Crochet a hyperbolic plane
The instructions for crocheting a hyperbolic plane are surprisingly easy given the complex surface which is the result. Just perform increases at regular intervals, and soon your crochet will start to resemble a coral or flower, showing the wiggly shape characteristic of hyperbolic geometry.
Knit a hyperbolic mushroom
The instructions for knitting a hyperbolic mushroom are the same as those for crocheting a coral: perform increases at regular intervals. These little mushrooms knit up quickly and show that the strange shape of fungi is just a result of a simple mathematical rule.
Fold a Sonobe cube
With a simple unit of modular origami called a Sonobe unit, you can make many interesting shapes. First look at the instructions for making a single Sonobe unit, then follow the instructions for assembling 6 Sonobe units into a cube.
If you become a master of the Sonobe cube, why not try to assemble 12 or 30 Sonobe units into stellated octahedra and icosahedra (respectively)?
Fold an origami dodecahedron
The instructions for this modular origami dodecahedron involve assembling 30 units of origami, and can be made using a set of sticky notes. For an extra challenge, try creating this model using only three colours, so that the three different colours meet at each corner. It's harder than it looks!
Stitch a number star
These instructions use simple number patterns to create beautiful stars, leaving lots of room for creativity where you can design your own patterns. But why stick with circular patterns, when you could also experiment with number patterns in different shapes?
(These activities were designed by Perth embroidery expert Ann-Marie Anderson-Mayes at Beautiful Stitches.)
A related craft idea is to create curved patterns from straight lines, in a technique called string art.
Make a Möbius strip
Möbius strips are weird and wonderful mathematical shapes which can be created using paper, crochet or knitting. Marvel at their strange one-sided nature, try to cut them in half, and experiment with what happens with different numbers of twists. Whatever you do, the Möbius strip will surprise you at every turn.
And if a single Möbius strip is not enough for your curiosity, try attaching two together to make these amazing Möbius hearts, as demonstrated by Matt Parker.