Generating 2D Metallic-Mean Quasiperiodic Tilings with NKS Substitution Rules

This Demonstration shows an alternative, computationally more efficient way of generating the colored two-dimensional (2D) quasiperiodic square tilings reported in [1]. The computation of the tilings in this Demonstration is achieved by using a set of substitution rules on colored squares for the Fibonacci (or golden ratio) system based on A New Kind of Science [2, Notes. Chapter 5: Section 4].
In this Demonstration, the rules in [2] are generalized to obtain 2D quasiperiodic tilings based on other metallic mean family (MMF) numbers. Such numbers are usually indexed as , since they are the irrational positive solution of the quadratic equation with and positive integers. The positive solutions for are the golden number , the silver number , the bronze number and so on for higher values with . Another set of MMF numbers, comprising the copper and nickel numbers, are obtained by setting and or , respectively [3]. After setting the indices and to choose the desired MMF number, the "nesting index " control lets you generate 2D tiling patterns of different complexity that correspond to the structures obtained in [1] using 2D grid products of coordinate points. You can also check that a 2D Fibonacci tiling pattern is obtained that is very similar to that published in [2].
"grayscale": black, white and gray
"default colors": a default set of three colors defined by the Hue function computed at each third of the unit interval
"random": a random set of three colors defined by the RandomColor function computed for three random real numbers always in the range
Move the slider "random color seed" to change the randomly chosen set of colors. The "permute colors over tiles" control lets you cycle color permutations over tiles. As anticipated in [1], the number of colors always turns out to be three for these systems, since horizontal and vertical symmetry of rectangular bands is accounted for in the code (unlike the example in [2]). Hence, the number of color permutations over the tiles is .

SNAPSHOTS

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DETAILS

Snapshot 1: square tiling of a 2D quasiperiodic system obtained from Fibonacci substitution rule (), mesh edge highlighting enabled and grayscale set of colors as reported in [2]
Snapshot 2: square tiling of a 2D quasiperiodic system obtained from Fibonacci substitution rule () with nesting index and default set of colors
Snapshot 3: square tiling of a 2D quasicrystal obtained from silver-mean substitution rule () with nesting index and default set of colors
Snapshot 4: square tiling of a 2D quasicrystal obtained from bronze-mean substitution rule () with nesting index and random set of colors
References
[1] J. Alfonsi. "Coloring 2D Metallic-Mean Quasicrystal Tilings with a Mesh-Based Method" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/Coloring2DMetallicMeanQuasicrystalTilingsWithAMeshBasedMetho.
[2] S. Wolfram, A New Kind of Science, Champaign, IL: Wolfram Media, Inc., 2002. www.wolframscience.com/nks/notes-5-4--substitution-systems-based-on-other-shapes.
[3] V. W. De Spinadel, "The Metallic Means Family and Art," Journal of Applied Mathematics, 3(1), 2010 pp. 53–64. www.aplimat.com/files/Journal_volume_3/Number_1.pdf.
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