Granny’s Not SoSquare, After All:Hyperbolic Tilings

with Truly HyperbolicCrochet Motifs

Joshua Holden

Joint work with (andexecution by) Lana Holden

http://www.rose-hulman.edu/~holden

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A hyperbolic plane is a surface with constant negativecurvature.

[The Geometry Center]

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A positive curvature surface bends away from thesame side of its tangent plane in every direction.

[Wikipedia]

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A zero curvature surface is flat (in at least onedirection).

[Robert Gardner, ETSU]

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A negative curvature surface bends away fromdifferent sides of its tangent plane.

[Wikipedia]

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Also, on a negative curvature surface circles havelarger circumferences than they “should”.

(This is the Bertrand-Diquet-Puiseux Theorem.)

[Daina Taimina and the Institute For Figuring]

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So one way to produce a hyperbolic plane is toconstruct it in arcs of exponentially increasing length.

[Daina Taimina and the Institute For Figuring]

Daina Taimina realized that you could do this with crochetstitches.

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A regular tiling fills a surface completely withcongruent regular polygons.

[Lana Holden]

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A regular tiling of the hyperbolic plane has morepolygons around each vertex than it “should”.

[Wikipedia]

And correspondingly, the interior angles are smaller than they“should be”.

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We could construct the surface first and then tile it.

[Daina Taimina]

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Or we could make flat tiles and attach them in such away that they curve negatively.

[Helaman Ferguson and Jeffrey Weeks]

Helaman Ferguson did this with stretchy materials such aspolar fleece that distribute the curvature.

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Our goal is to make tiles which are the correct shapeand the correct curvature.

[Lana Holden]

(Daina Taimina previously made some progress towards this.)

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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 c13 / 22

The number of sides of the polygon and the number ofpolygons around a vertex determine the angles.

For our construction, we need to know the inradius, thecircumradius, and the side length.

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We will crochet the tiles using variations of the classic“granny square”.

[Purl Soho and purlbee.com]

The inradius determines the number of rounds.

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We need to vary the pattern to add the exponentiallyincreasing length.

We add exponentially spaced increases to achieve the desiredside length.

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We need to vary the pattern to add the exponentiallyincreasing length.

We substitute longer stitches to achieve the desiredcircumradius.

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Et voilà!

[Lana Holden]

Five of these hyperbolic squares go around each vertex, ratherthan the “usual” four.

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Other “granny polygons” are also found in moderncrochet.

[Lana Holden]

Here we have put three “granny hexagons” around each vertex.

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And we can make hyperbolic versions of some ofthem.

[Lana Holden]

Here we have put four “granny hexagons” around each vertex.

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In theory we could construct any hyperbolic tiling.

However, as the interior angles get sharper, it will become moreand more difficult to turn the corners.

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Hope you enjoyed the show!

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