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corrected the angles at which different rhombs are assigned
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RavenclawPrefect
RavenclawPrefect
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At stated in the comments, all Penrose tilings contain any finite patch infinitely often, so your criterion doesn't narrow things down much. But judging from the sort of intuitive notion you're describing, it sounds like the pentagrid method is what you're looking for? It produces every Penrose tiling using a 5-tuple of real numbers in $[0,1]$ (up to a measure-0 set of invalid choices), so it is very easy to generate samples from.

The idea is to place down infinite "ladders" of parallel lines of constant spacing, one each at angles of $0^\circ, 72^\circ,\ldots,288^\circ$:

                                        enter image description here

We then associate to every intersection of two lines a skinny rhomb (if the intersecting lines differ by $72^\circ$$36^\circ$) or a fat rhomb (if they differ by $36^\circ$$72^\circ$). This describes the adjacency graph of the tiles, and with a little more work you can place them at the right coordinates. (From there, of course, it's easy to transform into a kite-and-dart tiling.) By normalizing and rotating, you can ensure your tiling is drawn from $\mathcal{P}$.

You may find this MSE post useful to get more information about pentagrids and constructing tilings from them - it contains many informative links about the process.

At stated in the comments, all Penrose tilings contain any finite patch infinitely often, so your criterion doesn't narrow things down much. But judging from the sort of intuitive notion you're describing, it sounds like the pentagrid method is what you're looking for? It produces every Penrose tiling using a 5-tuple of real numbers in $[0,1]$ (up to a measure-0 set of invalid choices), so it is very easy to generate samples from.

The idea is to place down infinite "ladders" of parallel lines of constant spacing, one each at angles of $0^\circ, 72^\circ,\ldots,288^\circ$:

                                        enter image description here

We then associate to every intersection of two lines a skinny rhomb (if the intersecting lines differ by $72^\circ$) or a fat rhomb (if they differ by $36^\circ$). This describes the adjacency graph of the tiles, and with a little more work you can place them at the right coordinates. (From there, of course, it's easy to transform into a kite-and-dart tiling.) By normalizing and rotating, you can ensure your tiling is drawn from $\mathcal{P}$.

You may find this MSE post useful to get more information about pentagrids and constructing tilings from them - it contains many informative links about the process.

At stated in the comments, all Penrose tilings contain any finite patch infinitely often, so your criterion doesn't narrow things down much. But judging from the sort of intuitive notion you're describing, it sounds like the pentagrid method is what you're looking for? It produces every Penrose tiling using a 5-tuple of real numbers in $[0,1]$ (up to a measure-0 set of invalid choices), so it is very easy to generate samples from.

The idea is to place down infinite "ladders" of parallel lines of constant spacing, one each at angles of $0^\circ, 72^\circ,\ldots,288^\circ$:

                                        enter image description here

We then associate to every intersection of two lines a skinny rhomb (if the intersecting lines differ by $36^\circ$) or a fat rhomb (if they differ by $72^\circ$). This describes the adjacency graph of the tiles, and with a little more work you can place them at the right coordinates. (From there, of course, it's easy to transform into a kite-and-dart tiling.) By normalizing and rotating, you can ensure your tiling is drawn from $\mathcal{P}$.

You may find this MSE post useful to get more information about pentagrids and constructing tilings from them - it contains many informative links about the process.

Source Link
RavenclawPrefect
RavenclawPrefect
  • 3.1k
  • 14
  • 24

At stated in the comments, all Penrose tilings contain any finite patch infinitely often, so your criterion doesn't narrow things down much. But judging from the sort of intuitive notion you're describing, it sounds like the pentagrid method is what you're looking for? It produces every Penrose tiling using a 5-tuple of real numbers in $[0,1]$ (up to a measure-0 set of invalid choices), so it is very easy to generate samples from.

The idea is to place down infinite "ladders" of parallel lines of constant spacing, one each at angles of $0^\circ, 72^\circ,\ldots,288^\circ$:

                                        enter image description here

We then associate to every intersection of two lines a skinny rhomb (if the intersecting lines differ by $72^\circ$) or a fat rhomb (if they differ by $36^\circ$). This describes the adjacency graph of the tiles, and with a little more work you can place them at the right coordinates. (From there, of course, it's easy to transform into a kite-and-dart tiling.) By normalizing and rotating, you can ensure your tiling is drawn from $\mathcal{P}$.

You may find this MSE post useful to get more information about pentagrids and constructing tilings from them - it contains many informative links about the process.