Hyperbolic Afghan, {3, 7}

"Hyperbolic Afghan {3, 7}" illustrates the tiling of the hyperbolic plane by triangles, 7 at a vertex, in crocheted cotton. Adapting techniques developed by Joshua and Lana Holden, the piece is not assembled from flat triangles but instead approximates constant curvature over its entire surface. Its coloration, inspired by William Thurston's rendition of the heptagon tiling underlying the Klein quartic, suggests the identifications required to construct that surface as a quotient of the hyperbolic plane.

Photo of completed afghan. Materials:

Crochet hook: 2.0mm/US 0 or to gauge.

Yarn/embroidery needle

6 skeins Knit Picks Shine Sport Black
2 skeins Knit Picks Shine Sport Serrano
3 skeins Knit Picks Shine Sport Macaw
1 skein Knit Picks Shone Sport French Blue
or equivalent sport weight cotton yarn

Abbreviations:

Ch - chain stitch
Sc - single crochet
Dc - double crochet
Tc - triple crochet
Slst - slip stitch
Join - connect to complete the row using a slip stitch

Gauge:

This pattern should work for different gauges, as long as your ratio of stitch height to width roughly matches mine.

Photo of single triangle. stitch width: 1 stitch ~ 0.5 cm
stitch height:
10 rows Sc ~ 5 cm
10 rows Dc ~ 10.5 cm
10 rows Tc ~ 24 cm

Finished size:

17" diameter (approx)

Instructions:

Following the pattern below, crochet 7 blue triangles, 14 red triangles, 21 green triangles, and 77 black triangles.

Sew triangles together as shown above (following the color scheme in color plate 2 of The Eightfold Way by Silvio Levy). Specifically, join 7 blue triangles to form the center. Attach 5 black triangles at each vertex of the blue heptagon. In the next layer, join 7 triangles at each vertex with a pair of red triangles opposite each side of the blue heptagon and a trio of green triangles opposite each vertex; the remaining 42 triangles in that layer are black.

Triangle pattern:

Row 0: Ch 4. Join.
Row 1: Ch 2, Sc, Dc. Repeat (2 Sc, Dc) twice. 2 Sc in Sc. Join.
Row 3: Ch 2, Sc, Ch 1, Dc in Dc, Ch 1. Repeat (2 Sc in Sc, Ch 1, Dc in Dc, Ch1) twice. Join.
Row 4: Ch 2, 2 Sc, (Dc, Tc, Dc in Dc) in Dc. Repeat (6 Sc, (Dc, Tc, Dc in Dc)) twice. 3 Sc, Join.
Row 5: Ch 2, 2 Sc in Sc, Sc, 2Sc in Dc, (Dc, (*)Ch 1, Ch 1, Slst in (*), Dc) in Tc). Repeat (2 Sc in Dc, Sc, 2 Sc in Sc, 2 Sc, 2 Sc in Sc, Sc, 2 Sc in Dc, (Dc, (*)Ch 1, Ch 1, Slst in (*), Dc) in Tc) twice. 2 Sc in Dc, Sc, 2 Sc in Sc, 2 Sc. Join, bind off, sew in ends.

Pattern notes:

The important points in crocheting the triangles are making sure the edges and vertices of your triangle line up from row to row, and that you're increasing evenly around the triangle at a rate of approximately 50% per row. If, for example, you have to start with Ch 2 3 Sc at the start of row 4 and 2 Sc join at the end to accomplish this, do so.

The instruction "(*)Ch 1, Ch 1, Slst in (*)" is intended to create a point at the vertex of the triangle. The chain stitches extend the edge, and the slip stitch prevents gaps. It will be easier to join your triangles if you can slip stitch in the underside of the chain stitch.

Because each vertex has degree 7, sewing the triangles together requires many separate pieces of yarn. A seam the length of the edge of a triangle can be completed with a piece of yarn about the length of your forearm. Double stitch at the start and end of each seam; the afghan is heavy enough to pull itself apart if you do not.

I hope you enjoy this project!

Heidi Burgiel