# Prime Numbers and Architecture

Prime Numbers are widely used in technology and cryptography. They are sometimes also used in small scales such as building gears and evolution of life cycle of Cicada insect. Are they in any way used in buildings and architecture? Can the following properties and applications or any other property of prime numbers be applied in architecture?

$(1)$ Distribution and Pseudo-Randomness

$(2)$ Composition of Arbitrarily large arithmetic progression (Green-Tao Theorem)

$(3)$ Appearance of primes in logarithmic spirals related to a metallic ratio such as golden ratio etc.

$(4)$ Ulam's spiral, Prime Number Theorem, Quadratic appearances or Dickson's conjecture

$(5)$ Highly Composite Numbers, Divisibility and the Fundamental Thoerem of Arithmetic

$(6)$ Constructability of polygons with Fermat prime number of sides.

$(7)$ Sieve theory and Eratosthenes Sieve

$(8)$ GCD, LCM, Euler Totient, Tau, Sigma and Pi functions

$(9)$ Modular Congruences

$(10)$ Random Number Generators

$(11)$ Equiangular P-gons

Biologically and mathematically speaking, humans aren't great random number generators. Small numbers chosen by humans are usually odd, more specifically prime numbers or numbers with very few factors, more than the probability of it being prime in the specific range of values. Can this show why prime numbers appear in architecture and other human made random choices for construction, irrelevant of the property of being prime, more than usual?

Moreover, are there certain kinds of geometric constructions which involve primes such as $(6)$ which can be used in real life construction?

• Now posted to MO, mathoverflow.net/questions/300012/…, without notice to either site. May 12, 2018 at 12:59
• wonder why this got downvotes, +1 here, would love to see some answers to this... Jun 4, 2018 at 11:06
• You may be interested in a paper by Manfred R. Schroeder entitled "The unreasonable effectiveness of number theory in acoustics". More broadly there is a published proceedings of an AMS short course called "The unreasonable effectiveness of number theory" that I believe includes this topic. Jun 5, 2018 at 0:01
• The encyclopedia of architecture has some interesting remarks about the use of prime numbers. Jun 6, 2018 at 11:48
• An interesting geometric property is that "$p \gt 2$ is a prime number iff any equiangular p-gon with rational side lengths is regular" (T. Andreescu, B. Enescu, Mathematical Olympiad Treasures, Birkhäuser, 2004).
– dxiv
Jun 6, 2018 at 23:15