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The Ammann Chair can be used in an infinite dissection of a rectangle, where the pieces have a scaling factor of $ k = 1/\sqrt{\phi} = 0.786151...$. The largest piece has area $\sqrt{5}$ and longest edge 1. The full rectangle has area and longest edge $3\phi+1$.

InfiniteAmmannChairs

The first piece has sides multiplied by $k^0$, piece 1 has sides multiplied by $k^1$, and so on.

Is there a scaling factor $g$ for some other shape with $k < g <1$ that allows an infinite sequential rectangle packing?

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1 Answer 1

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Looks like the Amman chair is an unnecessary sophisticated tool which was not designed for this task (the way it stands now), nor is its use really justified. Dissecting a rectangle into an infinite series of scaled-down copies of the same shape is not a big deal, and can be done in a multitude of ways. Any real number from $(0,1)$ would do as the scaling factor. Dissection 1 Wait, we can make it even simpler than that: the building block itself can be a rectangle. (Of course in that case we have somewhat less freedom, since the shape of that rectangle is intimately related to that of the big rectangle, and to the scaling factor as well.) Dissection 2

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  • $\begingroup$ What are the scaling factors for each of these? $\endgroup$
    – Ed Pegg
    Commented Aug 11, 2016 at 2:48
  • $\begingroup$ Around 0.8. But like I said, it can be made anything. $\endgroup$
    – Ivan Neretin
    Commented Aug 11, 2016 at 5:58

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