Coordinate geometry — flashcards

Every continuous map f : Sⁿ → ℝⁿ sends some antipodal pair to the same point: f(p) = f(−p) for some p. The n in both places is the content — a phrasing that drops either one is a different, false claim.

A necklace with k types of jewels, each present in an even number of beads, can be fairly split between two thieves using at most k cuts — and k is tight: arranged as k contiguous blocks, one per type, every block needs an internal cut, so 'at most k' cannot be improved in general.

The Circular Unitary Ensemble (CUE — Haar-random N×N unitary matrices, eigenvalues on the unit circle, joint density ∝ ∏_{j<k}|e^{iθⱼ}−e^{iθₖ}|²) is not an analogy for the deterministic evenly-spaced-points construction used elsewhere on this map (st/wallis-product-three-routes, st/chord-rational-classification) — it is that construction's finite-temperature (β=2) Coulomb-gas generalization, with the deterministic case as its zero-temperature limit. The correspondence to zeta zeros is two separate claims of different strength, chained together: zeta zeros ↔ GUE (the 'GUE hypothesis' — conjectural, though Montgomery proved a restricted form of it, with Odlyzko's numerical verification near height 10^20 as its evidential support) and GUE ↔ CUE (a proven theorem: both converge to the same bulk-scaling-limit determinantal sine-kernel process). The original arc-vs-chord question this resolves: the CUE correlation kernel's denominator, sin((θ−φ)/2), is literally the Chord function already used throughout the Wallis-video material; the GUE-on-a-line kernel is the sine kernel sin(πx)/(πx); the two coincide in the bulk limit because chord and arc agree to leading order as the angular window shrinks — curvature is a finite-N correction, not a persistent distinction. The Keating–Snaith conjecture (moments of |ζ(1/2+it)| match moments of |det(I−U)| for Haar-random U(N)) is the natural next question once the kernels are known to match.

The reciprocal-square identity π²/sin²(πx) = ∑_{n=−∞}^{∞} 1/(x−n)², obtained by Wästlund's lighthouse argument: equal-brightness sources at every integer, brightness ∝ 1/distance², a circle-doubling construction using the inverse-Pythagorean identity 1/a²+1/b²=1/h² at a right angle, iterated to infinity. x = 1/2 gives ∑ 1/(n−1/2)² = π², which is 8 times the sum of reciprocal squares over the positive odd integers, so that sum is π²/8; since ζ(2) = (odd sum) + ¼ζ(2), this forces ζ(2) = π²/6 — Euler's destination, reached by an elementary route with no complex analysis.

The Wallis product 2/1·2/3·4/3·4/5·⋯ = π/2 has at least three independent derivations that manage error differently, not three restatements of one argument. (1) Analytic: it is the x=1/2 evaluation of Euler's undifferentiated sine product sin(πx) = πx·∏(1−x²/n²) (st/basel-wallis-same-product) — a substitution into an already-established product, with no limiting process of its own beyond the product's own convergence. (2) Geometric/algebraic: N evenly-spaced points on a circle as the roots of x^N−1, giving two exact finite-N lemmas (observer halfway between two points: distance product = 2; observer replacing one point: distance product = N) whose ratio N·(distance)/2 is an *exact* invariant at every finite N — a single limit, taken once at the very end as N→∞, converts that invariant into π/2 via chord-length-to-arc-length convergence. (3) Classical calculus: Wallis's own route via the reduction formula ∫₀^(π/2) sin^n θ dθ, giving the asymptotic (2n)!!/(2n−1)!! ∼ √(πn) — which stacks two separate approximations (a pointwise limit inside the reduction, and an asymptotic equivalence) that must cancel jointly, rather than carrying one exact quantity to a single final limit. Routes (2) and (3) both pass through the same trigonometric object — a quarter-circle arc — entered through different doors: chord geometry directly, versus a sine substitution turning an algebraic integral into an arc-length integral.

Chord(f) = 2sin(fπ) reduces the question 'which rational f give a rational chord length?' to Niven's theorem: rational values of sin(θ) at rational-multiple-of-π angles θ are confined to {0, ±1/2, ±1}. Translated back, Chord(f) is rational for rational f∈[0,1] only at f=0, 1/6, 1/2, 5/6, 1 — f=1/2 (Wallis' own case) is not the only one, contrary to the natural first guess. At f=1/6 (Chord=1), the same product identity underlying st/wallis-product-three-routes' route (2), ∏_{k≠0} k/(k−f) = f·2π/Chord(f), generalizes cleanly to a second explicit Wallis-type product converging to π/3.

Asking what an irrational f means for the Chord(f) construction splits into two genuinely different questions, not one. (1) The value: for algebraic irrational f, the Gelfond–Schneider theorem characterizes Chord(f)=2sin(fπ) as transcendental. (2) The configuration: N points at an irrational rotation α (rather than N-th roots of unity) obey the three-distance theorem (Steinhaus) — at most three distinct gap lengths between consecutive points on the circle at any N, collapsing to exactly two precisely at the continued-fraction convergents of α. Verified directly for the golden ratio (whose continued fraction is all 1s): exactly two gap lengths at each Fibonacci count N=3,5,8,13,21,34.

Karol Borsuk, conjectured by Stanisław Ulam, 1933.

Charles Goldberg and Douglas West, 1985.

Johan Wästlund, 2010 (Wästlund's paper); popularized 2018 by 3Blue1Brown.

Hausdorff dimension takes values in a continuum and integer values are the exceptional slice. The familiar case is the degenerate one, which is the cleanest instance of this schema on the map.

Euclidean geometry is the curvature-zero member of a family. It looks canonical because it is flat, and flat is a constraint rather than an absence of one.

Affine algebraic sets ↔ radical ideals in a polynomial ring. Geometry of solutions ↔ algebra of the equations defining them.

The origin of the schema. Accept the fifth postulate or deny it; both worlds are consistent, and neither is the true one. This became the tool you reached for four more times, each at a higher level.

Euclidean geometry recast with vectors and inner products, following Descartes' translation into algebra.