Crocheted Model of the Hyperbolic Plane

This model of the hyperbolic plane was crocheted by the Latvian-born mathematician Daina Taimina in about 2002. Although called a model of a plane, it is not flat like a Euclidean plane and its lines are not straight. However, lines on any plane, Euclidean or hyperbolic, are still the shortest paths along the plane connecting two points.

The distinguishing difference between a hyperbolic plane and a Euclidean plane is that on a hyperbolic plane there are infinitely many lines parallel to a given line through a given point not on the given line. In this model lines are shown in yellow. The given line is the one closest to the top of the photograph and the given point is where the four other lines meet. None of those four lines will ever meet the given line, so they are all parallel to it.

On page 27 of her book, Crocheting Adventures with Hyperbolic Planes, (Wellesley, MA: A. K. Peters, 2009), Taimina has a photograph of a similar model, with only three yellow lines through the given point. On page 28 she has another photograph of that model with the caption: “The red line is a common perpendicular to only two of these yellow lines.” That photograph illustrates that on a hyperbolic plane, just as on a Euclidean plane, there is only one line through a given point not on a given line that is perpendicular to the given line.

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