Technique · Fields
Quasicrystal waves
Plane waves in five, seven or more directions added together: patterns with rotational symmetry that never repeat, sliding and re-forming.
Quasicrystal waves is a field drawn for every pixel by the graphics chip. Every piece on Synapse Drip is made from a recipe that layers techniques like this one, and you can change any of it in the Studio.
Good to know
- Roger Penrose published his non-repeating tilings in 1974, long before anyone knew that a real material could behave the same way.
- A repeating crystal can have two, three, four or six-fold rotation symmetry and no other; five-fold, seven-fold and ten-fold patterns can never repeat, which is why they were thought impossible in a crystal.
- In 2007 Peter Lu and Paul Steinhardt argued that girih tile patterns in medieval Islamic architecture, such as those in the Darb-i Imam shrine in Isfahan (1453), come very close to quasicrystalline order.
- Dan Shechtman saw a ten-fold diffraction pattern in an aluminium-manganese alloy in 1982, which no ordinary crystal can make; he was awarded the 2011 Nobel Prize in Chemistry for the discovery.
- Natural quasicrystals exist: a mineral found in the Khatyrka meteorite, in the mountains of far-eastern Russia, was identified in 2009 as one.
The mathematics
- Each of the 7 waves is cos(k (x cos a + y sin a) + phase), with the directions a spread evenly around a circle; because 7-fold symmetry cannot tile the plane, the sum is never periodic, though it comes arbitrarily close to repeating.
- Penrose tilings and patterns like these are slices of a higher-dimensional lattice: a five-fold tiling is a thin slice of a cubic lattice in five dimensions, projected down to the plane.
- Shifting every phase by the same amount slides the pattern, but shifting them by different amounts changes it, which is why the pattern keeps re-forming rather than just drifting.
In art
- Muslim artists drew girih patterns of stars and pentagons with only a straightedge and compass, centuries before the mathematics of aperiodic order was written down.
- Op art painters and later digital artists used sums of waves to make surfaces that seem to pulse, since the eye cannot find a repeating cell to rest on.
More ways to use it
Related techniques
Nebula and stars · Liquid · Droplets · Network · Terrain and maps · Neuron · Order to chaos · Fractal tree