Chapter 5

Hyperbolic Planes

[To son János:] For God's sake, please give it up. Fear it no less than the sensual passion, because it, too, may take up all your time and deprive you of your health, peace of mind and happiness in life.

[Bolyai's father urging him to give up work on non-Euclidean geometry.]

Wolfgang Bolyai (1775-1856)

[SE: Davis and Hersch, page 220

Hyperbolic geometry, discovered more than 170 years ago by C.F. Gauss (1777-1855, German), János Bolyai (1802-1860, Hungarian), and N.I. Lobatchevsky (1792-1856, Russian), is special from a formal axiomatic point of view because it satisfies all the axioms of Euclidean geometry except for the parallel postulate. In hyperbolic geometry there are infinitely many straight lines through a given point that do not intersect a given line. (See Figure 5.1.)

Figure 5.1. Two geodesics through a point parallel to a given line.

Hyperbolic geometry and non-Euclidean geometry are considered in many books as being synonymous, but as we have seen there are other non-Euclidean geometries, particularly spherical geometry. It is also not accurate to say (as many books do) that non-Euclidean geometry was discovered about 170 years ago. Spherical geometry (which is clearly not Euclidean) was in existence and studied by at least the ancient Babylonians, Indians, and Greeks more than 2,000 years ago. Spherical geometry was of importance for astronomical observations and astrological calculations. Even Euclid in his Phaenomena [AT: Euclid] (a work on astronomy) discusses propositions of spherical geometry. Menelaus, a Greek of the first century, published a book Sphaerica, which contains many theorems about spherical triangles and compares them to triangles on the Euclidean plane. (Sphaerica survives only in an Arabic version. For a discussion see [NE: Kline, page 120].)

Most texts and popular books introduce hyperbolic geometry either axiomatically or via "models" of the hyperbolic geometry in the Euclidean plane. These models are like our familiar map projections of the earth and (like these maps of the earth) intrinsic straight lines on the hyperbolic plane (surface of the earth) are not, in general, straight in the model (map) and the model (map) also, in general, distorts distances and angles. We will return to the subject of projection, maps, and models in Chapter 16.

In this chapter we will introduce the geometry of the hyperbolic plane as the intrinsic geometry of a particular surface in 3-space, in much the same way that we introduced spherical geometry by looking at the intrinsic geometry of the sphere in 3-space. Such a surface is called an isometric embedding of the hyperbolic plane into 3-space. We will construct such a surface in the next section. Nevertheless, many texts and popular books say that David Hilbert (1862-1943, German) proved in 1901 that it is not possible to have an isometric embedding of the hyperbolic plane onto a closed subset of Euclidean 3-space. These authors miss what Hilbert actually proved. In fact, Hilbert [NE: Hilbert] proved that there is no real analytic isometry (that is, an isometry defined by real-valued functions which have convergent power series), but his arguments work in class C4 (that is, functions whose derivatives exist and are continuous up to the fourth derivative). Moreover, in 1955, N. Kuiper [NE: Kuiper] showed that there is a C1 isometric embedding onto a closed subset of 3-space, and in 1972 Tilla Milnor [NE: Milnor] proved that a C2 isometric embedding was not possible. The construction used here was shown to the author by William Thurston (1946- , American) in 19781; and it is not defined by equations at all, since it has no definite embedding in Euclidean space.

Constructions of Hyperbolic Planes

We will describe three different isometric constructions of the hyperbolic plane (or approximations to the hyperbolic plane) as surfaces in 3-space. Then we will connect them with the standard upper half plane model which is a non-isometric representation of the hyperbolic plane.

1. The Hyperbolic Plane From Paper Annuli

A paper model of the hyperbolic plane may be constructed as follows: Cut out many identical annular ("annulus" is the region between two concentric circles) strips as in the following Figure 5.2. Attach the strips together by attaching the inner circle of one to the outer circle of the other or the straight ends together. The resulting surface is of course only an approximation of the desired surface. The actual hyperbolic plane is obtained by letting d ® 0 while holding r fixed. Note that since the surface is constructed the same everywhere (as ® 0), it is homogeneous (i.e. intrinsically and geometrically, every point has a neighborhood that is isometric to a neighborhood of any other point). We will call the results of this construction the annular hyperbolic plane.

I strongly suggest that the reader at this point take the time to cut out carefully several such annuli and to tape them together as indicated.

Figure 5.2. Annular strips for making an annular hyperbolic plane.

2. How to Crochet the Hyperbolic Plane

Once you tried to make your annular hyperbolic plane from paper annuli you will certainly realize that it will take a lot of time. Also, later you will have to play with it carefully because it is fragile and tears and creases easily -- you may want just to have it sitting on your desk. But there is another way to get a sturdy model of the hyperbolic plane which you can work and play with as much as you wish. This is the crocheted hyperbolic plane.

In order to make the crocheted hyperbolic plane you need just a very basic crocheting skills. All you need to know is how to make a chain (to start) and how to single crochet. That's it! Now you can start. See Figure 5.3 for a picture of these stitches, which will be described further in the next paragraph.

Figure 5.3. Crochet stitches for the hyperbolic plane.

First you should chose a yarn which will not stretch a lot. Every yarn will stretch a little but you need one which will keep its shape. Now you are ready to start the stitches:

  1. Make your beginning chain stitches (Figure 5.3a). (Topologists may recognize that as the stitches in the Fox-Artin wild arc!) About 20 chain stitches for the beginning will be enough.
  2. For the first stitch in each row insert the hook into the 2nd chain from the hook. Take yarn over and pull through chain, leaving 2 loops on hook. Take yarn over and pull through both loops. One single crochet stitch has been completed. (Figure 5.3b.)
  3. For the next N stitches proceed exactly like the first stitch except insert the hook into the next chain (instead of the 2nd).
  4. For the (N+1)st stitch proceed as before except insert the hook into the same loop as the N-th stitch.
  5. Repeat Steps 3 and 4 until you reach the end of the row.
  6. At the end of the row before going to the next row do one extra chain stitch.
  7. When you have the model as big as you want, you can stop by just pulling the yarn through the last loop.
Be sure to crochet fairly tight and even. Check if you are putting a hook through the chain exactly in the way that it is shown in Figure 5.3.b. In the beginning of the model while you do not have a lot of stitches in a row, you can start increasing closer to the beginning of the row. For a smoother hyperbolic plane, try to alternate the places where you do the first increasing of stitches in each row so they are not on a top of each other. But remember, that after the first increase the next one should be exactly after N stitches!

That's all you need from crochet basics. Now you can go ahead and make your own hyperbolic plane. You have to increase (by the above procedure) the number of stitches from one row to the next in a constant ratio, N to N+1 the ratio determines the radius (the r in the annular hyperbolic plane) of the hyperbolic plane. You can experiment with different ratios BUT not in the same model. You will get a hyperbolic plane ONLY if you will be increasing the number of stitches in the same ratio all the time. If you are doing this for the first time and want to see the exponential growth faster, you can try a ratio 5 to 6 (this means N = 5 and you are increasing the number of stitches after every 5th stitch). But you will realize soon how fast the number of stitches in each row is growing.

Crocheting will take some time but later you can work with this model without worrying about destroying it. The completed product is pictured in Figure 5.4.

Figure 5.4. A crocheted annular hyperbolic plane.

3. Polyhedral Constructions.

A polyhedral model can be constructed from equilateral triangles by putting 7 triangles at every vertex. This is called the {3,7} polyhedral model, because triangles (3-gons) are put together 7 at a vertex. This model has the advantage of being constructable more easily than the models above; however, one can not make better and better approximations by decreasing the size of the triangles. This is true because at each vertex the cone angle is 420° no matter what the size of the triangles are; whereas the hyperbolic plane in the small looks like the Euclidean plane. Another disadvantage of the polyhedral model is that it is not easy to describe coordinates.

We can avoid some of the problems of the {3,7} model by putting 7 triangles together only at every other vertex and 6 triangles together at the others. The precise construction can be described in three different (but, in the end, equivalent) ways:

1. Construct polyhedral annuli as in Figure 5.5 and then tape them together as with the annular hyperbolic plane.

Figure 5.5. Polyhedral annulus.

2. You can construct two annuli at a time by using the shape in Figure 5.6 and taping one of them to the next by joining: a®A, b®B, c®C.


Figure 5.6. Shape to make two annuli.

3. The quickest way is to start with many strips as pictured in Figure 5.7a these strips can be as long as you wish. Then add four of the strips together as in Figure 5.7b using 5 additional triangles. Next, add another strip every place there is a vertex with 5 triangles and a gap (as at the marked vertices in Figure 5.7b). Every time a strip is added an additional vertex with 7 triangles is formed.

Figure 5.7a. Strips.

Figure 5.7b. Forming the polyhedral annular hyperbolic plane.

The center of each strip runs perpendicular to each annulus, and you can show that these curves (the center lines of the strip) are each geodesics because they have local reflection symmetry.

Hyperbolic Planes of Different Radii (Curvature)

Note that in the construction of a hyperbolic plane is dependent on the r (the radius of the annuli) which is often called the radius of the hyperbolic plane. As in the case of spheres, we get different hyperbolic planes depending on the value of r. In Figures 5.8a,b,c there are crocheted hyperbolic planes with radii approximately 4 cm, 8 cm, and 16 cm. the pictures were all taken from approximately the same perspective and in each picture there is a centimeter rule in order to indicate the scale.

Figure 5.8a. Hyperbolic plane with r = ~4 cm.

Figure 5.8b. Hyperbolic plane with r = ~8 cm.

Figure 5.8c. Hyperbolic plane with r = ~16 cm.

Note that as r increases the hyperbolic plane becomes flatter and flatter (has less and less curvature). For both the sphere and the hyperbolic plane as r goes to infinity they both become indistinguishable from the ordinary flat (Euclidean) plane. Thus, the plane can be called a sphere (or hyperbolic plane) with infinite radius. In Chapter 7, we will define the Gaussian Curvature and show that it is equal to 1/r2 for a sphere and -1/r2 for a hyperbolic plane.

Problem 5.1. What is Straight in a Hyperbolic Plane?

a. Argue that the curves on the annular hyperbolic plane, which run radially across each annular strip, are intrinsically straight (geodesics).

Look for the local intrinsic symmetries each annular strip and then global symmetries in the whole hyperbolic plane. Make sure you give a convincing argument why the symmetry holds in the limit as 0.

b. Find other geodesics on whichever physical hyperbolic surface you have. Use the properties of straightness (such as symmetries) which you talked about in Problems 1.1, 2.1, and 2.2.

Try holding two points between the index finger and thumb on your two hands. Now pull gently — a geodesic segment (with its reflection symmetry should appear between the two points. Also (if your surface is durable enough), try folding the surface along a geodesic. Also, use a ribbon to test for geodesics.

c. Use your intuitive manipulations of your physical hyperbolic surface to convince yourself as best you can that, in the hyperbolic plane:

In the next two problems we will develop the upper half plane model of the hyperbolic plane and will use it to describe precisely the geodesics and symmetries of the hyperbolic plane. I strongly urge you to at least look over these problems. These problems, though elementary (nothing technical beyond first semester calculus is assumed), involve some analytic complications that you may wish to avoid because of the time involved to work through them. If you wish to skip these two problems then you can go onto the most of the rest of the book, if you assume the results of Part c. These results in Part c are proved in Problem 5.3.

Using the results of Part c (and Problem 5.3) we can now show that our work in Chapter 2 also applies to the hyperbolic plane.

d. Show that that Problem 2.1 (VAT), Problem 2.3 (ITT), and Problem 2.4 (intersections of circles and constructions) hold on the hyperbolic plane.

If you check your arguments and modify them (if necessary) to only involve symmetries then you will be able to see that they hold also in a hyperbolic plane.

Problem 5.2. The Upper Half Plane Model.

We now will define coordinates on the annular hyperbolic plane which will helps us to study it. Let r be the fixed inner radius of the annuli and let Hd be the approximation of the annular hyperbolic plane constructed, as above, from annuli of radius r and thickness d. On Hd pick the inner curve of any annulus and calling it the base curve, pick a positive direction on this curve, and pick any point on this curve and call it the origin O. We can now construct an (intrinsic) coordinate system xdR2 ® Hd by defining xd(0,0) = O, xd(w,0) to be the point on the base curve at a distance w from O, and xd(w,s) to be the point at a distance s from xd(w,0) along the radial geodesic through xd(w,0), where the positive direction is chosen to be in the direction from outer- to inner-curve of each annulus. Such coordinates are often called geodesic rectangular coordinates. See Figure 5.8.

a. Show that the coordinate map x is one-to-one and onto from the whole of R2 onto the whole of the annular hyperbolic plane. What maps to the annular strips, and what maps to the radial geodesics?

Figure 5.8. Geodesic rectangular coordinates on annular hyperbolic plane.

b. Let l and m be two of the radial geodesics described in Part a. If the distance between l and m along the base curve is t, then show that the distance between them at a distance c = nd from the base curve is on the paper hyperbolic model:


Now take the limit as d ® 0 to show that the distance between l and m on the annular hyperbolic plane is


Thus, the coordinate chart x preserves (does not distort) distances along the (vertical) 2nd coordinate curves but at x(a,b) the distances along the 1st coordinate curve are distorted by the factor of exp(-b/r) when compared to the distances in R2. To be more precise:

Definition: Let y: A ® B be a map from one metric space to another, and let be a curve in A. Then, the distortion of y along l at the point l(0) is defined as:


In the case of the above coordinate curves above, l is the path in R2, or, and the distortions of x along the coordinate curves are:



We seek a change of coordinates which will distort distances equally in both directions. The reason for seeking this change is that (as we will see below) if distances are distorted the same in both coordinate directions then the chart will preserves angles. (We call such a chart conformal.)

We can not hope to have no distortion in both coordinate directions (if there were no distortion then the chart would be an isometry); so we try to make the distortion in the 2nd coordinate direction the same as the distortion in the 1st coordinate direction. After a little experimentation we find that the desired change is

z(x,y) = x(x, r ln(y/r)),

with the domain of z being the upper half plane

R2+ º { (x,yΠR|  > 0 },

where x is the geodesic rectangular coordinates defined above. This is usually called the upper half plane model of the hyperbolic plane. The upper half plane model is a convenient way to study the hyperbolic plane -- think of it as a map of the hyperbolic plane in the same way that we use planar maps of the spherical surface of the earth.

c. Show that the distortion of z along both coordinate curves

® z(x,b)  and  y ® z(a,y)

at the point z(a,b) is r/b.

It may be best to first try this for r = 1. For the first coordinate direction, use the result of Part c. For the second coordinate direction, use the fact that the second coordinate curves in geodesic rectangular coordinates are parametrized by arc-length. Use first semester calculus where necessary.

Lemma. If the distortion of z at the point p = (a,b) is the same (say D(p)) along each coordinate curve, then the distortion of z at (a,b) has the same value along any other curve l(t) = z(x(t),y(t)) which passes through p; and z preserves angles at p (that is, z is conformal).

Proof. Suppose that l(0) = (x(0),y(0)) = (a,b) = p, then, assuming that the annular hyperbolic plane can be locally isometrically (that is, preserving distances and angles) imbedded in 3-space (see Part d),the distortion of z along l at p is


In particular, along the 1st coordinate curve , the distortion is:


Similarly, the distortion along the 2nd coordinate curve is |z2(p)|. The velocity vector of the curve z(l(t)) = z(x(t),y(t)) at p is


Thus, the velocity vector, at t=0 is a linear combination of the partial derivative vectors, z1(p) and z2(p) and, thus, the velocity vectors of curves through p=l(0) all lie in the same plane called the tangent plane at z(p). Also, note that the velocity vector, depends only on the velocity vector, , and not on the curve l. Thus, z induces a linear map (called the differential dz) that takes vectors at p = l(0) to vectors in the tangent plane at z(p). This differential is a similarity that multiples all length by D(p) and thus preserves angles. Also, the distortion of z along l is also D(p).

Definition. In the above situation we call D(p) the distortion of the map z at the point p and denote it dist(z)(p).

*d. Show that locally the annular hyperbolic plane is isometric to portions of a (smooth) surface defined by revolving the graph of a continuously differentiable function of z about the z-axis.

Outline of proof.

  1. Argue that each point on the annular hyperbolic plane is like any other point.
  2. Start with one of the annular strips and complete it to a full annulus in a plane. Then, construct a surface of revolution by attaching to the inside edge of this annulus other annular strips as described in the construction of the annular hyperbolic plane. Note that the second and subsequent annuli form truncated cones. Finally, imagine the width of the annular strips, d, shrinking to zero. (See Figure 5.9.)
  3. Derive a differential equation representing the coordinates of point on the surface using either the geometry inherent in Figure 5.9 or by using Part c. If R(z) is the desired function then the differential equation should be (remember that r is a constant):


  4. Solve (using tables or computer algebra systems) the differential equation for z as a function of R and then argue (using first semester calculus) that R(z) is a continuously differentiable function.
This surface is usually called the pseudosphere.

Figure 5.9. Hyperbolic surface of revolution -- pseudosphere.

We can also crochet a pseudosphere by starting with 5 or 6 chain stitches continue in spiral fashion increasing as when crocheting the hyperbolic plane. See Figure 5.10. Note that, when you crochet beyond the annular strip that lays flat and forms a complete annulus, then surface forms ruffles and is no longer a surface of revolution (nor a smooth surface).

Figure 5.10. Crocheted psuedo-sphere.

Problem 5.3. Hyperbolic Isometries and Geodesics

We have seen in Problem 5.1 that there are reflections about the radial geodesics, but it is not clear yet that there are any other reflections or other geodesics. To assist us in looking at transformations of the annular hyperbolic space we use the upper half plane model. If f is a transformation taking the upper half plane R2+ to itself then from the diagram

we see that is a transformation from the annular hyperbolic plane to itself. We call g the transformation of H2 that corresponds to f. We will call f an isometry of the upper half plane model if the corresponding g is an isometry of the annular hyperbolic plane. To show that g is an isometry, you must show that the transformation preserves distances. Remember that distance along a curve is equal to the integral of the speed along the curve. Thus it is enough to check that the distortion of g at each point is equal to 1. Before we do this we must first show:

a. The distortion of an inversion iC with respect to a circle G at a point P, which is a distance s from the center C of G, is equal to R2/s2 , where R is the radius of the circle. See Figure 5.11.

Figure 5.11. Distortion of an inversion.

[Hint: Since the inversion is conformal, the distortion is the same in all directions. Thus check the distortion along the ray from C, the center of circle, through P. The distance along this ray of an arbitrary point can be parametrized by . Use the definition of distortion given in Problem 5.2.]

b. Let f be the inversion in a circle whose center is on the x-axis. Show that f takes R2+ to itself and that has distortion 1 at every point and is thus an isometry.

Outline of a proof:

  1. Note that each of the maps z, z-1, f are conformal and have at each point a distortion that is the same for all curves at that point. If dist(k)(p) denotes the distortion of the function k at the point p, then argue that dist(g)(p) = dist(z -1)(p) ´ dist(f) ´ dist(z).
  2. If z(a,b) = p, then show (using 5.1c) that dist(z-1)(p) = b/r.
  3. Show (using Part a) that dist(f)(z-1(p)) = R2/s2, where R is the radius of the circle C which defines f and s is the distance from the center of C to (a,b).
  4. Then show that dist(z)(f(z-1(p)) = .

We call these inversions (or the corresponding transformations in the annular hyperbolic plane) hyperbolic reflections. We also call reflections through vertical lines (radial geodesics) hyperbolic reflections.

Now, we can prove:

c. If g is a semicircle in the upper half plane with center on the x-axis or a straight line in the upper half plane perpendicular to x-axis, then z(g) is a geodesic in the annular hyperbolic plane.

Because of this, we say that such g are geodesics in the upper half plane model. Since the compositions of two isometries is an isometry, we see immediately that any composition of inversions in semicircles (whose centers are on the x-axis) is an isometry in the upper half plane model (that is, the corresponding transformation in the annular hyperbolic plane is an isometry). In Chapter 18 (Isometries and Patterns) we will show that these are the only isometries in the hyperbolic plane and that the only reflections are the above hyperbolic reflections; thus, no other paths can have the reflection symmetry of a geodesic.

d. Any similarity of the upper half plane corresponds to an isometry of the annular hyperbolic. Such similarities must have their centers on the x-axis. (Why?)

[Hint: Look at the composition of inversions in two concentric semicircles.]

e. If g is a semicircle in the upper half plane with center on the x-axis, then there is an inversion (in another semicircle) that takes g to a vertical line that is tangent to g.

[Hint: An inversion takes any circle through the center of the inversion to a straight line, see Problem 4.2.]

In the following three parts each are concerned with finding a geodesic. Each problem should be looked at in both the annular hyperbolic plane and in the upper half plane model. In a crocheted annular hyperbolic plane one can construct geodesics by folding much the same way you can on a piece of (planar) paper. Geodesics in the upper half plane model can be constructed using properties of circles and inversions, see Problem 4.2. You will also find Part e very useful.

f. Given two points A and B in the hyperbolic plane there is a unique geodesic joining A to B; and there is an isometry which takes this geodesic to a radial geodesic (or vertical line in the upper half plane model).

[Hint: In the upper half plane model, construct a circle with center on the x-axis that passes through A and B. Then use Part e.]

We use AB to denote the unique geodesic segment joining A to B.

g. Given a geodesic segment AB with endpoints points A and B in the hyperbolic plane there is a unique geodesic that is a perpendicular bisector of AB.

[Hint: Use appropriate folding in annular hyperbolic plane. In the upper half plane model, use Part f to transform AB to a vertical line and then search for the semicircle that is perpendicular bisector in the upper half plane model.]

h. Given an angle ÐABC in the annular hyperbolic plane, there is a unique geodesic which bisects the angle.

[Hint: In the upper half plane model, use Part f to make one side of the angle vertical. Then use the fact that in an inversion any circle through the center will be inverted to a straight line.]

1 The idea for this construction is also included in Thurston's recent book Three-Dimensional Geometry and Topology, Vol. 1 (Princeton University Press, 1997, pages 49 and 50), and is also discussed in the recent book by the author, Differential Geometry: A Geometric Introduction (Prentice-Hall, 1998, page 31) where it is proved that the construction actually results in the hyperbolic plane (as d ® 0).