Tag Archives: hyperbolic geometry

Understanding Geometry – 3

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In some of my past posts, I have mentioned “hyperbolic curvature“,”hyperbolic trigonometry” and “hyperbolic ideal points“. In this post I will share some artworks, based on hyperbolic geometry, by contemporary artists (from Tumblr):

To explain the mathematics behind the construction of these pictures I will quote Roger Penrose from pp. 34 of  “The Road to Reality“:

Think of any circle in a Euclidean plane. The set of points lying in the interior of this circle is to represent the set of points in the entire hyperbolic plane. Straight lines, according to the hyperbolic geometry are to be represented as segments of Euclidean circles which meet the bounding circle orthogonally — which means at right angles. Now, it turns out that the hyperbolic notion of an angle between any two curves, at their point of intersection, is precisely the same as the Euclidean measure of the angle between the two curves at the intersection point. A representation of this nature is called conformal. For this reason, the particular representation of hyperbolic geometry that Escher used is sometimes referred to as the conformal model of hyperbolic plane.

In the above-quoted paragraph, Penrose refers to Escher’s “Circle Limit” works, explained in detail by Bill Casselman in this article.

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Understanding Geometry – 1

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When we think about mathematics, what comes to our mind are the numbers and figures. The  study of numbers is called arithmetic and the study of figures is called geometry (in very crude sense!). In our high school (including olympiad level) and college curriculum we cover various aspects of arithmetic. I am very much satisfied with that treatment, and this is the primary reason for my research interests in arithmetic (a.k.a. number theory).

But, I was always unsatisfied with the treatment given to geometry in our high school curriculum. We were taught some plane Euclidean geometry (with the mention of the existence of non-euclidean geometries), ruler and compass constructions, plane trigonometry (luckily, law of cosines was taught), surface area & volume of 3D objects, 2D coordinate geometry, conic sections and 3D coordinate geometry.  In the name of Euclidean geometry some simple theorems for triangles, quadrilaterals and circles are discussed, like triangle congruence criterias, triangle similarity criterias, Pythagoras theorem, Mid-Point Theorem, Basic Proportionality Theorem, Thales’ Theorem, Ptolemy’s theorem, Brahmagupta theorem etc. are discussed. Ruler-compass constructions are taught as “practical geometry”. Students are asked to cram the formulas of area (including Brahmagupta’s formula and Heron’s formula) and volume without giving any logic (though in earlier curriculum teacher used to give the reasoning). Once coordinate geometry is introduced, students are asked to forget the idea of Euclidean geometry or visualizing 3D space. And to emphasize this, conic sections are introduced only as equations of curves in two dimensional euclidean plane.

The interesting theorems from Euclidean geometry like Ceva’s theorem,  Stewart’s Theorem,  Butterfly theorem, Morley’s theorem (I discussed this last year with high school students), Menelaus’ theorem, Pappus’s theorem,  etc. are never discussed in classroom (I came to know about them while preparing for olympiads). Ruler-compass constructions are taught without mentioning the three fundamental impossibilities of angle trisection, squaring a circle and doubling a cube. The conic sections are taught without discussing the classical treatment of the subject by Apollonius.

In the classic “Geometry and Imagination“, the first chapter on conic sections is followed by discussion of crystallographic groups (Character tables for point groups),  projective geometry (recently I discussed an exciting theorem related to this) and differential geometry (currently I am doing an introductory course on it). So over the next few months I will be posting mostly about geometry (I don’t know how many posts in total…), in an attempt to fill the gap between high-school geometry and college geometry.

I agree with the belief that algebraic and analytic methods make the handling of geometry problems much easier, but in my opinion these methods suppress the visualization of geometric objects. I will end this introductory post with a way to classify geometry by counting the number of ideal points in projective plane:

  • Hyperbolic Geometry (a.k.a. Lobachevsky-Bolyai-Gauss type non-euclidean geometry) which has two ideal points [angle-sum of a triangle is less than 180°].
  • Elliptic Geometry (a.k.a. Riemann type non-euclidean geometry) which has no ideal points. [angle-sum of a triangle is more than 180°]
  • Parabolic Geometry (a.k.a. euclidean geometry)  which has one ideal point. [angle-sum of a triangle is 180°]

Hyperbolic Plane Example

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Few months ago I gave a lecture on Non-euclidean geometry and it was a bit difficult for me to give audience an example of hyperbolic surface from their day-to-day life. While reading Donal O’ Shea’s book on Poincaré Conjecture I came across following interesting example on pp. 97 :

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Negatively curved cloth will drape a woman’s side (© Donal O’ Shea, 2007)

Estrogen causes fat to be stored in the buttocks, thighs, and hips in women. Thus females generally have relatively narrow waists and large buttocks, and this along with wide hips make for a wider hip section and a lower waist-hip ratio compared to men. The saddle-shaped area on a woman’s side above her hip has negative curvature.

One can imagine cloth (it is flexible but does not stretch, hence an isometry) that would drape it perfectly. Here the region inside a circle of given radius contains more material than the same circle on the plane, and to make the cloth the tailor might start with a flat piece of fabric, make a cut as if he/she were going to make a dart, but instead of stitching the cut edges together, insert an extra piece of fabric or a gusset. Negatively curved cloth would have lots of folds if one tried to lay it flat in  dresser.

If one tries to extend a cloth with constant positive curvature (like a cap), in all directions, it would close up, making a sphere. On the other hand, if one imagines extending a cloth with constant negative curvature in all directions, one gets a surface called hyperbolic plane.