Crocheting Hyperbolic Planes


This website was originally created for students in an upper division college geometry course, for a lesson on hyperbolic geometry. The lesson plan included learning to crochet hyperbolic planes. Anyone interested in the subjects of hyperbolic geometry, crocheting, math education, fiber art, sustainability, or the intersection of math & art, is welcome here!



You will see a carousel “scroll bar” across the top of this site.  Use that to navigate to the pages you need to access. It scrolls from side to side so be sure to click on the arrows to see everything.

STEP 1: Start by clicking on “Why Crochet Hyperbolic Planes?”.  Feel free to browse through that page and be sure to watch the video The Beautiful Math of Coral.

STEP 2: If you don’t know how to crochet, click on the link in the carousel labeled How To Crochet For Beginners.  Watch the 5 minute video and then begin by practicing the chain stitch.

STEP 3: Finally, click on Homework Instructions and create the two models you find there.  You are also welcome to try making the optional hyperbolic space models if you’d like.

The rest of the items in the carousel and the links on the sidebar are there for you to browse if you’re interested in further exploring the use of crochet artwork in math education.



STEP 1: Why Crochet Hyperbolic Planes?

In 1997 a math professor at Cornell University, Dr. Daina Taimina, discovered that it was possible to create a physical model of a hyperbolic plane using the art of crochet. Many mathematicians thought a physical model was impossible to create. A 2005 article in the New York Times explains: Professor Lets Her Fingers To The Talking.

In an interview for Cabinet magazine, Dr. Taimina and her husband, David Henderson, discuss crocheting hyperbolic planes: “Until the nineteenth century, mathematicians knew about only two kinds of geometry: the Euclidean plane and the sphere. It was a deep shock to their community to find that there existed in principle a completely other spatial structure, whose existence was only discerned by overturning a two-thousand-year-old prejudice about “parallel” lines. The discovery of hyperbolic space by the Hungarian mathematician Janos Bolyai and the Russian mathematician Nicholay Lobatchevsky in the 1820’s and 1830’s marked a turning point in mathematics and initiated the formal study of non-Euclidean geometry. Almost two centuries later, Daina Taimina a mathematician at Cornell University made a physical model of the hyperbolic plane – a feat many mathematicians had believed was impossible – using, of all things, crochet. Taimina and her husband David Henderson, a geometer at Cornell, are the co-authors of “Experiencing Geometry,” a classic textbook on Euclidean and non-Euclidean space.”

Dr. Taimina has written a book on the subject and also has a blogShe explains hyperbolic geometry using crocheted models in the video, Geometries Beyond Euclid from the Annenburg Learner website, Mathematics Illuminated.

In Chapter 9 of her award-winning book, Dr. Taimana discusses how hyperbolic geometry has applications in many fields and is of interest to those studying computer science, mathematics, biology, chemistry, medicine, network security, music, art, and physics. Hyperbolic geometry is even used to help us visualize the world wide web.




The Beautiful Math of Coral

Taking the concept of hyperbolic crochet models further, science writer Margaret Wertheim uses this technique to create museum exhibits of crocheted coral reefs.  She explains on this stunning Ted Talks video, The Beautiful Math of Coral, citing the work of Dr. Taimina.  She and her sister, Christine, founded the Institute for Figuring, which is “… is an educational organization dedicated to enhancing the public understanding of figures and figuring techniques. From the physics of snowflakes and the hyperbolic geometry of sea slugs, to the mathematics of paper folding and graphical models of the human mind, the Institute takes as its purview a complex ecology of figuring.

Here is their explanation of hyperbolic space: Hyperbolic Spaces Online Exhibit.

You might also want to check out Margaret Wertheim’s book, A Field Guide To Hyperbolic Space in which she explores the intersection of higher geometry and feminine handicraft.  This book includes patterns for crocheting a wide variety of hyperbolic models.

Are you ready to create your own crocheted hyperbolic plane?  Proceed to Step 2 if you need to learn to crochet.  If you already know how to crochet, proceed to Step 3Have fun!

STEP 3: Homework Instructions

Crochet the following 2 models:

1.  Crochet a plane – 12 x 12. Twelve rows with twelve stitches per row.

Detailed diagrams and instructions are in this PDF document:

Foundation Row: Crochet 12 chain stitches plus 1 turning chain.

Row 1: Turn your work so the last chain you made (the turning chain) is now on your right.  Skip the turning chain and make one single crochet (sc) stitch in each chain of the 12 chains.  Finish the row by making one turning chain.

Row 2: Turn your work so the last stitch you made (the turning chain) is now on your right. Skip the turning chain and make one sc in each of the 12 sc stitches from row 1. Finish the row with a turning chain.

Rows 3 through 12: Repeat Row 2.


2. Crochet a hyperbolic plane. Start with a foundation row of 12 chains and then  make 12 rows using single-crochet stitches.  Increase in every 4th stitch (N=4) in each row.

Detailed instructions and diagrams are in this PDF document:
Instructions for Crocheting a Hyperbolic Plane



Would you like to experiment with crocheting other varieties of hyperbolic space?  You can make hyperbolic pseudospheres (hyperbolic cones).  You can make double-sided hyperbolic planes. You can make VERY curly forms.  You can make very gradual negative curvature. Here are the instructions, including photos: IFF-CrochetReef-HowToHandout


Gallery of Hyperbolic Models

What if…?

What if you increased in every other stitch?

What if you increased in every stitch?

What if you increased at a different rate in each row?

What if you started out with a longer foundation row?

What if you increased at random rates?

What if you started with a circle of chains for a foundation row instead of a line?


Taking Crochet To A Higher Plane

Another version of instructions for crocheting a hyperbolic plane


The Hyperbolic Crochet Coral Reef Project

Margaret Wertheim and Christine Wertheim, founders of the Institute For Figuring, initiated the Hyperbolic Crochet Coral Reef project.  This has been displayed throughout the world, including The Smithsonian.

EXCERPT from their website:

The inspiration for making crochet reef forms begins with the technique of “hyperbolic crochet” discovered in 1997 by Cornell University mathematician Dr. Daina Taimina. The Wertheim sisters adopted Dr Taimina’s techniques and elaborated upon them to develop a whole taxonomy of reef-life forms. Loopy “kelps”, fringed “anemones”, crenelated “sea slugs”, and curlicued “corals” have all been modeled with these methods. The basic process for making these forms is a simple pattern or algorithm, which on its own produces a mathematically pure shape, but by varying or mutating this algorithm, endless variations and permutations of shape and form can be produced. The Crochet Reef project thus becomes an on-going evolutionary experiment in which the worldwide community of Reefers brings into being an ever-evolving crochet “tree of life.” 

Crochet Coral Reef Slideshow

Conservation Crochet.

The Beautiful Math Of Coral

Ted Talk Video of Margaret Wertheim explaining the coral reef project


Fiber Art Crochet

Enjoy some fun with fiber art crochet!

Gallery of Freeform Crochet by Prudence Mapstone

Check out the Crochet Art of Joana Vasconcelos


Check out photos of a variety of Crochet Sculptures


Check out these stunning crochet art images by artist Agata Olek: Olek Fiber Art


Check out Daina Taimina’s Fiber Sculptures


Gabrielle Meyers’ Hyperbolic Surfaces

Hyperbolic Surface

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Hyperbolic Geometry on Pinterest

Pinterest is an online pinboard for organizing and sharing things you love


Fiber Art Crochet Pieces

by Susan Darge Lombardo