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I am trying to use this tiling but I can't reconstruct it accurately and really have no knowledge of this type of math. I'm seriously really ignorant to how this can be produced.

If there is anyway someone can explain how to construct these tiles in somewhat of a simple manner, I would be very grateful.

Here is the link to the tiling: https://tilings.math.uni-bielefeld.de/substitution/conch/#ref-anchor-Rau82

Thank you!

Jimswid
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  • The substitution rules are given at the top of the page. Do you want an explanation for how to follow the substitution rules? – Dan Rust Feb 19 '19 at 18:49
  • Yes. I have no prior knowledge of tiling or substitution. – Jimswid Feb 20 '19 at 19:11
  • I would suggest checking out the wikipedia pages for Penrose tilings and substiution rules. Everything you should need to know will be there. If not, let me know :). – Dan Rust Feb 20 '19 at 19:27
  • I looked through those pages and also a paper, I found on the subject. I understand a bit more now, but am still struggling to understand how I can recreate these shapes accurately. – Jimswid Feb 22 '19 at 18:54
  • perhaps you could be more specific in what part of the process you don't understand – Dan Rust Feb 22 '19 at 20:25
  • So I'm confused on how the substitution rule actually works for this pattern. But, even beyond how certain tiles substitute, I do not understand how I can go about constructing the tiles themselves. I imagine that by understanding the substitution, I can began using that to determine the structure of the tiles. I'm also confused on how the inflation factor is used for formatting the patch. – Jimswid Feb 22 '19 at 23:09
  • I have an understanding of the basics of substitution, but that was using colored squares. Beyond that, I'm having trouble translating the concepts for this one. – Jimswid Feb 22 '19 at 23:14
  • Okay, so I'm picking more of it up. If the inflation factor is (x^4)-x+1=0, is "x" the level of inflation? And if that is the case, then would the first graphic of the "grey rhombus -> blue rhombus" mean that the blue rhombus is similar to the blue rhombus by an inflation of ((2)^4)-2+1=15? And does that mean 15%? If that is the case, I'm unsure how the graphic with the "red rhombus -> yellow rhombus" is actually an inflation, because they do not seem similar. – Jimswid Feb 23 '19 at 00:17
  • That's not quite what the inflation factor means. Actually this example is a bit more complicated than is standard, so I shouldn't have assumed that you could work it out just from the links. So the inflation factor is a normally a single number which is bigger than 1. If the inflation factor is 1.618, then that means that every tile is first inflated by a factor of 1.618 and then it replaced by tiles of the original size. In this case, the inflation factor isn't a number however, it's a linear map. If you're not familiar with matrices, then this could be difficult to explain. – Dan Rust Feb 23 '19 at 01:16
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    There seems to be very little in the way of literature on the Conch tiling. What I have found at least is an Oberwolfach mini-workshop report on the Pisot conjecture which contains a short report by Edmund Harris in which he gives a (very brief) description of how to substitute the prototiles for the Conch tiling in terms of edge substitutions of the tiles: https://www.mfo.de/document/0910b/OWR_2009_13.pdf This may be difficult to read without a sufficient background. If you don't have that background, I would maybe suggest contacting Edmund directly (he's active on twitter and easy to find). – Dan Rust Feb 23 '19 at 01:50
  • Thank you very much! I've been trying to figure this out for weeks. I appreciate the help. – Jimswid Feb 23 '19 at 03:21

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