A Penrose tiling

This project generates a SVG file with a Penrose tiling (an aperiodic tiling of the plane with two tiles), using the cairo-fortran bindings as a fpm (Fortran Package Manager) dependency.

The algorithm uses acute and obtuse golden triangles, with two interlaced recursive procedures. See the figures in https://fr.wikipedia.org/wiki/Pavage_de_Penrose to understand.

See the project https://github.com/vmagnin/hat_polykite for more information on the ein Stein problem and the tiling of the plane, and the most recent results in this domain.

Be careful with the number of recursions: the SVG file is growing like the Fibonacci sequence! 12 recursions will generate a 12.7 Mio SVG file. 15 recursions will give a 228 Mio file and your SVG viewer may freeze or crash...

Requirements and dependencies

You need:

• a modern Fortran compiler, for example GFortran or the Intel ifort/ifx compilers. See the Fortran-lang.org compilers page for other compilers.
• The Cairo development files (libcairo2-dev package in Ubuntu).
• The Fortran Package Manager fpm.

Running the program

Just type fpm run and fpm will manage the dependencies and build and run the program:

$ fpm run
 + mkdir -p build/dependencies
Initialized empty Git repository in penrose_tiling/build/dependencies/cairo-fortran/.git/
remote: Enumerating objects: 23, done.
remote: Counting objects: 100% (23/23), done.
remote: Compressing objects: 100% (22/22), done.
remote: Total 23 (delta 0), reused 14 (delta 0), pack-reused 0
Unpacking objects: 100% (23/23), 82.99 KiB | 1.51 MiB/s, done.
From https://github.com/vmagnin/cairo-fortran
 * branch            HEAD       -> FETCH_HEAD
cairo-enums.f90                        done.
cairo-auto.f90                         done.
triangles.f90                          done.
libpenrose_tiling.a                    done.
main.f90                               done.
penrose_tiling                         done.
[100%] Project compiled successfully.
 ----------------------------------
           8 recursions
 These are Fibonacci numbers:
         610 blue acute triangles
         377 yellow obtuse triangles
 ----------------------------------
 Output file: penrose_tiling.svg
 ----------------------------------

Licenses

This project is licensed under the GNU General Public License version 3 or later.

The documentation is under the GNU Free Documentation License 1.3 or any later version. The figures are under CC-BY-NC-SA 4.0 license.

Cairo documentation

• https://cairographics.org/documentation/
• API Reference Manual: https://cairographics.org/manual/

Bibliography

• Roger Penrose, Set of tiles for covering a surface, Patent GB1548164A (US-4133152-A), filed 1975-06-25.
• Hargittai, Istvan, et Balazs Hargittai. « 2020 Physics Nobel Laureate Roger Penrose and the Penrose Pattern as a Forerunner of Generalized Crystallography ». Structural Chemistry 32, no.1 (1st February 2021): 1‑7. https://doi.org/10.1007/s11224-020-01669-8
• https://en.wikipedia.org/wiki/Einstein_problem
• https://en.wikipedia.org/wiki/Penrose_tiling
• https://fr.wikipedia.org/wiki/Pavage_de_Penrose
• https://en.wikipedia.org/wiki/Golden_ratio
• https://en.wikipedia.org/wiki/Golden_triangle_(mathematics)