Eisenstein Snowflakes

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

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


Snapshots


Details

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.



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send