I'm thinking of developing a rectangular tiling based on the Padovan sequence (think of the Fibonacci mosaic). It seems like something that should exist, but I can't find anything in the literature. Do you know of something like this and where I can find it?
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1$\begingroup$ I can't think of "the Fibonacci mosaic" since I've never heard of it. Reference? Link? $\endgroup$– Gerry MyersonCommented Mar 21, 2020 at 22:59
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$\begingroup$ "Padovan tiling" turns up on top of page 7 of people.csail.mit.edu/ddeford/DeFord_Enumerating_Distinct.pdf but with no illustration. $\endgroup$– Gerry MyersonCommented Mar 21, 2020 at 23:03
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$\begingroup$ @GerryMyerson See the two top tilings here: en.wikipedia.org/wiki/Fibonacci_number. I'm using the term mosaic because different size tiles are used. And thanks for the link, but it's not quite what I'm after. $\endgroup$– Cye WaldmanCommented Mar 22, 2020 at 0:01
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$\begingroup$ OK. You wouldn't take a tiling with equilateral triangles, would you? en.wikipedia.org/wiki/Padovan_sequence $\endgroup$– Gerry MyersonCommented Mar 22, 2020 at 0:11
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$\begingroup$ No. That one is very well known. I've done that one and the plastic pentagon gnomon as well. $\endgroup$– Cye WaldmanCommented Mar 22, 2020 at 2:56
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Can you please define rectangular tiling?
The Fibonacci numbers count number of ways to tile a $1\times n$-strip with rectangles of size $1 \times 1$ and $1\times 2$. This is easy to see from the recurrence.
Now, the Padovan sequence count number of ways to tile a $1\times n$ rectangle with rectangles of size $1\times 2$ and $1\times 3$. One needs to be a bit careful with initial conditions, so that there is one way to tile the $1\times 1$-rectangle.
answered Mar 21, 2020 at 20:47
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$\begingroup$ By rectangular tiling I mean a whorled figure where the tiles circle around and increase in size according to some sequence or numbers raised to successive powers. See en.wikipedia.org/wiki/Fibonacci_number for an example with the Fibonacci sequence. Of course, the Padovan sequence would require rectangles rather than squares. $\endgroup$ Commented Mar 22, 2020 at 0:04
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$\begingroup$ @CyeWaldman Which figure do you mean? (There are 10.) "Rectangular tiling" suggests the third one whose caption begins "Thirteen $(F_7)$ ways" showing tilings of a $1 \times 6$ board with $1\times1$ and $1\times 2$ tiles, in which case Per's answer seems appropriate. $\endgroup$ Commented Mar 23, 2020 at 13:30
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$\begingroup$ @BrianHopkins Thanks for looking. I'm talking about the first and second figures on that web page. The point being to create a tessellated figure using only the integers in the sequence. $\endgroup$ Commented Mar 23, 2020 at 15:12
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$\begingroup$ @CyeWaldman So you want a tiling of the plane using rectangles whose side lengths are sequential Padovan numbers, hopefully in some sort of spiral. $\endgroup$ Commented Mar 23, 2020 at 19:25
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$\begingroup$ @BrianHopkins Well, that's exactly what I'm talking about and I've developed it already. I'm merely trying to find out if it's been done before. I didn't want to go posting it anywhere without verifying that it hasn't been done before. I apologize if I've caused any misrepresentations. $\endgroup$ Commented Mar 23, 2020 at 21:20