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:

print copy
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


Infinite Acceleration: Risset Rhythms

In our most recent post, we took a look at and a listen to Shepard tones and their cousins, Shepard-Risset glissandos, which are tones or sequences of tones that create the illusion of perpetually rising (or falling) pitch. The illusion is created by overlaying a number of tones, separated by octaves, rising in unison. The volumes gradually increase from low pitch to middle pitch and gradually decrease from middle pitch to high pitch, leading to a fairly seamless continuous tone.

The same idea can be applied, mutatis mutandis, to percussive loops instead of tones, and to speed instead of pitch, thus creating the illusion of a rhythmic track that is perpetually speeding up (or slowing down). (The mechanism is exactly the same as that of the Shepard tone, so rather than provide an explanation here, I will simply refer the reader to the previous post.) Such a rhythm is known as a Risset rhythm.

I coded up some very basic examples on Supercollider. Here’s an accelerating Risset rhythm:

And a decelerating Risset rhythm:

Here’s a more complex Risset rhythm:

And, finally, a piece of electronic music employing Risset rhythms: “Calculus,” by Stretta.


Infinite Ascent: Shepard Tones

Have you ever been watching a movie and noticed that the musical score was seeming, impossibly, to be perpetually rising, ratcheting up the intensity of the film more and more? Or perhaps it seemed to be perpetually falling, creating a deeper and deeper sense of doom onscreen? If so, it is likely that this effect was achieved using a Shepard tone, a way of simulating an unbounded auditory ascent (or descent) in a bounded range.

To understand how Shepard tones work, let’s look at a simplified implementation of one. We will have three musical voices (middle, low, and high), with an octave between successive voices. The voices then start to move, in unison, and always an octave apart, up through a single octave, over, say, five seconds. As they go, though, they also change their volumes: the middle voice stays at full volume the whole time, the low voice gradually increases from zero volume to full volume, and the high voice gradually decreases from full volume to zero volume. The result will simply sound like a tone rising through an octave, and it can be represented visually as follows.


This by itself is nothing special, though. The trick of the Shepard tone is that this pattern is then repeated over, and over, and over again. Each repetition of the pattern sounds like a tone ascending an octave, but, because of the volume modulation, successive patterns are aurally glued together: the low voice from one cycle leads seamlessly to the middle voice of the next, the middle voice from one cycle leads seamlessly to the high voice of the next, and the high voice simply fades away. The result sounds like a perpetually increasing tone.


Note the similarity to the visual barber pole illusion, in which a rotating pole causes stripes to appear to be perpetually rising. Also, this whole story can be turned upside down, which will lead to a perpetually falling tone.

Let’s hear some Shepard tones in action! Now, in practice, using only three voices does not create a particularly convincing illusion, so, to make these sounds, I used nine voices, spread across nine octaves. Also, linearly varying the volume, as in the above visualization, seems to make it more noticeable when voices enter or fade away, so I used something more like a bell curve.

(Technical notes: These Shepard tones were created in Supercollider, using modified code written by Eli Fieldsteel, from whose YouTube tutorials I have learned a great deal of what I know about Supercollider. Also, I used a formant oscillator instead of the more traditional sine oscillator.)

First, a simple ascending Shepard tone:

The effect becomes more convincing, and the tone more interesting, if multiple Shepard tones are played simultaneously at a fixed interval. Here, we have two ascending Shepard tones separated by a tritone, a.k.a. the devil’s interval, a.k.a. half an octave:

Next, three descending Shepard tones, arranged in a minor triad:

Finally, two Shepard tones, with one ascending and the other descending:

The origins of the Shepard tone lie with Roger Shepard, a 20th-century American cognitive scientist, as a sequence of discrete notes. The continuous Shepard scale, or Shepard-Risset glissando, which our code approximates, was introduced by French composer Jean-Claude Risset, who perhaps most notably used it in his Computer Suite from Little Boy from 1968.

More recently, it has prominently been deployed by Christopher Nolan and Hans Zimmer, as the basis for the Batpod sound in The Dark Knight and in the Dunkirk soundtrack.

Cover image: M.C. Escher, Waterfall

Surreal Numbers (Universal Structures III)

“You get surreal numbers by playing games. I used to feel guilty in Cambridge that I spent all day playing games, while I was supposed to be doing mathematics. Then, when I discovered surreal numbers, I realized that playing games IS mathematics.”

-John Horton Conway

First issue of VVV, published by David Hare in collaboration with Marcel Duchamp, André Breton, and Max Ernst
“Julius Caesar,” by Man Ray
“Untitled,” by Joseph Cornell
“Las Meninas,” by Salvador Dali, after Diego Velasquez
“The Evil Genius of a King,” by Giorgio de Chirico
“Imaginary Numbers,” by Yves Tanguy
“Untitled,” by Joan Miró
“Möbius Strip II,” by M.C. Escher
“The Surrealist,” by Victor Brauner
“The Pleasure Principle (Portrait of Edward James),” by René Magritte

Dante, Einstein, and the Shape of the World

Last week, we began a series of posts dedicated to thinking about immortality. If we want to even pretend to think precisely about immortality, we will have to consider some fundamental questions. What does it mean to be immortal? What does it mean to live forever? Are these the same thing? And since immortality is inextricably tied up in one’s relationship with time, we must think about the nature of time itself. Is there a difference between external time and personal time? What is the shape of time? Is time linear? Circular? Finite? Infinite?

Of course, we exist not just across time but across space as well, so the same questions become relevant when asked about space. What is the shape of space? Is it finite? Infinite? It is not hard to see how this question would have a significant bearing on our thinking about immortality. In a finite universe (or, more precisely, a universe in which only finitely many different configurations of matter are possible), an immortal being would encounter the same situations over and over again, would think the same thoughts over and over again, would have the same conversations over and over again. Would such a life be desirable? (It is not clear that this repetition would be avoidable even in an infinite universe, but more on that later.)

Today, we are going to take a little historical detour to look at the shape of the universe, a trip that will take us from Ptolemy to Dante to Einstein, a trip that will uncover a remarkable confluence of poetry and physics.

One of the dominant cosmological views from ancient Greece and the Middle Ages was that of the Ptolemaic, or Aristotelian, universe. In this image of the world, Earth is the fixed, immobile center of the universe, surrounded by concentric, rotating spheres. The first seven of these spheres contain the seven “planets”: the Moon, Mercury, Venus, the Sun, Mars, Jupiter, and Saturn. Surrounding these spheres is a sphere containing the fixed stars. This is the outermost sphere visible from Earth, but there is still another sphere outside it: the Primum Mobile, or “Prime Mover,” which gives motion to all of the spheres inside it. (In some accounts the Primum Mobile is itself divided into three concentric spheres: the Crystalline Heaven, the First Moveable, and the Empyrean. In some other accounts, the Empyrean (higher heaven, which, in the Christianity of the Middle Ages, became the realm of God and the angels) exists outside of the Primum Mobile.)

An illustration of the Ptolemaic universe from The Fyrst Boke of the Introduction of Knowledge by Andrew Boorde (1542)

This account is naturally vulnerable to an obvious question, a question which, though not exactly in the context of Ptolemaic cosmology, occupied me as a child lying awake at night and was famously asked by Archytas of Tarentum, a Greek philosopher from the fifth century BC: If the universe has an edge (the edge of the outermost sphere, in the Ptolemaic account), then what lies beyond that edge? One could of course assert that the Empyrean exists as an infinite space outside of the Primum Mobile, but this would run into two objections in the intellectual climate of both ancient Greece and Europe of the Middle Ages: it would compromise the aesthetically pleasing geometric image of the universe as a finite sequence of nested spheres, and it would go against a strong antipathy towards the infinite. Archytas’ question went largely unaddressed for almost two millennia, until Dante Alighieri, in the Divine Comedy, proposed a novel and prescient solution.

Before we dig into Dante, a quick mathematical lesson on generalized spheres. For a natural number n, an n-sphere is an n-dimensional manifold (i.e. a space which, at every point, locally looks like n-dimensional real Euclidean space) that is most easily represented, embedded in n+1-dimensional space, as the set of all points at some fixed positive distance (the “radius” of the sphere) from a given “center point.”

Perhaps some examples will clarify this definition. Let us consider, for various values of n, the n-sphere defined as the set of points in (n+1)-dimensional Euclidean space at distance 1 from the origin (i.e. the point (0,0,…,0)).

If n=0, this is the set of real numbers whose distance from 0 is equal to 1, which is simply two points: 1 and -1.

If n=1, this is just the set of points (x,y) in the plane at a distance of 1 from (0,0). This is the circle, centered at the origin, with radius 1.

A 1-sphere

If n=2, this is the set of points (x,y,z) in 3-dimensional space at a distance 1 from the point (0,0,0). This is the surface of a ball of radius 1, and is precisely the space typically conjured by the word “sphere.”

A 2-sphere

0-, 1-, and 2-spheres are all familiar objects; beyond this, we lose some ability to visualize n-spheres due to the difficulty of considering more than three spatial dimensions, but there are useful ways to think about higher-dimensional spheres by analogy with the more tangible lower-dimensional ones. Let us try to use these ideas to get some understanding of the 3-sphere.

First, note that, for a natural number n, the non-trivial “cross-sections” of an n+1-sphere are themselves n-spheres! For example, if a 1-sphere (i.e. circle) is intersected with a 1-dimensional Euclidean space (a line) in a non-trivial way, the result is a 0-sphere (i.e. a pair of points). If a 2-sphere is intersected with a 2-dimensional Euclidean space (a plane) in a non-trivial way, the result is a 1-sphere (this is illustrated above in our picture of a 2-sphere). The same relationship holds for higher dimensional spheres: if a 3-sphere is intersected with a 3-dimensional Euclidean space in a non-trivial way, the result is a 2-sphere.

Suppose that you are a 2-dimensional person living in a 2-sphere universe. Let’s suppose, in fact, that you are living in the 2-sphere pictured above, with the 1-sphere “latitude lines” helpfully marked out for you. Let’s suppose that you begin at the “north pole” (i.e. the point at the top, in the center of the highest circle) and start moving in a fixed direction. At fixed intervals, you will encounter the 1-sphere latitude lines. For a while, these 1-spheres will be increasing in radius. This will make intuitive sense to you. You are moving “further out” in space; each successive circle “contains” the last and thus should be larger in radius. After you pass the “equator,” though, something curious starts happening. Even though you haven’t changed direction and still seem to be moving “further out,” the radii of the circles you encounter start shrinking. Eventually, you reach the “south pole.” You continue on your trip. The circles wax and wane in a now familiar way, and, finally, you return to where you started.

A similar story could be told about a 3-dimensional being exploring a 3-sphere. In fact, I think we could imagine this somewhat easily. Suppose that we in fact live in a 3-sphere. For illustration, let us place a “pole” of this 3-sphere at the center of the Earth. Now suppose that we, in some sort of tunnel-boring spaceship, begin at the center of the Earth and start moving in a fixed direction. For a while, we will encounter 2-sphere cross-sections of increasing radius. Of course, in the real world these are not explicitly marked (although, for a while, they can be nicely represented by the spherical layers of the Earth’s core and mantle, then the Earth’s surface, then the sphere marking the edge of the Earth’s atmosphere) but suppose that, in our imaginary world, someone has helpfully marked them. For a while, these successive 2-spheres have larger and larger radii, as is natural. Eventually, of course, they will start to shrink, contracting to a point before expanding and contracting as we return to our starting point at the Earth’s core.

Dante’s Divine Comedy, completed in 1320, is one of the great works of literature. In the first volume, Inferno, Dante is guided by Virgil through Hell, which exists inside the Earth, directly below Jerusalem (from where I happen to be writing this post). In the second volume, Purgatorio, Virgil leads Dante up Mount Purgatory, which is situated antipodally to Jerusalem and formed of the earth displaced by the creation of Hell. In the third volume, Paradiso, Dante swaps out Virgil for Beatrice and ascends from the peak of Mount Purgatory towards the heavens.

Dante’s universe. Image by Michelangelo Caetani.

Dante’s conception of the universe is largely Ptolemaic, and most of Paradiso is spent traveling outward through the larger and larger spheres encircling the Earth. In Canto 28, Dante reaches the Primum Mobile and turns his attention outward to what lies beyond it. We are finally in a position to receive an answer to Archytas’ question, and the answer that Dante comes up with is surprising and elegant.

The structure of the Empyrean, which lies outside the Primum Mobile, is in large part a mirror image of the structure of the Ptolemaic universe, a revelation that is foreshadowed in the opening stanzas of the canto:

When she who makes my mind imparadised
Had told me of the truth that goes against
The present life of miserable mortals —

As someone who can notice in a mirror
A candle’s flame when it is lit behind him
Before he has a sight or thought of it,

And turns around to see if what the mirror
Tells him is true, and sees that it agrees
With it as notes are sung to music’s measure —

Even so I acted, as I well remember,
While gazing into the bright eyes of beauty
With which Love wove the cord to capture me.

When Dante looks into the Empyrean, he sees a sequence of concentric spheres, centered around an impossibly bright and dense point of light, expanding to meet him at the edge of the Primum Mobile:

I saw a Point that radiated light
So sharply that the eyelids which it flares on
Must close because of its intensity.

Whatever star looks smallest from the earth
Would look more like a moon if placed beside it,
As star is set next to another star.

Perhaps as close a halo seems to circle
The starlight radiance that paints it there
Around the thickest mists surrounding it,

As close a ring of fire spun about
The Point so fast that it would have outstripped
The motion orbiting the world most swiftly.

And this sphere was encircled by another,
That by a third, and the third by a fourth,
The fourth by a fifth, the fifth then by a sixth.

The seventh followed, by now spread so wide
That the whole arc of Juno’s messenger
Would be too narrow to encompass it.

So too the eighth and ninth, and each of them
Revolved more slowly in proportion to
The number of turns distant from the center.

This seemingly obscure final detail, that the spheres of the Empyrean spin increasingly slowly as they increase in size, and in distance from the point of light, turns out to be important. Dante is initially confused because, in the part of the Ptolemaic universe from the Earth out to the Primum Mobile, the spheres spin faster the larger they are; the fact that this is different in the Empyrean seems to break the nice symmetry he observes. Beatrice has a ready explanation, though: the overarching rule governing the speed at which the heavenly spheres rotate is not based on their size, but rather on their distance from God.

This is a telling explanation and seems to confirm that the picture Dante is painting of the universe is precisely that of a 3-sphere, with Satan, at the center of the Earth, at one pole and God, in the point of light, at the other. If Dante continues his outward journey from the edge of the Primum Mobile, he will pass through the spheres of the Empyrean in order of decreasing size, arriving finally at God. Note that this matches precisely the description given above of what it would be like to travel in a 3-sphere. Dante even helpfully provides a fourth dimension into which his 3-sphere universe is embedded: not a spatial dimension, but a dimension corresponding to speed of rotation!

(For completeness, let me mention that the spheres of the Empyrean are, in order of decreasing size and hence increasing proximity to God: Angels, Archangels, Principalities, Powers, Virtues, Dominions, Thrones, Cherubim, and Seraphim.)

Dante’s ingenious description of a finite universe helped the Church to argue against the existence of the infinite in the physical world. Throughout the Renaissance, Scientific Revolution, and Enlightenment, this position was gradually eroded in favor an increasingly accepted picture of infinite, flat space. A new surprise awaited, though, in the twentieth century.

‘Paradiso II’; Dante and Beatrice in the sphere of the moon, with Beatrice explaining the nature of the heavens; illustration by Sandro Botticelli, circa 1490
Beatrice explaining the nature of the heavens to Dante. Drawing by Botticelli.

In 1917, Einstein revolutionized cosmology with the introduction of general relativity, which provided an explanation of gravity as arising from geometric properties of space and time. Central to the theory are what are now known as the Einstein Field Equations, a system of equations that describes how gravity interacts with the curvature of space and time caused by the presence of mass and energy. In the 1920s, an exact solution to the field equations, under the assumptions that the universe is homogeneous and isotropic (roughly, has laws that are independent of absolute position and orientation, respectively), was isolated. This solution is known as the Friedmann-Lemaître-Robertson-Walker metric, after the four scientists who (independently) derived and analyzed the solution, and is given by the equation,

ds^2 = -dt^2 + R^2(t)\left(\frac{dr^2}{1-kr^2} + r^2(d\theta^2 + \sin^2\theta d\phi^2)\right),

where k is a constant corresponding to the “curvature” of the universe. If k = 0, then the FLRW metric describes an infinite, “flat” Euclidean universe. If k < 0, then the metric describes an infinite, hyperbolic universe. If k>0, though, the metric describes a finite universe: a 3-sphere.

PS: Andrew Boorde, from whose book the above illustration of the Ptolemaic universe is taken, is a fascinating character. A young member of the Carthusian order, he was absolved from his vows in 1529, at the age of 39, as he was unable to adhere to the “rugorosite” of religion. He turned to medicine, and, in 1536, was sent by Thomas Cromwell on an expedition to determine foreign sentiment towards King Henry VIII. His travels took him throughout Europe and, eventually, to Jerusalem, and led to the writing of the Fyrst Boke of the Introduction of Knowledge, perhaps the earliest European guidebook. Also attributed to him (likely without merit) is Scoggin’s Jests, Full of Witty Mirth and Pleasant Shifts, Done by him in France and Other Places, Being a Preservative against Melancholy, a book which, along with Boord himself, plays a key role in Nicola Barker’s excellent novel, Darkmans.

Further Reading:

Mark A. Peterson, “Dante and the 3-sphere,” American Journal of Physics, 1979.

Carlo Rovelli, “Some Considerations on Infinity in Physics,” and Anthony Aguirre, “Cosmological Intimations of Infinity,” both in Infinity: New Research Frontiers, edited by Michael Heller and W. Hugh Woodin.

Cover Image: Botticelli’s drawing of the Fixed Stars.

The Infinite Art of Barcelona

I spent part of the last two weeks savoring the art and architecture of Barcelona, entranced by the work of two artistic giants of Catalonia: Joan Miró and Antoni Gaudí. Both artists seem to revel in dualities. They take the straight lines of the manmade world and bend them. They take the unruly forms of nature and impose upon them an order. They seek to contain the universe in a box, to express infinity in their necessarily finite works.

This is already too much writing from me. Simply enjoy the words and works of these two artists. (The Miró here is all from the Fundació Joan Miró and the Gaudí from the Sagrada Família.)

Look for the noise hidden in silence, the movement in immobility, life in inanimate things, the infinite in the finite, forms in a void, and myself in anonymity.

-Joan Miró

-Towards Infinity, Joan Miró



The straight line is the expression of the infinite.

-Antoni Gaudí


The straight line belongs to Man. The curved line belongs to God.

-Antoni Gaudí



Infinite Picture Books

I’ve been traveling for the last month or so and thus have not been posting here. I have returned, though, so I will try to begin regularly updating again.

Today, just a short post to tell you to go have a look at some fantastic work by Richard Evan Schwartz of Brown University, who has two delightful (and occasionally terrifying) PDF picture books about infinity on his website: