Experimenting with the Ulam Spiral

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

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

The Ulam spiral is named after Stanislaw Ulam, who discovered it by chance in 1963 while doodling on scratch paper at a scientific meeting.


Integers are placed on a square spiral. By marking the prime numbers, you can see that they tend to form patterns lining up in diagonal, horizontal, and vertical lines. This phenomenon can be best observed by drawing a large number of primes.

When starting the spiral with 41, a long diagonal line is clearly visible.

Triangular numbers (of the form ) as well as pentagonal numbers (of the form ) give rise to pinwheel patterns on the spiral.

Some things to try

• Play "number of integers" and/or "start from" to see how spiral patterns evolve.

• Set "number of integers" to its maximum and observe the pattern, a very regular, pleasant, spiral shape.

• Slide "start from" to its maximum. Notice the pattern disappears. It looks just like random noise.

• Play "number of integers" and/or "start from" to watch spiral patterns evolve.


Contributed by: Giovanna Roda (March 2011)
Open content licensed under CC BY-NC-SA



See also:

Triangle Numbers on Ulam Spiral by Adrian Leatherland http://yoyo.cc.monash.edu.au/~bunyip/primes/triangleUlam.htm

Ulam Spiral Images by Matthew M. Conroy at http://www.madandmoonly.com/doctormatt/mathematics/ulamSpirals/ulamSpirals.htm

Ulam spiral at http://en.wikipedia.org/wiki/Ulam_spiral

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.