Windows onto the infinite

What’s happening. The 3-sphere lives in four dimensions, so I’ve projected it into three. Hopf showed in 1931 that it splits exactly into circles, one for each point on an ordinary sphere. The small globe in the corner is the key: each dot there corresponds to one loop here, and the white pair is highlighted so you can follow it. Loops from the same latitude lie on a shared torus, and the tori nest inside one another.

Why it matters. It was the first example of a map from a higher-dimensional sphere onto a lower one that cannot be continuously undone, and it opened a large family of problems. It also describes a qubit: a qubit’s states form this 3-sphere, and discarding the overall phase, which no experiment can detect, is exactly the Hopf map onto the Bloch sphere.

Devil’s advocate. This is a projection. In four dimensions every circle is the same size; the projection enlarges some, shrinks others, and sends one off to infinity as a straight line.

Rabbit hole. Homotopy groups of spheres: in how many different ways can one sphere wrap around another? The full table is unknown, and single entries have taken years of work. Deeper still: in 1960 Frank Adams proved that maps like this one exist only onto spheres of dimension 1, 2, 4 and 8. The same result implies that real division algebras exist only in dimensions 1, 2, 4 and 8: the real numbers, complex numbers, quaternions and octonions.

Sources

• H. Hopf, “Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche,” Mathematische Annalen 104 (1931).
• D. W. Lyons, “An elementary introduction to the Hopf fibration,” Mathematics Magazine 76 (2003).

What’s happening. Every tile here is the same size. They look smaller near the edge because distance is measured differently there, so a walk to the rim never ends. Regular heptagons, three at each corner, cannot tile a flat floor; in this plane they fit exactly.

Why it matters. Drop Euclid’s parallel postulate and this geometry follows. Bolyai and Lobachevsky found it around 1830, and it was largely ignored at the time. Thurston later conjectured that most three-dimensional spaces are hyperbolic, and Perelman’s 2003 proof of geometrization confirmed it. Escher saw a figure like this in a paper by Coxeter and used it for his Circle Limit prints.

Devil’s advocate. “Most spaces are hyperbolic” is a statement about mathematics, not about our universe, which measures as nearly flat. The geometry is also less exotic than it looks: ruffled lettuce, coral and sea slugs grow in roughly hyperbolic shapes, and Daina Taimina crochets models of it.

Rabbit hole. Mostow rigidity: for finite-volume hyperbolic spaces of dimension three or more, the topology determines the exact geometry. Deeper still: hyperbolic space has enough room to hold trees with almost no distortion. In 2017 Nickel and Kiela showed that word hierarchies embed far more faithfully in this disk than in flat space.

Sources

• H. S. M. Coxeter, “Crystal symmetry and its generalizations,” Transactions of the Royal Society of Canada (1957).
• W. P. Thurston, Three-Dimensional Geometry and Topology, Princeton University Press (1997).
• D. Taimina, Crocheting Adventures with Hyperbolic Planes, A K Peters (2009).

What’s happening. The curve traces ζ(½ + it) as t climbs from 0 to 60. Each time it passes through the center it has found a zero, and a mark lights on the rim. The first is at t ≈ 14.1347. The trace slows near the center so no zero is missed.

Why it matters. Riemann conjectured in 1859 that every nontrivial zero lies on this line. If he was right, the primes are spread as regularly as they can be, and the error in counting them stays below roughly √x log x. It is one of the Clay Institute’s million-dollar problems, and has been open for 167 years.

Devil’s advocate. More than ten trillion zeros have been checked, all on the line. That is weaker evidence than it sounds. The logarithmic integral overcounts the primes at every number anyone has tested, yet Littlewood proved in 1914 that the order flips infinitely often, the first time somewhere below about 10³¹⁶.

Rabbit hole. The zeros are spaced like the energy levels of a random quantum system, a connection Montgomery and Dyson noticed over tea in 1972. A quantum system whose energy levels are exactly the zeros would prove the hypothesis; none has been found. Deeper still: Riemann’s explicit formula, in which each zero contributes one wave, and the waves together rebuild the staircase that counts the primes exactly.

Sources

• B. Riemann, “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse” (1859).
• H. L. Montgomery, “The pair correlation of zeros of the zeta function,” Proc. Symp. Pure Math. 24 (1973).
• J. E. Littlewood, “Sur la distribution des nombres premiers,” Comptes Rendus 158 (1914).

What’s happening. A closed shape is a path that comes back to where it started. Fourier’s idea is that any such path is a sum of circles, each turning at a whole number of turns per lap, each with its own size and starting angle. The window works out those circles from 256 points along the shape, then chains them tip to tail, largest first. The last tip redraws the shape. Draw your own and it does the same with yours.

Why it matters. Joseph Fourier claimed in 1807, while working on heat, that any function can be built from waves. Lagrange objected, and it took most of a century to settle when the claim holds. It now runs much of daily life: JPEG pictures and MP3 sound keep only the largest waves, MRI scanners build images from them, and radios use them to pull one station out of many. The fast Fourier transform, published by Cooley and Tukey in 1965, made the sums quick enough for all of that. Gauss had found it in 1805 and left it unpublished.

Devil’s advocate. Fitting a shape with circles says nothing about what made it. Ptolemy’s epicycles predicted the planets well, and they had the sun going round the Earth; with enough circles, anything fits. Corners also never come out clean: tap down to a few circles and the horns of the moon ring. However many circles you add, the sum overshoots a sudden jump by about 9 percent of its height, a quirk J. Willard Gibbs explained in 1899.

Rabbit hole. The uncertainty principle is a fact about Fourier sums: a signal cannot be narrow in time and narrow in frequency at once, and in quantum mechanics a particle’s position and momentum are such a pair. Deeper still: Riemann’s explicit formula is a Fourier sum for the primes, with one wave for each zero of the zeta function.

Sources

• J. Fourier, Théorie analytique de la chaleur, Paris (1822).
• J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Mathematics of Computation 19 (1965).
• M. T. Heideman, D. H. Johnson and C. S. Burrus, “Gauss and the history of the fast Fourier transform,” IEEE ASSP Magazine 1 (1984).
• J. W. Gibbs, “Fourier’s series,” Nature 59 (1899).

What’s happening. As the view zooms out, scattered points become spirals, then 44 arms, then 710 nearly straight rays. Grant Sanderson of 3Blue1Brown popularized this plot in 2019.

Why it matters. The spirals come from π and the fractions close to it. 44 radians is almost exactly 7 turns (22/7 ≈ π), and 710 radians almost exactly 113 turns (355/113 ≈ π), so all whole numbers fall on the same arms. The primes only determine which arms stay empty, the ones sharing a factor with 44 or 710, and they fill the others in equal shares. Dirichlet proved in 1837 that every allowed arm holds infinitely many primes, and the equal shares were proved in 1896.

Devil’s advocate. Tap the window to run the control: it plots every whole number instead of only the primes. The same spirals appear, so they cannot be telling you anything about primes.

Rabbit hole. Chebyshev’s bias: primes that leave remainder 3 when divided by 4 are ahead of the remainder-1 primes most of the time, even though in the long run the two are equally common. Rubinstein and Sarnak computed the proportion in 1994: about 99.6%. Deeper still: the Ulam spiral, which Stanisław Ulam found in 1963 by writing numbers on a grid during a dull talk and marking the primes.

Sources

• 3Blue1Brown (G. Sanderson), “Why do prime numbers make these spirals?” (2019).
• P. G. L. Dirichlet, proof of primes in arithmetic progressions (1837).
• M. Rubinstein and P. Sarnak, “Chebyshev’s bias,” Experimental Mathematics 3 (1994).

What’s happening. I start 2,600 points within 0.00001 of each other in Edward Lorenz’s 1963 weather model. They move together as one white spark, then their separation grows about tenfold every 2.5 time units until they spread across the whole attractor.

Why it matters. Whether this shape is a true strange attractor or an artifact of rounding was one of Stephen Smale’s problems for the twenty-first century. Warwick Tucker settled it in 2002 with a computer-assisted proof. The same sensitivity is why weather forecasts lose their skill after about two weeks.

Devil’s advocate. Chaos does not mean anything can happen: no point ever leaves the attractor. Individual paths are unpredictable, but the shape and the statistics are stable. Every orbit drawn here is also wrong, because rounding error moves each one off its true path within seconds; the demonstration is subject to the effect it shows.

Rabbit hole. Shadowing: when does a computed orbit, full of rounding errors, stay close to some true orbit? It is the main justification for simulating chaotic systems at all, and for Lorenz’s equations the answer is subtle. Deeper still: Lorenz found the effect by accident in 1961. He restarted a run from a printout rounded to three decimal places, went for coffee, and came back to a completely different forecast.

Sources

• E. N. Lorenz, “Deterministic nonperiodic flow,” Journal of the Atmospheric Sciences 20 (1963).
• W. Tucker, “A rigorous ODE solver and Smale’s 14th problem,” Foundations of Computational Mathematics 2 (2002).

What’s happening. For each point c, start at zero and repeat z → z² + c. Points whose orbit stays bounded are black; the colors show how quickly the rest escape. The zoom reaches about 25,000 times magnification, passing small copies of the whole set on the way.

Why it matters. Douady and Hubbard proved in 1982 that the set is connected: it is all one piece. Shishikura proved in 1998 that its boundary has Hausdorff dimension 2, the most a curve in the plane can have. Whether the set is locally connected (the MLC conjecture) is still open; a proof would settle which of these systems are stable under small changes.

Devil’s advocate. The pictures are much easier than the mathematics, and the MLC question concerns scales no zoom can reach. This zoom stops where the graphics card runs out of precision; deep zooms that use arbitrary-precision arithmetic go past 10¹⁰⁰⁰ times.

Rabbit hole. How Douady and Hubbard proved it is connected: they built a conformal map from the outside of the set to the outside of a disk. Deeper still: π appears in the set. Approach the pinch point at c = −0.75 from above and count how many steps orbits take to escape; that count multiplied by your distance from the point approaches π. David Boll found this in 1991.

Sources

• A. Douady and J. H. Hubbard, “Itération des polynômes quadratiques complexes,” C. R. Acad. Sci. Paris 294 (1982).
• M. Shishikura, “The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets,” Annals of Mathematics 147 (1998).

What’s happening. Each dot is one of E₈’s 240 roots, and 6,720 lines join each root to its nearest neighbors. I project them from eight dimensions to two and rotate slowly through the other six. Four times per loop the projection lines up into 30-fold symmetry: eight rings of 30.

Why it matters. The simple Lie groups, the basic continuous symmetries, fall into four infinite families and five exceptions. E₈ is the largest exception, a 248-dimensional symmetry. Its lattice is the densest possible packing of spheres in eight dimensions, as Maryna Viazovska proved in 2016; she received a Fields Medal in 2022. In 2010 physicists reported signs of E₈ symmetry in a cobalt niobate magnet.

Devil’s advocate. In 2007 Garrett Lisi proposed fitting all of known physics into E₈, and the idea drew wide press coverage. In 2010 Distler and Garibaldi proved that E₈ cannot hold three generations of matter in the way the proposal required.

Rabbit hole. Viazovska’s proof rests on a function built from modular forms, whose Fourier transform rules out any denser packing. Modular forms are also the subject of the next window. Deeper still: in 2007 a team computed E₈’s Kazhdan–Lusztig–Vogan polynomials, about 60 gigabytes of output. The announcement said that, written out in small print, it would cover Manhattan.

Sources

• M. Viazovska, “The sphere packing problem in dimension 8,” Annals of Mathematics 185 (2017).
• J. Distler and S. Garibaldi, “There is no ‘Theory of Everything’ inside E8,” Communications in Mathematical Physics 298 (2010).
• R. Coldea et al., “Quantum criticality in an Ising chain: experimental evidence for emergent E8 symmetry,” Science 327 (2010).

What’s happening. This is the j-function, drawn with hue for direction and brightness for size; faint lines trace the tiling. j is unchanged by an infinite family of transformations, so the same pattern repeats in every curved triangle out to the rim, and the animation can slide forever without the picture changing.

Why it matters. In 1978 John McKay noticed that j’s coefficient 196884 is one more than 196883, the smallest dimension in which the Monster group acts nontrivially. The Monster is the largest sporadic simple group, with about 8 × 10⁵³ elements, and the following coefficients decompose the same way. Richard Borcherds proved the connection in 1992 using ideas from string theory, and received a Fields Medal in 1998.

Devil’s advocate. In the prime spirals, a pattern that looked deep turned out to be simple arithmetic. Here, something that looked like numerology turned out to be deep. Meanwhile e^π − π ≈ 19.9991 appears to mean nothing, and there is no reliable way to tell these cases apart in advance.

Rabbit hole. Why the Monster? Moonshine is proved but not explained. Umbral moonshine (2012) pairs other groups with other functions, and Witten has asked whether the Monster is the symmetry of a theory of three-dimensional quantum gravity. Deeper still: e^(π√163) falls short of a whole number by less than a trillionth, and the j-function explains why. Martin Gardner claimed it was exactly a whole number in his April Fools’ column of 1975.

Sources

• J. H. Conway and S. P. Norton, “Monstrous moonshine,” Bulletin of the London Mathematical Society 11 (1979).
• R. E. Borcherds, “Monstrous moonshine and monstrous Lie superalgebras,” Inventiones Mathematicae 109 (1992).

What’s happening. Four mutually touching circles obey Descartes’ rule, (k₁ + k₂ + k₃ + k₄)² = 2(k₁² + k₂² + k₃² + k₄²), where k is one over the radius. Start at −3, 5, 8 and 8 (the outer circle counts as negative) and every new curvature is a whole number. Those are the labels.

Why it matters. The curvatures in a packing like this fall into only six of the 24 possible remainders after division by 24. For twenty years the local–global conjecture held that every large enough number with an allowed remainder eventually appears, and Bourgain and Kontorovich proved that almost all of them do.

Devil’s advocate. In 2023 Haag, Kertzer, Rickards and Stange, two of them undergraduates, disproved it. This packing contains no perfect squares, even though 36, 144, 324 and infinitely many other squares pass every local test. “Almost all” left room for an infinite family of exceptions.

Rabbit hole. Thin groups: number theory on sparse subgroups is a young field, and the disproof uses quadratic and quartic reciprocity in a way that resembles the Brauer–Manin obstruction in algebraic geometry. Deeper still: the Nobel chemist Frederick Soddy rediscovered Descartes’ rule in 1936 and published it in Nature as a poem, “The Kiss Precise.”

Sources

• S. Haag, C. Kertzer, J. Rickards and K. E. Stange, “The local-global conjecture for Apollonian circle packings is false,” Annals of Mathematics 200 (2024).
• R. L. Graham, J. C. Lagarias, C. L. Mallows, A. R. Wilks and C. H. Yan, “Apollonian circle packings: number theory,” Journal of Number Theory 100 (2003).
• J. Bourgain and A. Kontorovich, “On the local-global conjecture for integral Apollonian gaskets,” Inventiones Mathematicae 196 (2014).

What’s happening. Every number up to 5,000 traces its path to 1. Drawn backward from 1, the paths share their endings and form one tree, with even steps bending one way and odd steps the other. The white path is 27, which takes 111 steps and rises to 9,232 before falling to 1. Edmund Harriss popularized this way of drawing it.

Why it matters. Every starting number below 2⁶⁸, about 3 × 10²⁰, has been checked. In 2019 Terence Tao proved that almost all orbits eventually fall below any function that tends to infinity, however slowly. That is the strongest general result so far.

Devil’s advocate. The belief rests on a probabilistic argument: on average, each odd step and the halvings that follow shrink a number by a factor of about 3/4. Averages describe typical numbers, and the conjecture is about all of them; a single orbit that grows forever, or a single cycle other than 1, 4, 2, would disprove it. Erdős said mathematics may not yet be ready for such problems.

Rabbit hole. Conway showed in 1972 that slightly generalized Collatz rules can simulate any computer program, so some questions of exactly this shape are provably undecidable. Deeper still: the problem absorbed so much attention that in the 1960s Shizuo Kakutani joked it was a plot to slow down mathematical research in the United States.

Sources

• T. Tao, “Almost all orbits of the Collatz map attain almost bounded values,” Forum of Mathematics, Pi 10 (2022).
• J. H. Conway, “Unpredictable iterations,” Proceedings of the Number Theory Conference, Boulder (1972).
• J. C. Lagarias (ed.), The Ultimate Challenge: The 3x+1 Problem, American Mathematical Society (2010).

What’s happening. A live fluid on an 80 × 80 grid, stirred by three rotating jets of dye. The readout tracks the strongest rotation anywhere in the flow.

Why it matters. The Clay problem asks whether, in three dimensions, a smooth flow can develop infinite velocity gradients in finite time. Leray proved in 1934 that solutions exist, but only in a weak sense that could hide singularities. In 2022 Chen and Hou gave a computer-assisted proof that the frictionless Euler equations can blow up inside a cylinder with walls.

Devil’s advocate. This simulation is two-dimensional, where smoothness was settled decades ago, so it bears on the Clay problem only as an illustration. My solver also damps fine detail to stay stable, so it could never show a blowup. A real fluid would also stop behaving like a continuum, and start behaving like individual molecules, long before its velocity became infinite.

Rabbit hole. Tao’s fluid computer: in 2016 he proved that an averaged version of the equations blows up, by constructing a mechanism in the flow that pushes energy into ever-smaller eddies, and suggested the real equations might allow one too. Deeper still: turbulence, which Lewis Fry Richardson summarized in 1922 in verse: “Big whorls have little whorls / That feed on their velocity, / And little whorls have lesser whorls / And so on to viscosity.”

Sources

• C. L. Fefferman, “Existence and smoothness of the Navier–Stokes equation,” Clay Mathematics Institute problem description (2000).
• J. Stam, “Stable fluids,” Proceedings of SIGGRAPH (1999).
• T. Tao, “Finite time blowup for an averaged three-dimensional Navier–Stokes equation,” Journal of the AMS 29 (2016).
• J. Chen and T. Y. Hou, “Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data” (2022).

What’s happening. This is the mechanism behind the paradox: the free group, all sequences of two moves a and b and their inverses, drawn as a tree in hyperbolic space. Coloring each word by its first move gives four pieces. Shifting the blue piece by a turns it into three quarters of the tree, and adding the gold piece completes a full copy. The other two pieces make a second copy. Each shift is a rigid motion in this geometry, so nothing is stretched.

Why it matters. Two suitably chosen rotations of a sphere generate exactly this group, so the sphere inherits the decomposition. Banach and Tarski proved the result in 1924, and five pieces is the minimum. The proof relies on the axiom of choice, which selects one point from each of uncountably many sets without specifying how.

Devil’s advocate. The pieces are not measurable, so they have no volume, and nothing physical can be duplicated this way. Some take the paradox as a reason to reject the axiom of choice, but dropping choice causes other problems, such as vector spaces with no basis.

Rabbit hole. Amenability: the paradox fails in one and two dimensions because the groups of rigid motions there are amenable, as von Neumann showed in 1929. The plane has Tarski’s circle-squaring problem instead, solved by Laczkovich in 1990. Deeper still, the standard joke: what is an anagram of Banach–Tarski? Banach–Tarski Banach–Tarski.

Sources

• S. Banach and A. Tarski, “Sur la décomposition des ensembles de points en parties respectivement congruentes,” Fundamenta Mathematicae 6 (1924).
• S. Wagon, The Banach–Tarski Paradox, Cambridge University Press (1985).
• M. Laczkovich, “Equidecomposability and discrepancy; a solution of Tarski’s circle-squaring problem,” Journal für die reine und angewandte Mathematik 404 (1990).

What’s happening. 65,536 dark matter particles start almost evenly spaced, displaced by small ripples. As space expands, gravity amplifies the ripples: matter drains from voids into sheets, and sheets collapse into filaments and knots, where galaxies form. The motion follows Zel’dovich’s 1970 approximation, which describes the first collapse well.

Why it matters. The ripples are measured. The cosmic microwave background varies by about one part in 100,000, and galaxy surveys map this web across billions of light-years.

Devil’s advocate. Dark matter has not been detected directly. The evidence is strong and comes from many independent observations, but all of it is gravitational, and some physicists prefer to modify gravity instead. The approximation also fails visibly here: once streams cross, particles pass straight through filaments that would in reality hold them.

Rabbit hole. In 2025 the DESI survey reported evidence, not yet conclusive, that dark energy is weakening over time. If it holds, the standard model of cosmology will need revision. Deeper still: some reported structures may be too large for that model. The claimed Hercules–Corona Borealis Great Wall spans about 10 billion light-years, which, if real, strains the assumption that the universe is smooth on large scales.

Sources

• Ya. B. Zel’dovich, “Gravitational instability: an approximate theory for large density perturbations,” Astronomy and Astrophysics 5 (1970).
• DESI Collaboration, second data release results on baryon acoustic oscillations and dark energy (2025).

What’s happening. Each pixel is a ray of light traced backward through the curved space around a non-spinning black hole. The flat disk appears to arch over the top and under the bottom because gravity bends its light around the hole. One side is brighter because its gas is moving toward you at up to about half the speed of light.

Why it matters. Schwarzschild found this solution in 1915, weeks after Einstein published his field equations, while serving on the Russian front. For decades it was treated as a mathematical curiosity. LIGO detected two black holes merging in 2015, and the Event Horizon Telescope imaged the shadow of the one in M87 in 2019.

Devil’s advocate. That image is reconstructed from sparse radio data collected across the planet, and some researchers argue that parts of the ring could be artifacts of the reconstruction. The team has defended it with several independent analyses. Agreement with the theory does not by itself show that the image is accurate.

Rabbit hole. The information paradox: Hawking showed that black holes evaporate, which appears to destroy information and conflict with quantum mechanics. Recent work on “islands” suggests the information does escape, though how is not understood. Deeper still: Hawking bet on the question in 1997 and conceded in 2004, paying John Preskill with a baseball encyclopedia, “from which information can be retrieved at will.”

Sources

• K. Schwarzschild, “Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie” (1916).
• LIGO Scientific and Virgo Collaborations, “Observation of gravitational waves from a binary black hole merger,” Physical Review Letters 116 (2016).
• Event Horizon Telescope Collaboration, “First M87 Event Horizon Telescope results. I,” Astrophysical Journal Letters 875 (2019).

What’s happening. Chemical V converts U into more V, and both diffuse, V more slowly. From a few random specks the mixture organizes itself into spots that divide like cells, or stripes that wind into mazes, with no template. Each setting of the Gray–Scott model changes just two numbers: how fast U is fed in and how fast V is removed.

Why it matters. Alan Turing proposed in 1952, two years before his death, that biological patterns could form this way. The idea received little attention for decades. Since then, Turing-type mechanisms have been identified in zebrafish stripes, the spacing of hair follicles and the digits of the mouse hand.

Devil’s advocate. Much of biology does not work this way: a fruit fly embryo gets its stripes from chemical gradients and genetic switches. Resembling a Turing pattern says nothing about how a pattern actually formed.

Rabbit hole. Positional information and self-organization: Lewis Wolpert’s “French flag” model and Turing’s mechanism now appear to work together in many embryos. Deeper still: the mathematician James Murray used models like this one to argue that a spotted animal can have a striped tail, but a striped animal cannot have a spotted tail. Cheetahs and genets fit the prediction.

Sources

• A. M. Turing, “The chemical basis of morphogenesis,” Philosophical Transactions of the Royal Society B 237 (1952).
• J. E. Pearson, “Complex patterns in a simple system,” Science 261 (1993).
• J. Raspopovic et al., “Digit patterning is controlled by a Bmp-Sox9-Wnt Turing network,” Science 345 (2014).

What’s happening. 800 fireflies, each flashing at its own natural pace and each nudged toward the group’s average. The ring in the middle places every firefly by where it is in its cycle, and the arrow lengthens as they align. Once the coupling passes a critical value, disorder gives way quickly to a single shared pulse.

Why it matters. Yoshiki Kuramoto solved this model in 1975 and found a sharp threshold that behaves like a phase transition. The same mathematics describes the pacemaker cells that set your heartbeat, the synchronous fireflies of Southeast Asia and the Great Smoky Mountains, and the stability of power grids.

Devil’s advocate. When London’s Millennium Bridge swayed on its opening day in 2000, crowd synchronization became the standard explanation. Later work argued that the sway can begin without any walkers in step.

Rabbit hole. Chimera states: in 2002 Kuramoto and Battogtokh found that identical oscillators with identical coupling can split into a synchronized group and an incoherent one, possibly related to animals that sleep with one half of the brain at a time. Deeper still: Christiaan Huygens discovered synchronization in 1665 while ill in bed, watching two of his pendulum clocks on a shared beam fall into step. He called it “an odd kind of sympathy.”

Sources

• Y. Kuramoto, “Self-entrainment of a population of coupled non-linear oscillators,” Lecture Notes in Physics 39 (1975).
• S. H. Strogatz, Sync: The Emerging Science of Spontaneous Order, Hyperion (2003).
• I. Belykh et al., “Foot force models of crowd dynamics on a wobbly bridge,” Science Advances 3 (2017).

What’s happening. Each pixel is one of 36,864 small magnets that tend to align with their neighbors while heat flips them at random. When hot, the pattern is noise; when cold, one direction wins. At the critical temperature there are domains within domains at every size, with no typical scale.

Why it matters. Lars Onsager solved this model exactly in 1944, placing the critical temperature at 2 / ln(1 + √2) ≈ 2.269. Kenneth Wilson’s renormalization group (Nobel Prize, 1982) then explained universality: near their critical points, magnets, fluids and alloys share the same critical exponents, because the microscopic details stop mattering.

Devil’s advocate. The two-dimensional model is solved. The three-dimensional one, which describes real magnets and fluids, has resisted an exact solution for 80 years; its exponents are known to many decimal places, but not from a formula.

Rabbit hole. The conformal bootstrap derives critical exponents from symmetry and consistency alone, without a microscopic model, and gives the most precise three-dimensional values known. Deeper still: Ernst Ising solved the one-dimensional version in 1924, found no transition, and wrongly concluded there would be none in higher dimensions either. The model is still named after him.

Sources

• L. Onsager, “Crystal statistics. I. A two-dimensional model with an order-disorder transition,” Physical Review 65 (1944).
• K. G. Wilson, “The renormalization group: critical phenomena and the Kondo problem,” Reviews of Modern Physics 47 (1975).
• S. El-Showk et al., “Solving the 3D Ising model with the conformal bootstrap,” Physical Review D 86 (2012).

What’s happening. The glowing wave is a solution of Schrödinger’s equation for one particle: brightness is probability and color is phase. It passes through both slits and interferes with itself. Each dot on the right is one detection sampled from that probability, as if identical particles arrived one at a time.

Why it matters. With single electrons (Akira Tonomura’s team, 1989) the pattern builds up dot by dot. Each electron arrives at one spot, yet together they form fringes that only a wave passing through both slits could produce. The experiment has since been done with molecules of about 2,000 atoms.

Devil’s advocate. The equation predicts the pattern exactly and says nothing about which dot will appear. The Copenhagen, many-worlds and pilot-wave interpretations give very different accounts of what is real, and all of them predict these same pictures. Physicists disagree about whether choosing among them is physics or philosophy.

Rabbit hole. Bell tests: the 2022 Nobel Prize went to Aspect, Clauser and Zeilinger for experiments showing that no local hidden-variable theory can reproduce quantum predictions. Deeper still: in 1964 Feynman said he could safely say that nobody understands quantum mechanics.

Sources

• A. Tonomura et al., “Demonstration of single-electron buildup of an interference pattern,” American Journal of Physics 57 (1989).
• Y. Y. Fein et al., “Quantum superposition of molecules beyond 25 kDa,” Nature Physics 15 (2019).
• R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics, Vol. III (1965).

What’s happening. Each coloured region is every point of the globe that is closer to its dot than to any other dot, a Voronoi diagram on a sphere. A few times a second each dot moves part of the way to the middle of its own region, and the regions even out until the largest is barely bigger than the smallest. The full tool, with your own places on a real map, the statistics and a route finder, is at Vorotool, at legibleai.org/vorotool.

Why it matters. The same division answers which hospital, school, fire station or phone mast is nearest. John Snow drew one by hand in 1855, a line round the Broad Street pump enclosing the houses that were nearer to it than to any other pump, and the cholera deaths fell inside it. Weather services use the regions, called Thiessen polygons after Alfred Thiessen’s 1911 paper, to average rainfall from scattered gauges.

Devil’s advocate. Nearest as the crow flies is rarely nearest by road, across a river or over a border, and people do not always go to the closest place. Real catchments are weighted and uneven; the plain diagram is the baseline they are measured against.

Rabbit hole. Spread the dots as evenly as possible and the regions become mostly hexagons, but never all of them: when three regions meet at every corner and the rest are hexagons, Euler’s formula demands exactly twelve pentagons, as on a football or a geodesic dome. Deeper still: the best way to place a given number of points on a sphere is the Thomson problem, posed by J. J. Thomson in 1904 for electrons repelling each other, and for most numbers of points the best arrangement is still not proved.

Sources

• G. Voronoi, “Nouvelles applications des paramètres continus à la théorie des formes quadratiques,” Journal für die reine und angewandte Mathematik 133 (1908).
• A. H. Thiessen, “Precipitation averages for large areas,” Monthly Weather Review 39 (1911).
• J. Snow, On the Mode of Communication of Cholera, 2nd edition, John Churchill (1855).
• S. P. Lloyd, “Least squares quantization in PCM,” IEEE Transactions on Information Theory 28 (1982).
• J. J. Thomson, “On the structure of the atom,” Philosophical Magazine 7 (1904).