Eisenstein Snowflakes

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The Eisenstein integers are a ring of complex numbers that cover the Gaussian plane in a triangular lattice. Each number is the sum of a regular integer and an integral multiple of ω, the positive third root of unity. Eisenstein integers share many properties of the regular integers, including well-defined prime numbers and GCDs. This Demonstration shows the so-called totients of a given Eisenstein integer—that is, the integers below the number that share no factors with that number. It is simply a happy holiday coincidence that the resulting patterns resemble snowflakes.

Contributed by: John Wiltshire-Gordon (March 2011)
Open content licensed under CC BY-NC-SA



The controls allow you to change the two parts of the Eisenstein integer: the real part and the part that extends into the complex plane at a 120° angle.

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