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Mathematical Knitting

klein bottle hatOn the face of it, it’s hard to imagine two things as unrelated as mathematics and knitting.    And yet, here we have a website devoted to mathematical knitting:

A Klein-bottle hat (seen at right)?  A Mobius scarf?  Ingenious stuff, and a testament to our unlimited creativity and resourcefulness, if not our practicality.

The Klein bottle hat actually reminds me of the Klein bottle one of my students made out of plaster.


  1. Emily VA says:

    It’s only hard to imagine that math and knitting are unrelated if you’re not a knitter.

    The very first thing a knitter has to do when they sit down to make an object is “swatch”, which is make a small sample of the relevant stitch so they can see what their ratio of stitches and rows is to inches or cm. Making sure those ratios are right is the key to making a garment that fits.

    And people, being 3D objects not made of flat planes, also require curved surfaces in their knitting to make well-fitting garments. There’s tons of math in everyday knitting, even before you move into making abstract 3D objects (also cool).

  2. MrHonner says:

    Thanks for the thoughtful commentary, Emily! I’m admittedly not a knitter, but I definitely see the algorithms there, and I can imagine how precise execution of those algorithms, and understanding of how each part fits in the whole, is crucial to a knitting project.

    I like your description of the “swatch”, making sure the ratios all work out. Sounds like similarity and dilation play a part there!

    Be sure to check out this cool mathematical knitting I saw on the street:

    • Emily VA says:

      Yep – that’s actually the post that led me to this one… I subscribed to your blog in the last month or so — I love all the cool math photos and good thinking about teaching math, but missed your first knitting post until you linked to it today.

  3. Amanda says:

    I am an engineering major and a knitter. I am working on a project for my calculus class where we have been asked to find calculus in the world. I know that there are many talented knitter/mathematicians that have been producing some wildly complete math concepts in knitting. Are there Calculus concept knitting projects?
    If anyone has a pattern for one please email it to me. I have been researching calculus in knitting, but I can’t find any published patterns. I am not an advanced enough knitter to design my own pattern yet.
    Thank you.

  4. MrHonner says:

    Hi Amanda-

    I know of some “advanced” calculus knitting projects. The hyperolic plane, for example, is a surface in space that is of interest to Calculus students because of its strange geometry. You can find instructions on the website above (

  5. carrie says:

    I took up knitting in earnest when in my late 40s. And it was knitting that finally allowed me to fully understand (and to use daily) factorization! And though I knew of calculus, I had never studied it but recently had the occasion to peruse a calculus textbook and immediately saw the potential application to knitting project design. Thank you for your post; it’s good to know that others are thinking along these lines.

    Have you heard about or seen the “Crochet Coral Reef” project? Here is an excerpt from

    “The Crochet Reef Project was inspired by the technique of hyperbolic crochet originally developed by Dr Daina Taimina, a mathematician at Cornell. In 1997 Dr Taimina discovered how to make models of the geometry known as “hyperbolic space” using the art of crochet. Until that time many mathematicians believed it was impossible to construct physical models of hyperbolic forms; yet nature had been doing just that for hundreds of millions of years. It turns out that many marine organisms embody hyperbolic geometry in their anatomies – among them kelps, corals, sponges, sea slugs and nudibranchs. Thus the Crochet Reef not only looks like a coral reef, it draws on the same underlying geometry endemic in the oceanic realm.”

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