Lemma Lounge Mathematical Society
Two thousand years of mathematics, from Ptolemy's epicycles to Ramanujan's mock theta functions
Chapter I
c. 150 CE, Alexandria
Claudius Ptolemy puts the Earth at the center of everything and asks a precise question: where will each wandering star be tomorrow night? His answer, the Almagest, is the first great mathematical model of the sky.
Circles upon circles
Now and then a planet halts and drifts backwards against the stars. Ptolemy's remedy: the planet rides a small circle, the epicycle, whose center rides a larger one, the deferent. Watch the paths loop back on themselves.
Seven wanderers, seven metals
Alchemists paired each planet with a metal and gave both one sign. The model was wrong about the cosmos, yet it predicted the sky well enough to stand for fourteen centuries.
Chapter II
1665, Woolsthorpe
Plague closes Cambridge, and a young Isaac Newton goes home to think. To find the slope of a curve at a single point, slide a second point toward it. As the gap h shrinks toward nothing, the secant settles into the tangent.
dy/dx = limh→0 [ f(x + h) − f(x) ] / h
1684, Leipzig
Gottfried Wilhelm Leibniz publishes a calculus of his own, in the notation we still use. Fill an area with rectangles and let them thin: their sum converges to the integral. His ∫ is a stretched S, for summa.
∫ab f(x) dx = limn→∞ Σ f(xi) Δx
1687, the Principia
Differentiation and integration undo each other: the rate at which area accumulates is the height of the curve. With these ideas, Newton showed that one inverse-square law of gravity produces the planets' elliptical orbits.
d/dx ∫ax f(t) dt = f(x)
Chapter III
1807, Grenoble
Joseph Fourier, studying how heat spreads through a metal bar, makes an outrageous claim: any function can be written as a sum of sines and cosines.
Epicycles, returned
Written with complex numbers, each term of a Fourier series is a circle turning at its own whole-number speed. Chain them end to end and the tip traces a curve: Ptolemy's machinery, reborn as pure mathematics.
f(t) = Σ cn eint
More circles, sharper curves
With a handful of circles the drawing is a blur; with a hundred and more, it is our λ. Making Fourier's claim precise took a century of analysis, from Dirichlet to Riemann to Lebesgue.
1782, the sphere's own harmonics
A generation before Fourier, Laplace and Legendre, working out the gravity of the planets, found the sphere's version of sines and cosines. Today spherical harmonics describe the shapes of atomic orbitals, the Earth's gravity field, and the ripples in the cosmic microwave background. Keep scrolling to morph through them.
Ylm(θ, φ) ∝ Plm(cos θ) cos mφ
1926, the hydrogen atom
Solve Schrödinger's equation for hydrogen's one electron and the spherical harmonics appear unbidden: they are the angular shape of every orbital. The quantum numbers l and m of chemistry class are Laplace's indices, and the electron's probability cloud wears the same lobes as the sphere you just saw. Keep scrolling to climb from 1s up through the d and f orbitals.
ψnlm = Rnl(r) Ylm(θ, φ)
Chapter IV
Primes, from Euclid to Ulam
Around 300 BCE, Euclid proves there is no largest prime. In 1963, Stanisław Ulam, bored during a talk, writes the integers in a spiral and circles the primes. They gather along diagonals, a pattern still not fully explained.
Elliptic curves
A line through two points of this curve meets it exactly once more. Reflect that third point across the axis and you have added the first two: the points form a group. Elliptic curves carried Andrew Wiles' 1995 proof of Fermat's Last Theorem, and today they secure much of the internet.
1920, mock theta functions
In his last letter to G. H. Hardy, months before he died, Srinivasa Ramanujan described a strange new family of q-series. What they truly were stayed hidden until 2002, when Sander Zwegers showed they are the holomorphic parts of harmonic Maass forms.
f(q) = Σn≥0 qn² / (1 + q)²(1 + q²)² ⋯ (1 + qn)²
Chapter V
c. 300 BCE, the fifth postulate
Through a point off a line, exactly one parallel passes, and every triangle's angles sum to 180°. For two thousand years mathematicians tried to prove this postulate from Euclid's other four, and failed.
1854, Riemann
On a sphere the straightest paths are great circles, and any two of them meet. Two lines both perpendicular to the equator, parallel by Euclid's lights, cross at the pole: this triangle has three right angles. Riemann's curved geometry later became the language of Einstein's gravity.
1820s to 1882, the hyperbolic plane
Gauss, Bolyai and Lobachevsky bent space the other way: through the point P run infinitely many lines that never meet L. Poincaré fit the whole infinite plane inside a disk, where every triangle of this tiling is the same size.
1915, general relativity
Einstein read Riemann and saw the heavens anew: gravity is no force, but the curvature of spacetime itself. Each mass presses a well into the fabric, and the planets, the same seven wanderers Ptolemy tracked, coast along the straightest paths the curved geometry allows. The first test was already in the sky: a stubborn 43 arcseconds per century in Mercury's orbit that Newton could not explain.
Gμν = 8πG Tμν / c⁴
The branches keep dividing: topology, logic, probability, dynamics, and whatever gets proved next.
Some of it might be proved by you.
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