Square Matrix Permutations

Repeated application of a particular permutation of the elements of an matrix based on traversing diagonals results in the original matrix. The number of iterations for to complete the cycle is the same as the length of the period of the periodic sequence that results from the Fibonacci sequence modulus known as the Pisano period.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


For a 3×3 matrix the permutation takes into , for example.
The size of the matrix at the top left is given by the "size of first matrix" slider. This matrix and the next 19 matrices are represented at the same time. Holding the mouse over any matrix reveals its size and the corresponding Pisano period.
Use the "permutation" slider to rearrange the elements of each matrix and watch the matrices synchronize and desynchronize with each other. Choose a different "color scheme" to enhance viewing the weaving patterns.
The "cycle length" is the least common multiple of the 20 Pisano periods corresponding to the 20 displayed matrices.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+