row, the midpoint of each brick lines up with the edge between two bricks on the next row. If you consider carefully the lattice points of the brick lattice, the main observation to be made is that they are exactly the same as the lattice points of the 1 × 1 integer lattice.

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