[math-fun] Help wanted identifying a recursive tiling
I have a recursive substitution-rule to tile the plane, which seems simple enough but nevertheless does not seem to be discussed in the normal places (like the tilings encyclopedia, http://tilings.math.uni-bielefeld.de/substitution_rules). Google searches haven't helped either (example: an Image search with the keywords: tiling regular pentagon triangle). The rules for my tiling can be seen here: http://mrob.com/pub/math/images/penta-tiling.jpg I consider this a "simple" and "obvious" tiling because there are only two rules, and each rule tries to maximize the size of the pentagon(s) on the right-hand side. However, it is "non-simple" in the sense that, with repeated applications of the substitution rules, increasingly many different sizes of triangles and pentagons are produced. - Robert -- Robert Munafo -- mrob.com
While I'm not sure exactly how "tiling" should be defined, it must surely involve a finite number of tile types modulo isometry. As you remark, there are an infinite number of sizes of triangle involved in your dissection; so I doubt very much whether it would qualify. Perhaps you might explain in more detail about what exactly you want to know about it? Fred Lunnon On 9/5/10, Robert Munafo <mrob27@gmail.com> wrote:
I have a recursive substitution-rule to tile the plane, which seems simple enough but nevertheless does not seem to be discussed in the normal places (like the tilings encyclopedia, http://tilings.math.uni-bielefeld.de/substitution_rules). Google searches haven't helped either (example: an Image search with the keywords: tiling regular pentagon triangle).
The rules for my tiling can be seen here:
http://mrob.com/pub/math/images/penta-tiling.jpg
I consider this a "simple" and "obvious" tiling because there are only two rules, and each rule tries to maximize the size of the pentagon(s) on the right-hand side. However, it is "non-simple" in the sense that, with repeated applications of the substitution rules, increasingly many different sizes of triangles and pentagons are produced.
- Robert
-- Robert Munafo -- mrob.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Fred lunnon wrote:
[...] Perhaps you might explain in more detail about what exactly you want to know about it?
Good question Fred. I guess what I was looking for is a web site or a book where "my" tiling is described and it tells who originally discovered or invented it. Mike Stay wrote:
This isn't really a tiling: you can take nearly any shape and "tile" the plane with it in the way you describe--given any unfilled region, just scale down your shape until it fits and recurse. See the Apollonian gasket, for example.
As far as I can tell, my tiling is no different in that respect from the Ammann tilings, for example. (See http://tilings.math.uni-bielefeld.de/substitution_rules/ammann_a3) So, like I said above, I want some book or website that explains how and why they are different. Some kind of organized taxonomy for recursive substitution-systems for space-filling patterns. It should include everything on tilings.math.uni-bielefeld.de but should also include more information (that website seems to be an abandoned project)
On 9/5/10, Robert Munafo <mrob27@gmail.com> wrote:
The rules for my tiling can be seen here:
-- Robert Munafo -- mrob.com
Robert, I believe the distinction being made is this: For Ammann tilings, Penrose tilings, etc., the substitution rules, when applied properly, always result in a fixed number of different tiles. For example, with a Penrose tiling, you never need more than 2 tile types, regardless or how deeply you carry out the expansion. Or to put it another way, you can generate arbitrarily large tilings without ever needing more than 2 different tile types. This is because the tile sizes can always be made to "sync up", with as many old sizes being eliminated as new sizes being introduced. For your rules to qualify, I believe you would need to be able to provide some number n, which is the maximum number of tile types (taking size into account) ever needed to genererate arbitrarily large tilings. For your substitution rules, it appears that n is infinite. Tom Robert Munafo writes:
As far as I can tell, my tiling is no different in that respect from the Ammann tilings, for example. (See http://tilings.math.uni-bielefeld.de/substitution_rules/ammann_a3) So, like I said above, I want some book or website that explains how and why they are different.
In 1984 or so, somewhere in the UK, perhaps Nottingham University, or maybe Leeds, I attended a lecture by Roger Penrose in which he explained how he first found his aperiodic tilings. He essentially started with Robert Munafo's two diagrams, but then, just as Tom Karzes suggests below, he went on to describe how he struggled to reorganize/regroup the iteration of the diagrams to obtain only finitely many *sized* tile types. The first "success" involved 5 or 6 tiles I think, one of which he called the "witch's hat." I think the "kite and dart" (ie, two tile) tilings came later. I hesitate to send this because I'm sure others know this history much better than I do. On Sun, Sep 5, 2010 at 9:41 PM, Tom Karzes <karzes@sonic.net> wrote:
Robert, I believe the distinction being made is this: For Ammann tilings, Penrose tilings, etc., the substitution rules, when applied properly, always result in a fixed number of different tiles. For example, with a Penrose tiling, you never need more than 2 tile types, regardless or how deeply you carry out the expansion. Or to put it another way, you can generate arbitrarily large tilings without ever needing more than 2 different tile types. This is because the tile sizes can always be made to "sync up", with as many old sizes being eliminated as new sizes being introduced.
For your rules to qualify, I believe you would need to be able to provide some number n, which is the maximum number of tile types (taking size into account) ever needed to genererate arbitrarily large tilings. For your substitution rules, it appears that n is infinite.
Tom
Robert Munafo writes: > As far as I can tell, my tiling is no different in that respect from > the Ammann tilings, for example. (See > http://tilings.math.uni-bielefeld.de/substitution_rules/ammann_a3) So, > like I said above, I want some book or website that explains how and > why they are different.
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There's actually quite a lot of info on the wiki page: http://en.wikipedia.org/wiki/Penrose_tiling The first set of tiles he found, "P1", has 6 tile types (consisting of three different pentagon types, i.e. with difference edge-matching rules, and three other shapes): http://upload.wikimedia.org/wikipedia/commons/8/8c/Penrose_Tiling_%28P1%29.s... The different pentagon edge-matching rules are indicated by different colors, but to meet the strict definition, these actually need to be enforced by adding bumps to the edges to prevent disallowed combinations. Tom Thane Plambeck writes:
In 1984 or so, somewhere in the UK, perhaps Nottingham University, or maybe Leeds, I attended a lecture by Roger Penrose in which he explained how he first found his aperiodic tilings.
He essentially started with Robert Munafo's two diagrams, but then, just as Tom Karzes suggests below, he went on to describe how he struggled to reorganize/regroup the iteration of the diagrams to obtain only finitely many *sized* tile types. The first "success" involved 5 or 6 tiles I think, one of which he called the "witch's hat."
I think the "kite and dart" (ie, two tile) tilings came later.
I hesitate to send this because I'm sure others know this history much better than I do.
On Sun, Sep 5, 2010 at 9:41 PM, Tom Karzes <karzes@sonic.net> wrote:
Robert, I believe the distinction being made is this: For Ammann tilings, Penrose tilings, etc., the substitution rules, when applied properly, always result in a fixed number of different tiles. For example, with a Penrose tiling, you never need more than 2 tile types, regardless or how deeply you carry out the expansion. Or to put it another way, you can generate arbitrarily large tilings without ever needing more than 2 different tile types. This is because the tile sizes can always be made to "sync up", with as many old sizes being eliminated as new sizes being introduced.
For your rules to qualify, I believe you would need to be able to provide some number n, which is the maximum number of tile types (taking size into account) ever needed to genererate arbitrarily large tilings. For your substitution rules, it appears that n is infinite.
Tom
Robert Munafo writes: > As far as I can tell, my tiling is no different in that respect from > the Ammann tilings, for example. (See > http://tilings.math.uni-bielefeld.de/substitution_rules/ammann_a3) So, > like I said above, I want some book or website that explains how and > why they are different.
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This isn't really a tiling: you can take nearly any shape and "tile" the plane with it in the way you describe--given any unfilled region, just scale down your shape until it fits and recurse. See the Apollonian gasket, for example. On Sun, Sep 5, 2010 at 1:52 PM, Robert Munafo <mrob27@gmail.com> wrote:
I have a recursive substitution-rule to tile the plane, which seems simple enough but nevertheless does not seem to be discussed in the normal places (like the tilings encyclopedia, http://tilings.math.uni-bielefeld.de/substitution_rules). Google searches haven't helped either (example: an Image search with the keywords: tiling regular pentagon triangle).
The rules for my tiling can be seen here:
http://mrob.com/pub/math/images/penta-tiling.jpg
I consider this a "simple" and "obvious" tiling because there are only two rules, and each rule tries to maximize the size of the pentagon(s) on the right-hand side. However, it is "non-simple" in the sense that, with repeated applications of the substitution rules, increasingly many different sizes of triangles and pentagons are produced.
- Robert
-- Robert Munafo -- mrob.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
participants (5)
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Fred lunnon -
Mike Stay -
Robert Munafo -
Thane Plambeck -
Tom Karzes