Constructions of Hyperbolic Planes

We will describe four different isometric constructions of hyperbolic planes (or approximations to hyperbolic planes) 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 you a feel for hyperbolic planes 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 Figure 5.2. Attach the strips together by taping the inner circle of one to the outer circle of the other. It is crucial that all the annular strips have the same inner radius and the same outer radius, but the lengths of the annular strips do not matter. You can also cut an annular strip shorter or extend an annular strip by taping two strips together along their straight ends. 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 (as d ® 0) the same everywhere it is homogeneous (that is, 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 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 have 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 sturdier 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 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, and see their description in the list below. a b

Figure 5.3 Crochet stitches for the hyperbolic plane

First you should choose a yarn that will not stretch a lot. Every yarn will stretch a little but you need one that 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 tightly and evenly. 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-gons) 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 heptagons (7-gons) are put together 3 at a vertex. These models have the advantage of being constructed more easily than the annular or crocheted models; however, one cannot 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 that 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 five hexagons or two hexagons and one pentagon together around each vertex. The plane can be tiled by hexagons, each surrounded by six other hexagons. The hyperbolic plane can be approximately constructed by using heptagons (7-sided) surrounded by seven hexagons and two hexagons and one heptagon together around each vertex. See Figure 5.5. Because 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 you can find a variety on the supplements website). As with any polyhedral construction one cannot 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 seven triangles together only at every other vertex and six 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 five additional triangles. Next, add another strip every place there is a vertex with five triangles and a gap (as at the marked vertices in Figure 5.7b). Every time a strip is added an additional vertex with seven 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 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 5.85.10 (see next page) 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 to indicate the scale.

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. 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