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Updated: Jul 14, 2021

A self-avoiding walk (SAW) is a sequence of moves on a grid that does not visit the same point more than once.

SAWs are used to study how networks form, including social networks, biological networks and computer networks. They have provided inspiration to artists and designers.

Make a hand-embroidered self-avoiding walk

You will need:

  • An embroidery hoop

  • Fabric

  • Needle

  • Thread

  • Dice.


  1. Place your fabric in the embroidery hoop (Video: how to mount fabric in a hoop).

  2. Thread your needle and tie a knot at the end (Video: how to thread a needle).

  3. Use the needle to insert the thread into a hole in the centre of hoop.

  4. Make one stitch in the fabric using 'backstitch' (Video: how to do backstitch).

  5. Roll the dice to determine the direction of next stitch.

  6. Each dice number has a direction on the fabric:

    1. If you roll a 1 = stitch up 1 square

    2. 2 = stitch right 1 square

    3. 3 = stitch down 1 square

    4. 4 = stitch left 1 square

    5. If you roll a 5 or 6, ignore and roll again.

  7. Ignore any rolls that result in a new stitch crashing into the chain of stitches or going back on the chain of stitches.

  8. Keep going until you reach an edge or have no options to move because of surrounding stitches.

  9. Well done!

  10. Share a photo of your self avoiding walk on social media and use #SAWstitch .

For more help

  • You can view an information sheet below

saw stitch instructions
Download PDF • 428KB
  • You can view a short video below.


How many stitches (or steps) are in your SAWstitch?

Why did your SAWstitch stop? Did it get stuck?

With your remaining thread, can you make a second SAWstitch in the same fabric?

How do the SAWstitch patterns differ?

If you had more thread, how many different SAWstitch patterns do you think you could make? How would they be different?

Next time you go for a walk, could you use a SAW to find a new route?


In this video, you can see a self-avoiding walk with 100 million steps! It has been generated using a computational technique which uses an algorithm to roll the dice. Video credit: Nathan Clisby.

In this article called 'How to Avoid Yourself' from American Scientist, you can read more about self-avoiding walks. Article credit: Brian Hayes

How to Avoid Yourself
Download PDF • 1.84MB

This research paper explores the average number of steps that self-avoiding walks take on a square lattice. The paper contains some examples of what self-avoiding walks look like when they get stuck or ‘trapped’. ‘An average self-avoiding random walk on the square lattice lasts 71 steps’. Paper credit: S. Hemmer and P.C. Hemmer.

In this research paper scientists have explored self-avoiding walks in 2D, 3D and 4D. One of the images from the paper might look similar to your SAWstitch. 'Multifractality of Self-Avoiding Walks on Percolation Clusters'. Paper credit: Viktoria Blavatska and Wolfhard Janke


SAWstitch is a collaborative project by Lorna Dougan, Paul Beales, Christa Brown and Kalila Cook and benefited from many enjoyable conversations with the Physics Craft Gathering group. Our interest in SAWstitch stems from our research on network formation of protein hydrogels. We use biomolecules called proteins as our 'thread' and create networks by making them connect through sticky points on their surface, through photo-activated chemical cross-linking. This results in the creation of a network with a defined shape and material properties.

You can read more about our research here and in this Science highlight here.

We acknowledge funding from the Engineering and Physical Sciences Research Council (EPSRC), Royal Academy Engineering (RAE) and Economic and Social Research Council (ESRC).

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