decide - Cant Decide? Undecide! Chaim Goodman-Strauss In my...

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Can’t Decide? Undecide! Chaim Goodman-Strauss In my mathematical youth, when I frst learned oF G¨odel’s Theorem, and computational undecidability, I was at once Fascinated and strangely reassured oF our limited place in the grand universe: incredibly mathematics itselF establishes limits on mathematical knowledge. At the same time, as one digs into the Formalisms, this area can seem remote From most areas oF mathematics and irrelevant to the e±orts oF most work-a-day mathematicians. But that’s just not so! Undecidable problems surround us, everywhere, even in recreational mathematics! 1 Three Mysterious Examples Somehow these simple questions seem difficult to resolve: 1.1 Mysterious Example #1 Tilings are a rich source oF combinatorial puzzles. We can ask, For a given tile, whether or not it admits a tiling : that is, does there exist a tiling oF the plane by copies oF this tile? ²or many examples, this is utterly trivial: clearly the tile at leFt in the fgure below does admit a tiling, and the tile at middle leFt does not. One might discover a simple prooF that the tile at middle right does not admit a tiling either 1 , though it is more difficult to work out just how large a region you can cover beFore getting stuck. But it’s a reasonable bet that you will not be able to discover whether or not the tile at right, discovered by J. Myers in 2003, admits a tiling, at least not without resorting to some sort oF brute-Force calculation on a computer! Try this For yourselF! A downloadable fle with tiles to cut out and play with has been placed at http://mathfactor.uark.edu/downloads/myers tile.pdf In general, then, we have Input: At i le . and a Decision Problem: Does the given tile admit a tiling of the plane? 1 Hint: the tile, discovered by C. Mann, can be viewed as a cluster of hexagons, with some edges bulging inwards and some bulging out — but there are more bulging inwards than outwards. Etc. . . . 1
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With enough brute-force eFort, in some circumstances, we can answer this problem: We might simply enumerate all possible con±gurations admitted by the tile, covering larger and larger disks. If the tile does not admit a tiling, eventually there will be some sized disk we can no longer cover, we run out of con±gurations to enumerate, and we then know the answer to our problem: No , the tile fails to admit a tiling. If a tile does not admit a tiling, the “Heesch number” is a measure of the complexity of such a tile, as the largest possible combinatorial radius of disks it can cover (in other words, the maximum number of concentric rings copies of the tile can form); C. Mann discovered the current world record examples, with Heesch number 5 [22, 23]. But how can we determine whether an arbitrary given tile does admit a tiling?
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decide - Cant Decide? Undecide! Chaim Goodman-Strauss In my...

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