Number Spirals
See the patterns that underly algebra

The numbers in the rectangular grid are either increasing by rows, or arranged in a spiral from the centre.

Choose or enter a function (using numbers, the pronumeral n, operators +, -, *, / and brackets), and view the numbers or the patterns that result.

Zoom out to see more numbers or a larger pattern (zoom is continuous if you keep the button pressed), and press and hold on the image to magnify.

2 × 2

           

Select Function: 2n (even numbers) n (n - 1) (product of
consecutive numbers) Primes
2n - 1 (odd numbers) 4n2 + n Enter expression in n
n2 (perfect squares) 4n2 - n

About the activity

Algebra extends arithmetic since it gives us a way to describe and work with many numbers at once — but not just any numbers, but numbers that are related in very specific ways. An algebraic expression represents such relationships, and this activity gives the opportunity to visualise the pattern that results from particular expressions.

There are two views: a normal rectangular grid, and a spiral arrangement where the numbers start in the middle and spiral outwards. Changing from the regular grid view to the spiral view changes the layout of the numbers selected by an expression, and remarkable additional patterns are frequently revealed. Understanding the structure of these patterns, predicting various aspects, and uncovering related patterns and their associated expressions helps develop an understanding of algebra as generalisation.

Some particular problems:

  • Consider 2n, 3n, 4n etc in both grid and spiral views and note the differences in pattern. Choose a higher number, predict the pattern and then test your prediction.
  • Now consider n × n + n, n × n + 2n, n × n + 3n, etc in both views. Again choose a higher number, predict the pattern and then test your prediction.
  • Find the expression that gives the other half of the 4n2 ± n patterns.

The link with primes and number spirals is the most famous, thanks to Stanislaw Ulam. There is a intriguing predominance of specific diagonals in the distribution of the primes.

Euler found an equation that generates prime numbers for all n from 0 to 40. Try entering the expression n × n - n + 41 and comparing with the primes view.