Life is a full circle, widening until it joins the circle motions of the infinite.
No shape has captured the human imagination quite like the circle has. Its perfect symmetry and constant radius stand in contrast to the messy variability of our everyday lives. We have inner circles, vicious circles, fairy circles, crop circles, family circles, circles of influence, the circle of life. Circles permeate our conception of time, as they provide the shape of our clocks, and when things return to their original configuration, we say that they have come full circle.
We have seen intimate connections between circles and infinity in many previous posts. The circle is the one-point compactification of the infinite straight line, which itself can be thought of as a circle with infinite radius. The process of circle inversion provides a useful duality between the finite region inside a circle and the infinite region outside. The Poincaré disk model provides an elegant finite setting for a decidedly infinite instantiation of hyperbolic geometry.
In the next few posts, we will be exploring circles more deliberately, through lenses of mathematics, philosophy, linguistics, art, and literature. We will think about circular definitions and closed timelike curves, about causality and the Liar Paradox.
These are for future weeks, though. For today, to lay the foundations and whet the appetite, please enjoy these three essential pieces of circle-related culture:
Although popularly every one called a Circle is deemed a Circle, yet among the better educated Classes it is known that no Circle is really a Circle, but only a Polygon with a very large number of very small sides. As the number of the sides increases, a Polygon approximates to a Circle; and, when the number is very great indeed, say for example three or four hundred, it is extremely difficult for the most delicate touch to feel any polygonal angles. Let me say rather, it would be difficult: for, as I have shown above, Recognition by Feeling is unknown among the highest society, and to feel a circle would be considered a most audacious insult. This habit of abstention from Feeling in the best society enables a Circle the more easily to sustain the veil of mystery in which, from his earliest years, he is wont to enwrap the exact nature of his Perimeter or Circumference. Three feet being the average Perimeter, it follows that, in a Polygon of three hundred sides, each side will be no more than the tenth part of an inch; and in a Polygon of six or seven hundred sides the sides are little larger than the diameter of a Spaceland pin-head. It is always assumed, by courtesy, that the Chief Circle for the time being has ten thousand sides.
Edwin A. Abbott, Flatland: A Romance of Many Dimensions
Go, of course, is a notoriously difficult and abstract board game, invented in China over 2500 years ago and further developed in Korea and Japan. After millennia of being largely unknown outside of East Asia, it has in the last century become popular throughout the world (the publication of this book played a significant role in introducing it to France) and has even been in the news recently, as computer programs using neural networks have defeated some of the best professional go players in highly publicized matches.
The stated goal of Lusson, Perec, and Roubaud’s book is to “[provide], in a clear, complete, and precise manner, the rules of the game of GO” and to “heighten interest in this game.” As a practical manual on the rules and basic tactics and strategy of the game, the modern reader can do much better with other books. As a literary meditation on play, on art, and on infinity, it is dazzling. It is this latter aspect of the book that I want to touch on here today.
A theme running throughout the book is the idea that the practice of go is akin to a journey into infinity, and this theme is expressed both with respect to one’s relationship with other players and with one’s relationship to the game itself.
A joy of learning any game is developing relationships and rivalries with other players, and this is especially true with go, for two main reasons. First, an individual match is not simply won or lost but rather is won or lost by a certain number of points. Second, there is a robust handicapping system whereby a substantially weaker player can legitimately compete with a stronger player, in a match of interest to both players, by placing a specific number of pieces on specific points of the board before the first move. Through these two mechanisms, a rich and rewarding go relationship can thus develop, even between players of unequal skill, over not just one match but a progression of matches, during which handicaps can be finely calibrated and can, indeed, change over time, as the players learn more about each other’s play and about the game in general.
As such, GO, at its limits, constitutes the best finite simulation of an infinite game. The two adversaries battle on the Goban the way cyclists pursue each other in a velodrome.
This is not as much the case with, say, chess, in which the facts that the result of a game is purely binary and that handicap systems are clumsier and more substantially alter the character of the game mean that matches between players of unequal skill will frequently be frustrating for the weaker and boring for the stronger.
It is a cliché to say that one can never truly master a subject, that there is always more to learn. But the richness of go makes it especially true here, and, in a sense, quantifiably so. The number of legal positions in go is more than . This is a truly astounding number, dwarfing the estimated number of atoms in the observable universe () or the estimated number of legal positions in chess (a piddling ). The possibilities in go are, for all intents and purposes, infinite. No matter how much one learns, one knows essentially nothing.
Crucially, though, there is a well-developed hierarchy through which one progresses and by which one can measure one’s skill, even if remains practically zero when taking a wider view. Lusson, Perec, and Roubaud write about this better than I could in the following two excerpts, so let us simply listen to them:
The genius of GO stems precisely from what it hides as well as what it reveals, at any moment, at any level, in its different, hierarchized mysteries whose progressive mastery transforms the game every time:
A garden of bifurcating pathways, a labyrinth, the game of Babel, each step forward is decisive and each step forward is useless: we will never have finished learning..
(Note the surely intentional nod to Borges in the last sentence above).
From a beginner to a classified amateur in the bottom ranks of kyu, a player can rise to the top kyus and then, one by one, climb the six amateur echelons, approaching (but only approaching) the inaccessible regions where the true players reign, the professionals…
In this last excerpt, we hear echoes of a mathematical concept from set theory, my personal area of expertise. The authors temper the reader’s (and their own) dreams of go mastery by noting that, no matter how high an amateur go player may ascend in their study of go, they will still never reach the “inaccessible” realm of the true masters of the game. These professionals also inhabit a hierarchy, but it is a separate hierarchy, visible but unreachable from below.
This calls to mind the concept of an inaccessible cardinal, which is an uncountable cardinal number that cannot be reached from below through the standard procedures of climbing to the next cardinal, applying cardinal exponentiation, or taking unions of a small number of small sets. (More formally, is (strongly) inaccessible if it is regular, uncountable, and, for all , we have .)
It cannot be proven that inaccessible cardinals exist or are even consistent, and the assumption of the consistency of such cardinals has significant implications for what one can prove (see a previous post for more information about inaccessible and other “large cardinals”). On the simplest descriptive level, an inaccessible cardinal divides the hierarchy of infinite cardinals into two realms that cannot communicate via the standard operations of arithmetic: those above and those below.
(A modern version of the book would surely posit a third separate region of hierarchies in go: that of the neural networks that with stunning swiftness have become stronger than the strongest professionals.)
So why bother? If one can spend one’s whole life studying go and still hardly understand the game, if one can develop to one’s full potential and still be nowhere close to the level of professional players, let alone the newly ascendant artificial intelligences, then why start at all?
The authors consider this question, but ultimately they reject its premises. The study of go is not worthwhile in spite of the fact that it is an “infinite pathway.” It is worthwhile because of it.
And this clearly has implications outside of go. Why devote much of my life to mathematical research if I can never know more than a miniscule portion of what remains undiscovered? Why write if it is physically impossible to write more than about 10,000,000 words in a life, and if everything I may write is already contained in Borges’ Library of Babel anyway? Perhaps because the best life is a finite simulation of an infinite life.
Only one activity exists to which GO may be reasonably compared.
We will have understood it is writing.
PS: We have had occasion to mention chess and its relation to the infinite in a previous post. One of the joys of A Short Treatise… is the exaggerated contempt expressed by its authors for the game of chess. We end by offering you just a taste:
One of the best European players, Zoran Mutabzija, abandoned chess, to which he had devoted himself since the age of four, as soon as someone taught him GO!
In related news.
We just received a detailed report concerning a certain Louis A. caught in the act of robbing a gas station attendant on the Nationale 6. According to the report, whose source and information cannot be called into question, Louis A. is a notorious chess player.
I am no bird; and no net ensnares me: I am a free human being with an independent will.
-Charlotte Brontë, Jane Eyre
An object is independent from others if it is, in some meaningful way, outside of their area of influence. If it has some meaningful measure of self-determination. Independence is important. Nations have gone to war to obtain independence from other nations or empires. Adolescents go through rebellious periods, yearning for independence from parents or other authority figures. Though perhaps less immediately exciting, notions of independence permeate mathematics, as well. Viewed in the right light, they can even be seen as direct analogues of the more familiar notions considered above: in various mathematical structures, there is often a natural way of defining an area of influence of an element or subset of the structure. A different element is then independent of this element or subset if it is outside its area of influence. Such notions have proven to be of central importance in a wide variety of mathematical contexts. Today, in anticipation of some deeper dives in future posts, we take a brief look at a few prominent examples.
Graph Independence: Recall that a graph is a pair , where is a set of vertices and is a set of edges between these vertices. If is a vertex, then its neighborhood is the set of all vertices that are connected to in the graph, i.e., . One could naturally consider a vertex’s area of influence in the graph to consist of the vertex itself together with all of its neighbors. With this viewpoint, we can say that a vertex is independent from a subset if, for all , is not in the neighborhood of , i.e., is not in the area of influence of any element of . Similarly, we may say that a set of vertices is independent if each element of is independent from the rest of the elements of , i.e., if each is independent from .
Fun Fact: In computer science, there are a number of interesting computational problems involving independent sets in graphs. These problems are often quite difficult; for example, the maximum independent set problem, in which one is given a graph and must produce an independent set of maximum size, is known to be NP-hard.
Linear Independence: Let be a natural number, and consider the real -dimensional Euclidean space , which consists of all -tuples of real numbers. Given and in and a real number , we can define the elements and in as follows:
(In this way, becomes what is known as a vector space over ). Given a subset , the natural way to think about its lineararea of influence is as , which is equal to all -tuples which are of the form
where is a natural number, are real numbers, and are elements of .
In this way, we say that an -tuple is linearly independent from a set if is not in . A set is linearly independent if each element of is not in , i.e., if each element of is linearly independent from the set formed by removing that element from . It is a nice exercise to show that every linearly independent subset of has size at most and is maximal if and only if it has size equal to .
Fun Fact: Stay tuned until the end of the post!
Thou of an independent mind,
With soul resolv’d, with soul resign’d;
Prepar’d Power’s proudest frown to brave,
Who wilt not be, nor have a slave;
Virtue alone who dost revere,
Thy own reproach alone dost fear—
Approach this shrine, and worship here.
-Robert Burns, “Inscription for an Altar of Independence”
Algebraic Independence: If is a set of real numbers, then one can say that its algebraic area of influence (over , the set of rational numbers), is the set of all real roots of polynomial equations with coefficients in , i.e., the set of all real numbers that are solutions to equations of the form:
where is a natural number and are elements of . With this definition, a real number is algebraically independent (over ) from a set if is not the root of a polynomial equation with coefficients in . A set is algebraically independent (over ) if each element is algebraically independent from .
Fun Fact: Note that a 1-element set is algebraically independent over if and only if it is transcendental, i.e., is not the root of a polynomial with rational coefficients. and are famously both transcendental numbers, yet it is still unknown whether the 2-element set is algebraically independent over . It is not even known if is irrational!
Logical Independence: Let be a consistent set of axioms, i.e., a set of sentences from which one cannot derive a contradiction. We can say that the logical area of influence of is the set of sentences that can be proven from , together with their negations. In other words, it is the set of sentences which, if one takes the sentences in as axioms, can be proven either true or false. A sentence is then logically independent from if neither nor its negation can be proven from the sentences in .
Logical independence is naturally of great importance in the study of the foundations of mathematics. Much of modern set theory, and much of my personal mathematical research, involves statements that are independent from the Zermelo-Fraenkel Axioms with Choice (ZFC), which is a prominent set of axioms for set theory and indeed for all of mathematics. These are statements, then, that in our predominant mathematical framework can neither be proven true nor proven false. The most well-known of these is the Continuum Hypothesis (CH), which, in one of its formulations, is the statement that there are no infinite cardinalities strictly between the cardinality of the set of natural numbers and the cardinality of the set of real numbers. To prove that CH is independent from ZFC, one both produces a mathematical structure that satisfies ZFC and in which CH is true (which Kurt Gödel did in 1938) and produces a mathematical structure that satisfies ZFC and in which CH is false (which Paul Cohen did in 1963). Since Cohen’s result in 1963, a great number of natural mathematical statements have been proven to be independent from ZFC.
In our next post, we will consider a logical independence phenomenon of a somewhat simpler nature: the independence of Euclid’s parallel postulate from Euclid’s four other axioms for plane geometry, which will lead us to considerations of exotic non-Euclidean geometries.
Fun Fact: In the setting of general vector spaces, which generalize the vector spaces from the above discussion of linear independence, a basis is a linearly independent set whose span (what we referred to as its linear area of influence) is the entire vector space. A basis for is thus any linearly independent set of size . Using the Axiom of Choice, one can prove that every vector space has a basis. However, there are models of ZF (i.e., the Zermelo-Fraenkel Axioms without Choice) in which there are (infinite-dimensional) vector spaces without a basis. Thus, the statement, “Every vector space has a basis,” is logically independent from ZF.
…trying to master chess is like trying to master the infinite, and the psychological consequences can be transcendent or terrifying.
I’ve been busy traveling lately, so, in lieu of a new post, I’m just giving you a couple of literary links today.
First, a piece at The Millions about depictions of chess in literature. Chess, like the infinite, is often depicted in the popular imagination as an object of obsession, a pursuit that can lead either to transcendence or madness. This is often a little overwrought, but certainly entertaining.
A spa town in the Czech Republic, Marienbad was a favored vacation spot of European royalty and celebrities during the 19th and early 20th centuries. Some mathematicians came, too: Karl Weierstrass, Gösta Mittag-Leffler, and Sofia Kovalevskaya were all drawn by the combination of the restful atmosphere and sparkling social life that could be found at the spa.
Though its golden era ended in the early 20th century, Marienbad remained popular between the world wars. Among the visitors during that time was a young Kurt Gödel. According to some accounts, Gödel’s interest in the sciences was kindled by a teenage visit to Marienbad, during which he and his brother studied Goethe’s philosophical theory of color and found it lacking in comparison to Newton’s more strictly scientific account.
After the war we were in Marienbad quite often with my brother, and I remember that we once read Chamberlain’s biography of Goethe together. At several points, he took a special interest in Goethe’s theory of color, which also served as a source of his interest in the natural sciences. In any case, he preferred Newton’s analysis of the color spectrum to Goethe’s.
Goethe himself was a frequent visitor to Marienbad. During an 1823 trip, the 73-year-old Goethe became infatuated with the 18-year-old Baroness Ulrike von Levetzow. The pain caused by her rejection of his marriage proposal led him to write the famous Marienbad Elegy (updated in 1999 by the great W.G. Sebald).
…who dance, stroll up and down, and swim in the pool, as if this were a summer resort like Los Teques or Marienbad.
-Adolfo Bioy Casares, The Invention of Morel
In 1961, Last Year at Marienbad was released. Directed by Alain Resnais and written by Alain Robbe-Grillet, the film is as beautiful as it is inexplicable. On its face, the film is set at a resort hotel; an unnamed man (‘X’) becomes infatuated with an unnamed woman (‘A’) and attempts to convince her that they had an affair the previous year. The film unfolds in combinatorial play, with narration and scenes repeated in ever-evolving and bewildering variation.
A popular theory is that Last Year at Marienbad is actually an adaptation of Adolfo Bioy Casares’ The Invention of Morel, a novel of which our friend Jorge Luis Borges wrote, “To classify it as perfect is neither an imprecision nor a hyperbole.” I will not say much about this, so as not to spoil the book (you should go read it right now), but will only mention that Morel was, in a way, an homage to Louise Brooks, a Hollywood actress with whom Casares was somewhat obsessed and whose performance in Pandora’s Box provided a model for Delphine Seyrig’s performance as ‘A’ in Marienbad. The idea that there is a direct line from Casares and Brooks to the main characters in Morel to ‘X’ and ‘A’ in Marienbad, and that the obscurities of both novel and film are at heart simply odes to the power of cinema is an appealing one.
This theory about the connection between Marienbad and Morel was never acknowledged by the filmmakers (some say that the source text for the film is not Casares’ novel, but rather Wittgenstein’s Philosophical Investigations). Perhaps its fullest explication is given by this article in Senses of Cinema, in which the only sources given by the author are the dust jacket of a different Casares work and an Encyclopedia Britannica article which has since been removed from the online archives. Regardless of the theory’s truth, though, when one views the film through the lens of Morel, it comes tantalizingly close to making sense; the characters of the film lose their agency, consigned to repeating their roles ad infinitum.
At first sight, it seemed impossible to lose your way. At first sight…
–Last Year at Marienbad
The third main character in Marienbad is ‘M’, a man who may or may not be the husband of ‘A’. Throughout the film, we see ‘M’ playing a version of the mathematical game of Nim with ‘X’.
This version of Nim became known by the name “Marienbad” and was a brief craze in certain circles. It even got written about in Time:
Last week the Marienbad game was popping up at cocktail parties (with colored toothpicks), on commuter trains (with paper matches), in offices (paper clips) and in bars (with swizzle sticks). Only two can play, but any number can kibitz — and everyone, it seems, has a system for duplicating “X’s” talent for winning.
-“Games: Two on a Match,” Time, Mar. 23, 1962
The game is a theoretical win for the second player, although, as it is unlikely that a player will stumble upon the winning strategy by accident, ‘X’ is able to win even as the first player. We will return to general winning strategies for Nim and other games in later posts.
-The one who starts, wins.
-You must take an even number.
-You must take the smallest odd number.
-It’s a logarithmic series.
-You must switch rows as you go.
-And divide by three.
-Seven times seven is forty-nine.
-kibitzers in Last Year at Marienbad
I leave you now with Nick Cave’s exquisite “Girl in Amber.” Another secret adaptation of The Invention of Morel? Possible…
In Persian mythology, the Simurgh is a bird that lives in the mountains of Alborz. Sometimes she has the head or body of a dog, sometimes of a human. She has witnessed the destruction of the world three times. The wind of her beating wings is responsible for scattering seeds from the Tree of Life, creating all plants in the world.
The Simurgh is, in some tellings, the archetype of all birds. Her name resembles the Persian phrase si murg, meaning “thirty birds.”
In The Conference of the Birds, Farid ud-Din Attar’s 12th-century masterpiece, the birds of the world undertake a journey to find the Simurgh. And they succeed.
Their life came from that close, insistent sun
And in its vivid rays they shone as one.
There in the Simorgh’s radiant face they saw
Themselves, the Simorgh of the world – with awe
They gazed, and dared at last to comprehend
They were the Simorgh and the journey’s end.
They see the Simorgh – at themselves they stare,
And see a second Simorgh standing there;
They look at both and see the two are one,
That this is that, that this, the goal is won.
-Farid ud-Din Attar, The Conference of the Birds
The Simurgh is a bird that contains all birds. She is a universal bird.
Unsurprisingly, the Simurgh shows up a number of times in the works of Jorge Luis Borges, in both his short stories and his essays. One reference appears in the masterful story, “The Aleph,” a particularly rich and dense work which you should certainly read for yourself.
“The Aleph” is partly about how we create our own worlds, how we approximate the unknowable universe within our lives and our art. The narrator of the story, also named Borges, grieving the loss of his beloved Beatriz, pays repeated visits to the home of her father and her cousin, the poet Carlos Argentino Daneri. On one of these visits, Carlos Argentino takes Borges to his basement to show him the source of his poetry, the titular Aleph, a single point that contains the universe.
On the back part of the step, toward the right, I saw a small iridescent sphere of almost unbearable brilliance. At first I thought it was revolving; then I realised that this movement was an illusion created by the dizzying world it bounded. The Aleph’s diameter was probably little more than an inch, but all space was there, actual and undiminished. Each thing (a mirror’s face, let us say) was infinite things, since I distinctly saw it from every angle of the universe. I saw the teeming sea; I saw daybreak and nightfall; I saw the multitudes of America; I saw a silvery cobweb in the center of a black pyramid; I saw a splintered labyrinth (it was London); I saw, close up, unending eyes watching themselves in me as in a mirror; I saw all the mirrors on earth and none of them reflected me; I saw in a backyard of Soler Street the same tiles that thirty years before I’d seen in the entrance of a house in Fray Bentos; I saw bunches of grapes, snow, tobacco, lodes of metal, steam; I saw convex equatorial deserts and each one of their grains of sand; I saw a woman in Inverness whom I shall never forget; I saw her tangled hair, her tall figure, I saw the cancer in her breast; I saw a ring of baked mud in a sidewalk, where before there had been a tree; I saw a summer house in Adrogué and a copy of the first English translation of Pliny — Philemon Holland’s — and all at the same time saw each letter on each page (as a boy, I used to marvel that the letters in a closed book did not get scrambled and lost overnight); I saw a sunset in Querétaro that seemed to reflect the colour of a rose in Bengal; I saw my empty bedroom; I saw in a closet in Alkmaar a terrestrial globe between two mirrors that multiplied it endlessly; I saw horses with flowing manes on a shore of the Caspian Sea at dawn; I saw the delicate bone structure of a hand; I saw the survivors of a battle sending out picture postcards; I saw in a showcase in Mirzapur a pack of Spanish playing cards; I saw the slanting shadows of ferns on a greenhouse floor; I saw tigers, pistons, bison, tides, and armies; I saw all the ants on the planet; I saw a Persian astrolabe; I saw in the drawer of a writing table (and the handwriting made me tremble) unbelievable, obscene, detailed letters, which Beatriz had written to Carlos Argentino; I saw a monument I worshipped in the Chacarita cemetery; I saw the rotted dust and bones that had once deliciously been Beatriz Viterbo; I saw the circulation of my own dark blood; I saw the coupling of love and the modification of death; I saw the Aleph from every point and angle, and in the Aleph I saw the earth and in the earth the Aleph and in the Aleph the earth; I saw my own face and my own bowels; I saw your face; and I felt dizzy and wept, for my eyes had seen that secret and conjectured object whose name is common to all men but which no man has looked upon — the unimaginable universe.
-Jorge Luis Borges, “The Aleph”
Aleph () is of course the letter chosen by Georg Cantor to represent transfinite cardinals and the first letter of the Hebrew alphabet. It plays a special role in Kabbalah as the first letter in “Ein Sof,” roughly translated as “infinity,” and in “Elohim,” one of the names of the Hebrew god. We will surely return to these matters.
This is the first installment in a mini-series on what we will call “universal structures,” objects that contain all other objects of their type. We will continue to look at examples from literature and religion, and will delve into the existence of universal structures in mathematics, a topic which continues to drive cutting-edge research to this day. Next week, we will look at a particular universal structure in mathematics, the wonderfully named “random graph.” I hope you will join us.
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.)
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 , an -sphere is an -dimensional manifold (i.e. a space which, at every point, locally looks like -dimensional real Euclidean space) that is most easily represented, embedded in -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 , the -sphere defined as the set of points in -dimensional Euclidean space at distance 1 from the origin (i.e. the point (0,0,…,0)).
If , this is the set of real numbers whose distance from 0 is equal to 1, which is simply two points: 1 and -1.
If , this is just the set of points in the plane at a distance of 1 from . This is the circle, centered at the origin, with radius 1.
If , this is the set of points in 3-dimensional space at a distance 1 from the point . This is the surface of a ball of radius 1, and is precisely the space typically conjured by the word “sphere.”
0-, 1-, and 2-spheres are all familiar objects; beyond this, we lose some ability to visualize -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 , the non-trivial “cross-sections” of an -sphere are themselves -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 -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 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.
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,
where is a constant corresponding to the “curvature” of the universe. If , then the FLRW metric describes an infinite, “flat” Euclidean universe. If , then the metric describes an infinite, hyperbolic universe. If , 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.
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.