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- 1. Granny’s Not So Square, After All: Hyperbolic Tilings with Truly Hyperbolic Crochet Motifs Joshua Holden Joint work with (and execution by) Lana Holden http://www.rose-hulman.edu/ ~holden 1 / 22
- 2. A hyperbolic plane is a surface with constant negative curvature. [The Geometry Center] 2 / 22
- 3. A positive curvature surface bends away from the same side of its tangent plane in every direction. [Wikipedia] 3 / 22
- 4. A zero curvature surface is ﬂat (in at least one direction). [Robert Gardner, ETSU] 4 / 22
- 5. A negative curvature surface bends away from different sides of its tangent plane. [Wikipedia] 5 / 22
- 6. Also, on a negative curvature surface circles have larger circumferences than they “should”. (This is the Bertrand-Diquet-Puiseux Theorem.) [Daina Taimina and the Institute For Figuring] 6 / 22
- 7. So one way to produce a hyperbolic plane is to construct it in arcs of exponentially increasing length. [Daina Taimina and the Institute For Figuring] Daina Taimina realized that you could do this with crochet stitches. 7 / 22
- 8. A regular tiling ﬁlls a surface completely with congruent regular polygons. [Lana Holden] 8 / 22
- 9. A regular tiling of the hyperbolic plane has more polygons around each vertex than it “should”. [Wikipedia] And correspondingly, the interior angles are smaller than they “should be”. 9 / 22
- 10. We could construct the surface ﬁrst and then tile it. [Daina Taimina] 10 / 22
- 11. Or we could make ﬂat tiles and attach them in such a way that they curve negatively. [Helaman Ferguson and Jeffrey Weeks] Helaman Ferguson did this with stretchy materials such as polar ﬂeece that distribute the curvature. 11 / 22
- 12. Our goal is to make tiles which are the correct shape and the correct curvature. [Lana Holden] (Daina Taimina previously made some progress towards this.) 12 / 22
- 13. To calculate the correct shape, we use the (Second) Hyperbolic Law of Cosines: [Wikipedia] cos C = − cos A cos B + sin A sin B cosh c 13 / 22
- 14. The number of sides of the polygon and the number of polygons around a vertex determine the angles. For our construction, we need to know the inradius, the circumradius, and the side length. 14 / 22
- 15. We will crochet the tiles using variations of the classic “granny square”. [Purl Soho and purlbee.com] The inradius determines the number of rounds. 15 / 22
- 16. We need to vary the pattern to add the exponentially increasing length. We add exponentially spaced increases to achieve the desired side length. 16 / 22
- 17. We need to vary the pattern to add the exponentially increasing length. We substitute longer stitches to achieve the desired circumradius. 17 / 22
- 18. Et voilà! [Lana Holden] Five of these hyperbolic squares go around each vertex, rather than the “usual” four. 18 / 22
- 19. Other “granny polygons” are also found in modern crochet. [Lana Holden] Here we have put three “granny hexagons” around each vertex. 19 / 22
- 20. And we can make hyperbolic versions of some of them. [Lana Holden] Here we have put four “granny hexagons” around each vertex. 20 / 22
- 21. In theory we could construct any hyperbolic tiling. However, as the interior angles get sharper, it will become more and more difﬁcult to turn the corners. 21 / 22
- 22. Hope you enjoyed the show! 22 / 22

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