Problem 17: Infinite Hyperbolic Geometries

We saw in Problem 16 that the axioms for a hyperbolic plane were:

HYPERBOLIC PLANE

1. Two distinct points are contained in a unique line.

2. Associated to any non-incident point-line pair (p, l) is a set consisting of two or more lines, each line of which is incident with p but not l.

3. If a subspace X (X will be a subspace if p and q in X implies that the line pq is also in X) contains a triangle, then X must be the entire set of points.

We have also seen how there are geometries with finitely many points that are projective planes, and at least one geometry (the real projective plane) with infinitely many points that satisfies the axioms for a projective plane. So it is natural to ask if there is a hyperbolic plane with infinitely many points. The answer is yes, and as with projective planes the infinite model was historically the first model.

A fantastic realization (thanks to the University of Minnesota Geometry Center, http://www.geom.umn.edu) of the hyperbolic plane, tiled with congruent copies of simple radial shape, is shown below.

Wow! You will find a plethora of material on the real hyperbolic plane on the Web. But we will give you a short introduction below using the software program Cinderella.

The model of the real hyperbolic plane we will be using is called the Poincare Disk Model. It consists of the interior region of a disk of radius 1, excluding the boundary circle. This boundary circle is to be thought of as a circle at infinity: although from our outsider's perspective a point in the hyperbolic plane is at a distance of less than 1 unit from the nearest point on the boundary circle, in hyperbolic geomtry this distance is to be considered infinite. This is why in the picture above (the boundary circle is not shown but can be imagined) the radial shapes appear to us to shrink in size as they near the circle at infinity, but in reality they remain the same size. Points are, well, points, and lines will consist of circular arcs which intersect the circle at infinity at angles, including diameters.

Below we see the classical construction in euclidean geometry showing that the angle bisectors of a triangle are concurrent. You may click and drag the vertices of the triangle to see this.

Please enable Java for an interactive construction (with Cinderella).

Created with Cinderella

Could the same be true in the real hyperbolic plane?

Please enable Java for an interactive construction (with Cinderella).

Created with Cinderella

Similarly we can first look at the euclidean construction showing that the three medians of any
triangle are concurrent,

Please enable Java for an interactive construction (with Cinderella).

Created with Cinderella

and then look at the corresponding hyperbolic analogue:

Please enable Java for an interactive construction (with Cinderella).

Created with Cinderella

Lastly, recall that in euclidean plane geometry, the interior angles of a triangle must sum to 180
degrees, independent of the area of the triangle.

Please enable Java for an interactive construction (with Cinderella).

Created with Cinderella

PROBLEM 17: Investigate the relationship (if any) between the sum of the interior angles of a hyperbolic triangle and its size, using the Poincare disk model. (The first image shows the hyperbolic plane as it sits in the euclidean plane, and the second image shows the Poincare disk model.)

Please enable Java for an interactive construction (with Cinderella).

Please enable Java for an interactive construction (with Cinderella).

Created with Cinderella

Reading Assingment : Please read Chapter 4 of Yaglom and answer the following questions:

  1. 1. Are affine and projective geometries non-euclidean geometries? Explain.
  3. 2. What are the five axioms, or postulates, of euclidean geometry?
  5. 3. What is the angular defect in hyperbolic planar geometry?
  7. 4. In hyperbolic planar geometry, is it true that cosh2x + sinh2x = 1? Is it true that cosh2x - sinh2x = 1?
  9. 5. What separated J. Bolyai, Gauss, and Lobachevsky from Lambert, Saccheri, and Schweikart?
  11. 6. Did Gauss put geometry on a par with arithmetic? Explain.
  13. 7. Has our approach to geometry been akin to that of Gauss' and Lobachevsky's, or rather of J. Bolyai's? Explain.
  15. 8. True or False: J. Bolyai was an excellent swordsman.
  17. 9. Einstein's general theory of relativity is based the geometric works of what mathematician?
  19. 10. What is meant by the term local properties?
  21. 11. What is meant by the term intrinsic geometry?
  23. 12. True or False: Riemann's elliptic space is the same as thing as the real projective plane/
  25. 13. What were Klein and Riemann both trying to accomplish in the field of geometry? How did their accomplishments differ?
  27. 14. Read pages 66-70 VERY CAREFULLY !!