Chapter 5

Straightness on
Hyperbolic Planes

[To son János:] For God's sake, please give it [work on hyperbolic geometry] 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.

Wolfgang Bolyai (1775-1856)

[SE: Davis and Hersch, page 220

We now study hyperbolic geometry. This chapter may be skipped if the reader will not be covering geometric manifolds and the shape of space in Chapters 16 and 22 and if in the remainder of this book the reader leaves out all mentions of hyperbolic planes. However, to skip studying hyperbolic planes would be to skip an important notion in the history of geometry and to skip the geometry which may be the basis of the geometry of our physical universe.

As with the cone and cylinder we must use intrinsic point of view on hyperbolic planes. This is especially true because there is no standard extrinsic embedding of a hyperbolic plane into 3-space.

A Short History of Hyperbolic Geometry

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 postulates (axioms) of Euclidean geometry except for the parallel postulate. In hyperbolic geometry straight lines can converge toward each other without intersecting (violating Eulcid's Fifth Postulate), and there are more than one straight lines through a given point that do not intersect (are parallel to) a given line (violating Playfair's Parallel Postulate). (See Figure 5.1.)

The reader can explore more details of the axiomatic nature of hyperbolic geometry in Chapter 18. Note that the 450° cone also violates the two parallel postulates mentioned above. Thus the 450° cone has many of the properties of the hyperbolic plane.

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

Hyperbolic geometry has turned out to be useful in various branches of higher mathematics. Also, the geometry of binocular visual space appears experimentally to be best represented by hyperbolic geometry (see [NE: Zage]). In addition, hyperbolic geometry is one of the possible geometries for our three-dimensional physical universe — this connection we will explore more in Chapters 16 and 22.

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. In Aristotle we can find evidence that non-Euclidean geometry was studied even before Euclid. (See [Hi: Heath, page 57] and [Hi: Toth].) 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 [Hi: Kline, page 119-120].)

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, in general, distorts distances and angles. We will return to the subject of projection and models in Chapter 15.

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, no isometry defined by real-valued functions which have convergent power series). In 1972, Tilla Milnor [NE: Milnor] extended Hilbert's result by proving that there is no isometric embedding defined by functions whose 1st and 2nd derivatives are continuous. Without giving an explicit construction, N. Kuiper [NE: Kuiper] showed in 1955 that there is a differentiable isometric embedding onto a closed subset of 3-space.

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. In Problem 5.2 we will show that our isometric model is locally isometric to a certain smooth surface of revolution called the pseudosphere which is well known to locally have hyperbolic geometry. Later, in Chapter 15 we will explore the various (non-isometric) models of the hyperbolic plane (these models are the way that hyperbolic geometry is presented in most texts) and prove that these models and the isometric constructions here produce the same geometry.

Constructions of Hyperbolic Planes

We will describe four different isometric constructions of the hyperbolic plane (or approximations to the hyperbolic plane) as surfaces in 3-space. It is very important that you actually perform at least one of these constructions. The act of constructing the surface will give a feel for hyperbolic plane that is difficult to get any other way. Templates for all the paper constructions (and information about possible availability of crocheted hyperbolic planes) can be found at the supplements site:

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 the radius r fixed. Note that since the surface is constructed the same everywhere (as d ® 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 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). 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. 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 and size of the yarn determine 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. We suggest that you start with a ratio of 5 to 6. You will get a hyperbolic plane ONLY if you will be increasing the number of stitches in the same ratio all the time.

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. {3,7} and {7,3} Polyhedral Constructions.

A polyhedral model can be constructed from equilateral triangles by putting 7 triangles together at every vertex, or by putting 3 regular heptagons (7-sided) together at every vertex. These are called the {3,7} polyhedral model and the {7,3} polyhedral model because triangles (3-gons) are put together 7 at a vertex [or 7-gons put together 3 at a vertex]. These 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 (7 ´ p/3) = 420° or (3 ´ 5p/7) = 385.71¼°), no matter what the size of the triangles and heptagons are; whereas the hyperbolic plane in the small looks like the Euclidean plane with 360° cone angles. Another disadvantage of the polyhedral model is that it is not easy to describe the annuli and related coordinates.

You can make these models less "pointy" by replacing the sides of the triangles with arcs of circles in such a way the new vertex angles are 2p/7 or by replacing the sides of the heptagons with arcs of circles in such a way that the new vertex angles are 2p/3. But then the model is less easy to construct because you are cutting and taping along curved edges.

See Problems 10.6 and 21.5 for more discussions of regular polyhedral tilings of plane, spheres, and hyperbolic planes.

4. Hyperbolic Soccer Ball Construction

We now explore a polyhedral construction that involves two different regular polygons instead of the single polygon used in the {3,7} and {7,3} polyhedral constructions. A spherical soccer ball (outside the USA, called a football) is constructed by using pentagons surrounded by 5 hexagons or two hexagons and one pentagon together around each vertex. The plane can be tiled by hexagons each surrounded by 6 other hexagons. The hyperbolic plane can be approximately constructed by using heptagons (7-sided) surrounded by 7 hexagons and two hexagons and one heptagon together around each vertex. See Figure 5.5. Since a heptagon has interior angles with 5p/7 radians (= 128.57¼°), the vertices of this construction have cone angles of 368.57¼° and thus are much smoother than the {3,7} and {7,3} polyhedral constructions. It also has a nice appearance if you make the heptagons a different color from the hexagons. It is also easy to construct (as long as you have a template there is variety on the supplements web site.) As with any polyhedral construction one can not get closer and closer approximations to the hyperbolic plane. There is also no apparent way to see the annuli.

The hyperbolic soccer ball construction is related to the {3,7} construction in the sense that if a neighborhood of each vertex in the {3,7} construction is replaced by a heptagon then the remaining portion of each triangle is a hexagon

Figure 5.5. The Hyperbolic Soccer Ball.

5. "{3,6½}" Polyhedral Construction

We can avoid some of the disadvantages of the {3,7} and soccer ball constructions by constructing a polyhedral annulus. In this construction we have 7 triangles together only at every other vertex and 6 triangles together at the others. This construction still has the disadvantage of not being able to produce closer and closer approximations and it also is more "pointy" (larger cone angles) than the hyperbolic soccer ball.

The precise construction can be described in two different (but, in the end, equivalent) ways:

1. Construct polyhedral annuli as in Figure 5.6 and then tape them together as with the annular hyperbolic plane.

Figure 5.6. Polyhedral annulus.

2. 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.

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.

Figure 5.7a. Strips.

Figure 5.7b. Forming the polyhedral annular hyperbolic plane.

Hyperbolic Planes of Different Radii (Curvature)

Note that 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 5.8-10 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 5.8. Hyperbolic plane with r = ~4 cm.

Figure 5.9. Hyperbolic plane with r = ~8 cm.

Figure 5.10. 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. On a hyperbolic plane, consider the curves which run radially across each annular strip. Argue that these curves are intrinsically straight. Also, show that any two of them are asymptotic, in the sense that they converge toward each other but do not intersect.

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.

We shall say that two geodesics which converge in this way are asymptotic geodesics. Note that there are no geodesics (straight lines) on the plane which are asymptotic.

b. Find other geodesics on your physical hyperbolic surface. Use the properties of straightness (such as symmetries) which you talked about in Problems 1.1, 2.1, and 4.1.

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. If your surface is durable enough, try folding the surface along a geodesic. Also, you may use a ribbon to test for geodesics.

c. What properties do you notice for geodesics on a hyperbolic plane? How are they the same as geodesics on the plane or spheres, and how are they different from geodesics on the plane and spheres.

Explore properties of geodesics involving intersecting, uniqueness, and symmetries. Convince yourself as much as possible using your model full proofs for some of the properties will have to wait until Chapter 15.

*Problem 5.2. The Pseudosphere is Hyperbolic

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. This is the surface usually called the pseudosphere.

Outline of proof.

  1. Argue that each point on the annular hyperbolic plane is like any other point. (Think of the annular construction.)
  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. (See Figure 5.11.) Note that the second and subsequent annuli form truncated cones. Finally, imagine the width of the annular strips, d, shrinking to zero.
  3. Derive a differential equation representing the coordinates of point on the surface using the geometry inherent in Figure 5.11. If f(r) is the height (z-coordinate) of the surface at a distance of r from the z-axis, then the differential equation should be (remember that r is a constant):


Figure 5.11. Hyperbolic surface of revolution -- pseudosphere.

This surface is usually called the pseudosphere. The term "pseudosphere" seems to have originated with Hermann von Helmholtz (1821-1894, German) who was contrasting spherical space with what he called pseudospherical space. However, Helmholtz did not actually find a surface with this geometry. Eugenio Beltrami (1835-1900, Italian) actually constructed the surface, which is called the pseudosphere, and showed that its geometry is locally the same as (locally isometric to) the hyperbolic geometry constructed by Lobatchevsky. (For more historical discussion, see [Hi: Katz], page 781-783.) Mathematicians searched further for a surface (in those days "surface" meant "real analytic surface") that would be the whole of the hyperbolic plane (as opposed to only being locally isometric to it). This search was halted when Hilbert proved that such a surface was impossible in his theorem discussed on page 39 of this chapter at the end of the section A Short History of Hyperbolic Geometry.

We can also crochet a pseudosphere by starting with 5 or 6 chain stitches and continue in spiral fashion increasing as when crocheting the hyperbolic plane. See Figure 5.12. 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 5.12. Crocheted pseudosphere.

Problem 5.3. Rotations & Reflections on Surfaces

On the plane or on spheres rotations and reflections are both intrinsic and extrinsic (in the sense that they are also symmetries of the plane or sphere), and thus they are particularly easy to see. In addition, on the plane and sphere all rotations and reflections are global in the sense that they are symmetries of the whole space (the plane or sphere). On cylinders and cones, intrinsic rotations and reflections exist locally because cones and cylinders are locally isometric with the plane. However:

a. Which intrinsic rotations and reflections on which cones and cylinders are also extrinsic? Which are global?

Be sure to look at the cone points. The answers to the two questions are not exactly the same.

Now, we can see from our physical hyperbolic planes that geodesics exist joining every pair of point and that these geodesics each have reflection-in-themselves symmetry. (If you did not see this in Problem 5.1c, then go back and explore some more with your physical model. In Chapter 15 we will prove rigorously that this is in fact true by using the upper half plane model.) In Chapter 15 we will show that these reflections are global reflections of the whole hyperbolic space. However, note that there do not exist extrinsic reflections of the hyperbolic plane (in Euclidean 3-space). However, given all this, it is not clear that there exist intrinsic rotations, nor is it necessarily clear what exactly intrinsic rotations are.

b. Let l and m be two geodesics on the hyperbolic plane which intersect at the point P. Look at the composition of the reflection Rl through l with the reflection Rm through m. Show that this composition RmRl deserves to be called a rotation about P. What is the angle of the rotation?

Let A be a point on l and B be a point on m, and let Q be an arbitrary point (not on l or m). Investigate where A, B, and Q are sent by Rl and then by RmRl . See Figure 5.13.

Figure 5.13. Composition of two reflections is a rotation.

We will study symmetries and isometries in more detail in Chapter 10. In that chapter we will show that every isometry (on the plane, spheres, and hyperbolic planes) in a composition of one, two, or three reflections.

c. Show that that Problem 3.1 (VAT) holds on cylinders, cones (including the cone points), and hyperbolic planes.

If you check your proof(s) of 3.1 and modify them (if necessary) to only involve symmetries then you will be able to see that they hold also on the other surfaces.

d. Define "rotation about P through an angle q " without mentioning reflections in your definition.

What does a rotation do to a point not at P?

e. A popular high school textbook series defines a rotation as the composition of two reflections. Is this a good definition? Why? or Why not?

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.