Quasicrystals are mathematical entities that are used for example in cosmology and quantum computing. Is it possible that quasicrystals can be used as data compression also? I don t know because I am not a mathematician. From community.wolfram netpage: “1D quasicrystals by fibonacci substitution and lattice projection”. Fibonacci numbers can be tribonacci (order-3 Fibonacci) numbers etc. Have quasicrystals capacity to store information greater than order-3 Fibonacci, Zero displacement ternary etc. number systems? Or more information storing capacity than bihederitary / tree- based number systems (Paul Tarau), or number systems represented in netpage “Novaloka maths on the number horizon - beyond the abacus”. Can quasicrystal contain more information? Strange concepts such as “time crystals” from quasicrystals have been proposed. Nothing to do with previous but “Mathematics without quantors is possible” by Erkki Hartikainen 2015, unfortunately this text is buried inside his very long internet publication that has nothing to do with math, so it is bit difficult to find. Nevermind his ideological stuff, focus to his scientific text only. He has also written “A new proposal for empirical geometry” 2015, and then “An ontological anti-relativist postulate in physics” which is easy to find. If quasicrystals can be used in video, audio, text or other data compression I don t know. Or space probe communication etc. where high data compresssion efficiency is needed. I don t actually know mathemathical description of quasicrystal, not even understand it so I don t know. Wofram graph: “Tridimensional trivalent graph”. “From prime numbers to nuclear physics and beyond” 2013, "Critical -wave functions and Cantor-set spectrum of a one-dimensional quasicrystal model"1987. “Electronic energy spectrum of cubic Fibonacci quasicrystal” 2001, “Between order and disorder, Hamiltonians for…” (chemnitz netpage). Multifractal system is relative to quasicrystal and multifractals have been used in data compression. “Super-resolution reconstruction of remote sensing image using multifractal…” and then experimental M3F multifractal image compression and other multifractal applications. Amplituhedron is used in quantum physics, it makes long equations shorter. From futurism netpage: “New discovery simplifies quantum physics” 2013. Amplituhedron is a sort of data compression method, so it can be used as data compression perhaps, and perhaps quasicrystals also. Also number system based on quasicrystal or amplituhedron could be made perhaps. It would be like index-calculus based number systems, based on differential equations or something like that, I don t know, I am not a mathematician. If amplituhedron makes equations much shorter, number system based on equations is then used in “amplituhedron number system”, if quasicrystal makes numerical information packed in small space, quasicrystal can also be used as base of some exotic number system. If that makes efficient data compression possible. Also “Base infnity number system” Eric James Parfitt, if amplituhedron or quasicrystal and base infinity number system are combined, very large numerical base can be represented as short amplituhedron or quasicrystal. That would perhaps lead to “infinity computer” or similar concepts. If amplituhedron and quasicrystal can be used as similar number base as infinity number system. It would perhaps not be infinite base number system, but very large number base system, “almost infinite range number system”. Floating point numbers have extremely large range already, but smaller accuracy. If accuracy can be very large also if aplituhedron or quasicrystal is used in some number system. Also “TGD as generalized number theory” by Matti Pitkänen has something about p-adic numbers and TGD theory of physics. If p-adic numbers can offer way to make “true compact numbers” (in “Floating point numbers as information storage and data compression” netpage) perhaps this text about TGD, or other texts relative to it, like concept of “fuzzy topologies” can offer way to make “true compact numbers” (“Topologies created by the fuzzy numbers” 2017 and “Connecting fuzzyfying topologies and generalized ideals by means of fuzzy preorders”, etc.), and “supra fuzzy topologies” (“A note on intuitionistic supra fuzzy soft topological spaces”, “Supra fuzzy topological spaces: fuzzy topology”). Or fuzzy topologies and/or p-adic numbers could make very efficient number system, like amplituhedron or quasicrystal. Again I don t know because I am not a mathematician. There is also skyrmion and knotted skyrmion, “Skyrmion resuffler comes to aid stochastic computing”. If skyrmion stochastic electronic circuit makes possible stochastic computing, which in turn perhaps makes possible to use “true compact numbers”, if stochastic computing is possible to combine with “true compact numbers”. I don t know, I am not a mathematician. Perhaps at some accuracy, because 100% or not even near to 100% accuracy is not needed, if it is possible to use “true compact numbers”, in stochastic or some other way, even very coarse accuracy is acceptable if “true compact numbers” can be used in computing, saving in information space can be huge, but if 100% accuracy is needed true compact numbers cannot be used perhaps. Stochastic computing, fuzzy topologies, supra fuzzy topologies, amplituhedrons, quasicrystals, skyrmions, knotted skyrmions are perhaps suitable for searches of true compact numbers that can be used in (fuzzy logic / stochastic etc.) computing. There is Benford s law also, maybe it can be used in data compression (it is already used in data compression in some applications) and also as “true compact number” making. “A simple explanation of Benford s law”, “Benford s law, Zipf s law and the Pareto distribution”, math stackexchange netpage: “Why does Benford s law (Zipf s law) hold?” 2010. Benford s law is somehow relative to “Borel sets” and “descriptive set theory”. “The modulo 1 central limit theorem and Benford s law”, “Leading digit laws on linear Lie groups”, “Order statistics and Benford s law”. “Python - Benford s law number generator inequality”, “Python - is there a random number generator that obey Benford s law?” If Benford s law can be used in the making of “true compact numbers”, or in other data compression. Borel sets / descriptive set theory etc. (I don t understand them because I am not a mathematician) may also help. Or theories that are similar to descriptive set theory. In Steven W. Smith: “DSP guide chapter 34: Explaining Benford s law” is that Benford s law is simple (or complex) antilogarithm and logarithm thing, and there is nothing mystic or unexplained about it. Something similar is sigma algebra, Zech logarithm, and ergodic theory. “Ergodic theory: interactions with combinatorics and number theory”. Math stackexchange questions tagged ercodic theory, “Combinatorially designed LDPC codes using…”, Borel hierarchy, “Expansions of Black-Scholes processes and Benford s law”. If anything (ergodic theory etc.) helps in data compession or making of “true compact numbers”. There is winding number, nonzero rule, residue theorem, and point in polygon principles, if any of them helps to make “true compact numbers” or helps in data compression. In the book “God of the universe” by L. Charles Arnold is about logarithmic spiral and relationship between logarithmic spiral / golden ratio and Fibonacci series. Golden ratio and Fibonacci series are in some way connected to each other.