Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our User Agreement and Privacy Policy.

Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our Privacy Policy and User Agreement for details.

Successfully reported this slideshow.

1,599 views

Published on

Published in:
Education

No Downloads

Total views

1,599

On SlideShare

0

From Embeds

0

Number of Embeds

10

Shares

0

Downloads

4

Comments

0

Likes

1

No embeds

No notes for slide

- 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

No public clipboards found for this slide

Be the first to comment