Pick a starting shape and an increase rate. Increase every 6th stitch and the surface stays flat. Go below 6 and it has to ruffle outward to fit the extra material — the same mechanism Daina Taimiņa used to crochet the hyperbolic plane.
For over a century after hyperbolic geometry was discovered in the 1820s and 1830s, mathematicians had no physical model of it — paper and tape versions existed but were fragile and tore easily. In 1997, Cornell mathematician Daina Taimiņa realized that the same exponential growth could be made durable and tactile with a crochet hook: increase the stitch count in every round by a constant amount, and the fabric is forced to ruffle outward to make room for itself. Her book, Crocheting Adventures with Hyperbolic Planes, turned that into a teaching tool now used in geometry classrooms worldwide.
This page is a small simulation of that idea: instead of crocheting it by hand, it computes where every stitch should sit in 3D space, given only a starting shape and how often you increase. The stitch lines are always visible; toggle "shaded surface" to fill in the fabric between them so it reads as a solid object rather than a wireframe.
Pick "circle," set how many stitches to start with, and pick an increase rate of N. Every round after the first adds one extra stitch for every N stitches of the round before it — the same logic as a real pattern instruction like "2 sc in next st, 1 sc in next, repeat around." The starting count and the increase rate are independent: a ring of 5 stitches increasing every 4th and a ring of 7 increasing every 4th follow the same rule but grow at different paces, since the increase count each round depends on how many stitches are already there. The rows slider controls how many rounds to build out, so you can watch a slow ruffle develop over many rounds or jump straight to a dramatic one.
The number 6 is the dividing line. A flat crochet circle follows the textbook rule of one increase per 6 stitches (the familiar "+6 per round"). Set the slider to 6 and the model has just enough new material each round to stay flat. Below 6, you're adding increases faster than flat needs — the surface has nowhere to put the extra fabric except up and down, which is exactly the ruffling Taimiņa used to model the hyperbolic plane. Above 6, the round comes up short of what flat would need, so it pulls inward instead, the same way decreases close up the top of a sphere.
For each round r, the model compares the actual increase density to the flat reference ratio of 1-in-6:
When xr is positive, round r has more increases than the flat rule allows, and the excess accumulates round over round into the ruffle amplitude you see on the model. When it's negative, the round is short of what flat needs, and the same accumulation pulls the surface into a dome instead. This is a discrete approximation of Gaussian curvature — in the continuous version of this idea, a surface's curvature at a point is exactly what governs whether geodesics on it spread apart (hyperbolic, negative curvature), stay parallel (flat, zero curvature), or converge (spherical, positive curvature). The ripple pattern itself is drawn with as many peaks as there are real increase points in that round, so the visual frequency isn't decorative — it's tied to where the stitches actually split. Those increase points are spaced evenly around each round rather than clustered at one spot, the same way a real pattern spreads its increases out so the piece doesn't pull lopsided.
A few real, documented crochet pieces that use this exact mechanism, for comparison against the model above:
images aren't reproduced here for copyright reasons — each card links to the original source.