Crocheting Adventures with Hyperbolic Planes – Daina Taimiņa

Thoughts: Weird book, though worthwhile. I now have a much better sense of how to define curvature, and of the quirks of surfaces with negative curvature.

(The notes below are not a summary of the book, but rather raw notes - whatever I thought, at the time, might be worth remembering.)

Taimiņa, Daina. 2009. Crocheting Adventures with Hyperbolic Planes. A K Peters.

Introduction

Chapter 1: What Is the Hyperbolic Plane? Can We Crochet It?

• 12: Curvature of a plane: we can measure the curvature of a line at a single point by finding the largest possible circle that lies tangent to it - if a segment has a positive curvature, the circle will fit beneath it (think top of ridge); if it has a negative curvature, the circle will fit above it (think bottom of valley).
  • We can measure of the curvature of a plane at a point by measuring the curvature of two perpendicular lines passing through that point, and multiplying the measurements. if the curvatures of the two lines have the same sign, the plane has a positive at that point point (think top of hill, or middle of depression); if If the curvature of the two lines have different signs, the plane has a negative curvature (think pringle). If any one of the lines has zero curvature, the surface has zero curvature (think cylinder)
• 13-14: Surfaces with positive curvature will close in on themselves (like a sphere), while surfaces with negative curvature flare out to infinity (think kale). If you try to project a surface with positive curvature onto a surface with zero curvature, as you move away from the centre of the projection, you must either stretch or tear the surface (think trying to flatten an orange peel), while if you try to project a negative-curvature surface onto a zero curvature, you will have to either squish or fold the surface as you move away from the centre (think of trying to flatten a frilly doily)
  • j: this can, I think, be generalized to surfaces with greater or lesser curvature than each other - think of trying to project a flat dust cover onto the curved headrest of a squishy chair - the dust cover forms creases near its edges
• 23: Surfaces with negative curvature grow exponentially as you move away from a particular point - in contrast, the area of a circle grows linearly as a function of the circle’s radius (this means you need a lot more yarn to knit a hyperbolic plane with a slightly larger radius)

Chapter 2: What Can You Learn from Your Model?

• 26-27: on a hyperbolic plane, you can have infinitely many lines passing through a given point that are parallel to a given line (i.e. they won’t intersect the given line). Some of the lines will approach the given line as an asymptote, while others will diverge from it in both directions.

Chapter 3: Four Strands in the History of Geometry

• 41: to learn more about: the geometric sona patterns of the Tchokwe people - curved lines that enclose grids of points, used in storytelling
• 57: there’s a misconception that the Greeks didn’t know how to trisect an angle. They couldn’t do it with just a straightedge and compass, but they could accomplish it using a linkage.

Chapter 4: Tidbits from the History of Crochet

Chapter 5: What is Non-Euclidean Geometry?

Chapter 6: How to Crochet a Pseudosphere and a Symmetric Hyperbolic Plane

Chapter 7: Metamorphoses of the Hyperbolic Plane

• 91: geodesic: an “intrinsically straight line”.
  • e.g. take a sheet of paper and draw a vertical, horizontal and diagonal line on in. Roll it into a cylinder. all the lines are still geodesics, even though one has been transformed into a circle and one into a helix
• 94: The Klein Bottle was initially called a Kleinsche Fläsche in German (i.e. Klein’s surface). It was incorrectly translated into English as Kleinsche Flasche (i.e. Klein’s Bottle).

Chapter 8: Other Surfaces with Negative Curvature: Catenoid and Helicoid

Chapter 9: Who Is Interested in Hyperbolic Geometry Now and How Can It Be Used?

• 124: Hyperbolic planes can be useful in modelling networks, e.g. of connected pages on the internet. As you consider nodes that are connected by a series of n edges, the number of nodes tends to increase exponentially as n increases. Hyperbolic planes have a similar property where area increases exponentially as you move away from a point
• 124: fisheye lenses are like projections of a hyperbolic plane onto a flat surface (cf. various Escher prints): centre is highly magnified, while toward the edges of the field, images become increasingly smaller
• 125: pitch perception has the unique property of being both logarithmic and periodic
• 125: to do: check out Dmitri Tymoczko’s work in describing chords on orbifolds