Life on the Poincaré Disk

Just at this time I left Caen, where I was then living, to go on a geological excursion under the auspices of the school of mines. The changes of travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of  non-Euclidean geometry. I did not verify the idea; I should not have had time, as, upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience’ sake I verified the result at my leisure.

-Henri Poincaré, Science and Method

You’re out for a walk one day, contemplating the world, and you suddenly have an out-of-body experience, your perspective floating high above your corporeal self. As you rise, everything seems perfectly normal at first, but, when you reach a sufficient altitude, you notice something strange: your body appears to be at the center of a perfect circle, beyond which there is simply…nothing!

You watch yourself walk towards the edge of the circle. It initially looks like you will reach the edge in a surprisingly short amount of time, but, as you continue watching, you notice yourself getting smaller and slowing down. By the time you are halfway to the edge, you are moving at only 3/4 of your original speed. When you are 3/4 of the way to the edge, you are moving at only 7/16 of your original speed. Maybe you will never reach the edge after all? What is happening?

At some point, you see your physical self notice some friends, standing some distance away in the circle. You wave to one another, and your friends beckon you over. You start walking toward them, but, strangely, you walk in what looks not to be a straight line but rather an arc, curving in towards the center of the circle before curving outward again to meet your friends. And, equally curiously, your friends don’t appear to be surprised or annoyed by your seemingly inefficient route. You puzzle things over for a few seconds before having a moment of insight. ‘Oh!’ you think. ‘My physical body is living on a Poincaré disk model for hyperbolic geometry, which my mind has somehow transcended during this out-of-body experience. Of course!”

The Poincaré disk model, which was actually put forth by Eugenio Beltrami, is one of the first and, to my mind, most elegant models of non-Euclidean geometry. Recall from our previous post that a Euclidean geometry is a geometry satisfying Euclid’s five postulates. The first four of these postulates are simple and self-evident. The fifth, known as the Parallel Postulate (recall also that two lines are parallel if they do not intersect), is unsatisfyingly complex and non-immediate. To refresh our memories, here is an equivalent form of the Parallel Postulate, known as Playfair’s Axiom:

Given any line \ell and any point P not on \ell, there is exactly one line through P that is parallel to \ell.

A non-Euclidean geometry is a geometry that satisfies the first four postulates of Euclid but fails to satisfy the Parallel Postulate. Non-Euclidean geometries began to be seriously investigated in the 19th century; Beltrami, working in the context of Euclidean geometry, was the first to actually produce models of non-Euclidean geometry, thus proving that, supposing Euclidean geometry is consistent, then so is non-Euclidean geometry.

The Poincaré disk model, one of Beltrami’s models, is a model for hyperbolic geometry, in which the Parallel Postulate is replaced by the following statement:

Given any line \ell and any point P not on \ell, there are at least two distinct lines through P that are parallel to \ell.

Points and lines are the basic objects of geometry, so, to describe the Poincaré disk model, we must first describe the set of points and lines of the model. The set of points of the model is the set of points strictly inside a given circle. For concreteness, let us suppose we are working on the Cartesian plane, and let us take the unit circle, i.e., the circle of radius one, centered at the origin, as our given circle. The points in the Poincaré disk model are then the points in the plane whose distances from the origin are strictly less than one.

Lines in the Poincaré disk model (which we will sometimes call hyperbolic lines) are arcs formed by taking one of the following type of objects and intersecting it with the unit disk:

  1. Straight lines (in the Euclidean sense) through the center of the circle.
  2. Circles (in the Euclidean sense) that are perpendicular to the unit circle.

(These can, of course, be seen as two instances of the same thing, if one takes the viewpoint that, in Euclidean space, straight lines are just circles of infinite radius.)

D, D1, and D2 are all lines in the Poincaré disk model. By Jean-Christophe BENOIST, Own work – CC BY 3.0

It’s already pretty easy to see that this geometry satisfies our hyperbolic replacement of the Parallel Postulate. In fact, given a line \ell and a point P not on \ell, there are infinitely many lines through P parallel to \ell. Here’s an illustration of a typical case, with three parallel lines drawn:

Three lines passing through a given point, parallel to a given line. Source: William Barker

We’re not quite able right now to prove that the disk model satisfies the first four of Euclid’s postulates, in part because we haven’t yet specified what it means for two line segments in the model to be be congruent (we don’t, for example, have a notion of distance in our model yet). We’ll get to this in just a minute, but let us first show that our model satisfies the first postulate: Given any two distinct points, there is a line containing both of them.

To this end, let A and B be two points in the disk. If the (Euclidean) line that contains A and B passes through the center of the disk, then this is also a line in the disk model, and we are done. Otherwise, the (Euclidean) line that contains A and B does not pass through the center of the disk. In this case, we use the magic of circle inversion, which we saw in a previous post. Let A' by the result of inverting A across the unit circle. Now A, A', and B are distinct points in the Cartesian plane, so there is a unique circle (call it \gamma) containing all three. Since A and A' are both on the circle, it is perpendicular to the unit circle. Therefore, its intersection with the unit disk is a line in the disk model containing both A and B. Here’s a picture:

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Hyperbolic line containing A and B. Source: Euclid and Beyond by Robin Hartshorne

We turn now to distance in the Poincaré disk model. And here, for the sake of brevity, I’m not even going to try to explain why things are they way they are but will just give you a formula. Given two points A and B in the disk, consider the hyperbolic line containing them, and let P and Q be the points where this line meets the boundary circle (with P closer to A and Q closer to B). Then the hyperbolic distance between A and B is given by:

d(A,B) = \mathrm{ln}(\frac{|PB|\cdot|AQ|}{|PA|\cdot|BQ|}).

This is likely inscrutable right now. That’s fine. Let’s think about what it means for this to be the correct notion of distance, though. For one thing, it means that, given two points in the disk model, the shortest path between them is not, in general, the straight Euclidean line that connects them, but rather the hyperbolic line that connects them. This explains your body’s behavior in the story at the start of this post. When you were walking over to your friends, what appeared to your mind (which was outside the disk, in the Euclidean realm) as a curved arc, and therefore an inefficient path, was in fact a hyperbolic line and, because your body was inside the hyperbolic disk, the shortest path between you and your friends.

This notion of distance also means that distances inside the disk which appear equal to an external Euclidean observer in fact get longer and longer the closer they are to the edge of the disk. This is also consistent with the observations at the beginning of the post: as your body got further toward the edge of the disk, it appeared from an external viewpoint to be moving more and more slowly. From a viewpoint inside the disk, though, it was moving at constant speed and would never reach the edge of the disk, which is infinitely far away. The disk appears bounded from the external Euclidean view, but from within it is entirely unbounded and limitless.

Let’s close by looking at two familiar shapes, interpreted in the hyperbolic disk. First, circles. Recall that a circle is simply the set of points that are some fixed distance away from a given center. Now, what happens when we interpret this definition inside the hyperbolic disk? Perhaps somewhat surprisingly, we get Euclidean circles! (Sort of.) To be more precise, hyperbolic circles in the Poincaré disk model are precisely the Euclidean circles that lie entirely within the disk. (I’m not going to go through the tedious calculations to prove this; I’ll leave that up to you…) Beware, though! The hyperbolic center of the circle is generally different from the Euclidean center. (This should make sense if you think about our distance definition. The hyperbolic center will be further toward the edge of the disk than the Euclidean center, coinciding only if the Euclidean center of the circle is in fact the center of the hyperbolic disk.)

Next, triangles. A triangle is, of course, a polygon with three sides. This definition works perfectly fine in hyperbolic geometry; we simply require that our sides are hyperbolic line segments rather than Euclidean line segments. If we assume the first four of Euclid’s postulates, then the Parallel Postulate is actually equivalent to the statement that the sum of the interior angles of a triangle is 180 degrees. In the Poincaré disk model (and, in fact, in any model of hyperbolic geometry) all triangles have angles that sum to less than 180 degrees. This should be evident if we look at a typical triangle:

A typical triangle in the Poincaré disk model.

Things become interesting when you start to ask how much less than 180 degrees a hyperbolic triangle has. The remarkable fact is that the number of degrees in a hyperbolic triangle is dependent entirely on its (hyperbolic) area! The smaller a triangle is, the larger the sum of its interior angles: as triangles get smaller and smaller, approaching a single point, the sum of their angles approaches 180 degrees from below. Correspondingly, as triangles get larger, the sum of their angles approaches 0 degrees. In fact if we consider an “ideal triangle”, in which the three vertices are in fact points on the bounding circle (and thus not real points in the disk model), then the sum of the angles of this “triangle” is actually 0 degrees!

“Ideal” triangle with interior angles adding to zero.

A consequence of this is the fact that, in the Poincaré disk model, if two triangles are similar, then they are in fact congruent!

This leads us to our final topic: one of the perks of living in a Poincaré disk model. Perhaps the most frequent complaint I hear from people living on a Euclidean plane is that there aren’t enough ways to tile the plane with triangles. Countless people come up to me and say, “Chris, I want to tile the plane with triangles, and I want this tiling to have the following two pleasing properties:

  1. All of the triangles are congruent, they don’t overlap, and they fill the entire plane.
  2. At every vertex of the tiling, all angles meeting that vertex are the same.

But there are only four essentially different ways of doing this, and I’m tired of all of them! What should I do?”

(Exercise for the reader: Find all four such tilings!)

It just so happens that I have a simple answer for these people: “Move to a Poincaré disk model, where there are infinitely many tilings with these properties!” Here are just a few (all by Tamfang and in the public domain):

Right triangles. The smallest, in fact, that can tile the Poincaré disk model.
Larger right triangles.
The largest right “triangles”, each with two “ideal” vertices on the edge of the disk.
The dual to a tiling hidden in Escher’s Circle Limit III, the cover image to this post.
Equilateral triangles.
The largest “triangles”, each with three ideal vertices.

I’ll leave you with that! Hyperbolic geometry is fascinating, and I encourage you to investigate further on your own. The previous mentioned Euclid and Beyond, by Hartshorne, is a nice place to start.

This also wraps up (for now, at least) a couple of multi-part investigations here at Point at Infinity: a look at the interesting geometry of circles, which started in our post on circle inversion, and a look at various notions of independence in mathematics, the other posts being here and here. Join us next time for something new!

Cover Image: M. C. Escher, Circle Limit III


Infinity in the Classroom II

We’re between posts about non-Euclidean geometry here at Point at Infinity. In the meantime, take a look at this article about a fascinating connection between hyperbinary numbers and the countability of the rationals, and about exploring this connection in the classroom. Enjoy!

Parallel Lines

Detest it as lewd intercourse, it can deprive you of all your leisure, your health, your rest, and the whole happiness of your life.

Do not try the parallels in that way: I know that way all along. I have measured that bottomless night, and all the light and all the joy of my life went out there.

-Letters from Farkas Bolyai to his son, János, attempting to dissuade him from his investigations of the Parallel Postulate.

In our previous post, cataloging various notions of mathematical independence, we introduced the idea of logical independence. Briefly, given a consistent set of axioms, T, a sentence \varphi is independent from T if it can be neither proven nor disproven from the sentences in T. Today, we discuss one of the most prominent and interesting instances of logical independence: Euclid’s Parallel Postulate.

Among the most famous sets of axioms (top 5, certainly) are Euclid’s postulates, five statements underpinning (together with 23 definitions and five other statements putting forth the properties of equality) the mathematical system of Euclidean geometry set forth in the Elements and still taught in high school classrooms to this day. (We should note here that, from a modern viewpoint, Euclid’s proofs do not always strictly conform to the standards of mathematical rigor, and some of his results rely on methods or assumptions not justified by his five postulates. This has been fixed, for example by Hilbert, who gave a different set of axioms for Euclidean geometry in 1899. Now that we have noted this, we will proceed to forget it for the remainder of the post.)

Euclid’s first four postulates are quite elegant in their simplicity and self-evidence. Reformulated in modern language, they are roughly as follows:

  1. Given any two points, there is a unique line segment connecting them.
  2. Given any line segment, there is a unique line (unbounded in both directions) containing it.
  3. Given any point P and any radius r, there is a unique circle of radius r centered at P.
  4. All right angles are congruent.

The fifth postulate, however, which is known as the Parallel Postulate, is, quite unsatisfyingly, markedly more complicated and less self-evident:

  1. If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines must intersect one another on that side.

A picture might help illustrate this postulate:

The two indicated angles sum to less than two right angles, so, by the Parallel Postulate, the two lines, if extended far enough, will intersect on that side of the third line. By Harkonnen2, CC BY-SA 3.0

This postulate doesn’t seem to be explicitly about parallel lines, so the reader may be wondering why it is often called the Parallel Postulate. The reason becomes evident, though, when considering the following statement and learning that, in the context of the other four postulates, it is in fact equivalent to the Parallel Postulate:

  1. Given any line \ell and any point P not on \ell, there is exactly one line through P that is parallel to \ell.

(Recall that two lines are parallel if they do not intersect.) This reformulation of the Parallel Postulate is often named Playfair’s Axiom, after the 18th-century Scottish mathematician John Playfair, though it was stated already by Proclus in the 5th century.

The Parallel Postulate was considered undesirably unwieldy and less satisfactory than the other four postulates, even by Euclid himself, who made a point of proving the first 28 results of the Elements without recourse to the Parallel Postulate. The general opinion among mathematicians for the next two millennia was that the Parallel Postulate should not be an axiom but rather a theorem; it should be possible to prove it using just the other four postulates.

Many attempts were made to prove the Parallel Postulate, and many claimed success at this task. Errors were then inevitably discovered by later mathematicians, many of whom subsequently put forth false proofs of their own. The aforementioned Proclus, for example, after pointing out flaws in a purported proof of Ptolemy, gives his own proof, which suffers from two instructive flaws. The first is relatively minor: Proclus assumes a consequence of Archimedes’ Axiom, which essentially states that, given any two line segments, there is a natural number n such that n times the length of the shorter line segment will exceed the length of the longer. (We encountered Archimedes’ Axiom in a previous post, about infinitesimals, which the reader is invited to revisit.) Archimedes’ Axiom seems like an entirely reasonable axiom to assume, but it notably does not follow from Euclid’s postulates.

Proclus’ more serious error, though, is that he makes the assumption that any two parallel lines have a constant distance between them. But this does not follow from the first four postulates. In fact, the statement, “The set of points equidistant from a straight line on one side of it form a straight line,” known as Clavius’ Axiom, is, in the presence of Archimedes’ Axiom and the first four postulates, equivalent to the Parallel Postulate. Proclus’ proof is therefore just a sophisticated instance of begging the question.

In the course of the coming centuries’ attempts to prove the Parallel Postulate, a number of other axioms were unearthed that are, at least in the presence of Archimedes’ Axiom and the first four postulates, equivalent to the Parallel Postulate. In addition to Playfair’s Axiom and Clavius’ Axiom, these include the following:

  • (Clairaut) Rectangles exist. (A rectangle, of course, being a quadrilateral with four right angles.)
  • (Legendre) Given an angle \alpha and a point P in the interior of the angle, there is a line through P that meets both sides of the angle.
  • (Wallis) Given any triangle, there are similar triangles of arbitrarily large size.
  • (Farkas Bolyai) Given any three points, not all lying on the same line, there is a circle passing through all three points.

A key line of investigation into the Parallel Postulate was carried out, probably independently, by Omar Khayyam, an 11th-century Persian mathematician, astronomer, and poet, and by Giovanni Gerolamo Saccheri, an 18th-century Italian Jesuit priest and mathematician. For concreteness, let us consider Saccheri’s account, which has the wonderful title, “Euclid Freed from Every Flaw.”

Saccheri and Khayyam were, similarly to their predecessors, attempting to prove the Parallel Postulate. Their method of proof was contradiction: assume that the Parallel Postulate is false and derive a false statement from it. To do this, they considered figures that came to be known as Khayyam-Saccheri quadrilaterals.

To form a Khayyam-Saccheri quadrilateral, take a line segment (say, BC). Take two line segments of equal length and form perpendiculars, in the same direction, at B and C (forming, say, AB and DC). Now connect the ends of those two line segments with a line segment (AD) to form a quadrilateral. A picture is given below.

A Khayyam-Saccheri quadrilateral. By HR – Own work, CC BY-SA 3.0

By construction, the angles at B and C are right angles, but the angles at A and D are unclear. Saccheri proves that these two angles are equal. He also proves that, if these angles are obtuse, then they are obtuse for every such quadrilateral; if they are right, then they are right for every such quadrilateral; and, if they are acute, then they are acute for every such quadrilateral. This then naturally divides geometries into three categories: those satisfying the Obtuse Hypothesis, those satisfying the Right Hypothesis, and those satisfying the Acute Hypothesis. (These types of geometries subsequently became known as semielliptic, semieuclidean, and semihyperbolic, respectively.)

At this point, Saccheri attempts to prove that the Obtuse Hypothesis and the Acute Hypothesis both lead to contradiction. (Note that a geometry satisfying all five of Euclid’s postulates must satisfy the Right Hypothesis. The converse is not true, so even a successful refutation of the Obtuse and Acute Hypotheses would not be enough to establish the Parallel Postulate.) Saccheri is able to prove (in the presence of Archimedes’ Axiom) that the Obtuse Hypothesis leads to the conclusion that straight lines are finite, thus contradicting the second postulate. He is unable to obtain a logical contradiction from the Acute Hypothesis, though. Instead, he derives a number of counter-intuitive statements from it and then concludes that the Acute Hypothesis must be false because it is “repugnant to the nature of a straight line.”

The next big steps towards the establishment of the independence of the Parallel Postulate were made by Nikolai Lobachevsky and János Bolyai (who fortunately did not heed his father’s letters quoted at the top of this post), 19th-century mathematicians from Russia and Hungary, respectively. (Similar work was probably done by Gauss, as well, though it was never published.) Their work entailed a crucial shift in perspective – rather than attempt to prove the Parallel Postulate from the others, the mathematicians seriously considered the possibility that it is not provable and thought of non-Euclidean geometries (i.e., those failing to satisfy the Parallel Postulate) as legitimate objects of mathematical study in their own right. In particular, they were interested in hyperbolic geometry, in which the Parallel Postulate is replaced by the assertion that, given any line \ell and any point P not on the line, there are at least two distinct lines passing through P and parallel to \ell. (Not surprisingly, considering the nomenclature, hyperbolic geometries are semihyperbolic, i.e., they satisfy the Acute Hypothesis.) This viewpoint was vindicated when, in 1868, Eugenio Beltrami produced a model of hyperbolic geometry. This shows that, as long as Euclidean geometry is consistent, then the Parallel Postulate is independent of the other four postulates: all five postulates are true, for example, in the Cartesian plane, while the first four are true and the Parallel Postulate is false in any model of hyperbolic geometry.

A number of other models for hyperbolic geometry are now known. In our next post, we will look at a particularly elegant one: the Poincaré disk model.

Cover Image: Michael Tompsett, Parallel Lines

For more information on this and many other geometric topics, I highly recommend Robin Hartshorne’s excellent book, Geometry: Euclid and Beyond.