pi.md

--- title: Pi out of the stack lead: Pi recovered from the density of coprime points in the stacked set, two disjoint ways. figure: research-pi slug: pi ---

Pi is not inside a single carpet's area, which is rational at every level. Pi is in the stack of carpets, and it comes out as a counted number rather than an assumed one.

Every claim below is tagged. Proved means proved or classical; Verified means recomputed here and reported as measured; Conjecture means supported and open. The Farey sequence page draws the stack this page counts, with phi(n) new nodes at each scale and the primes at maximal novelty.

A single carpet cannot hold pi

Take any self-similar fractal that keeps fill of the base^dim subcells at every level. Its filled fraction at level level is exactly (fill/base^dim)^level - rational at every level, and its limit is 0 or 1. The carpet keeps 8 of 9, the sponge 20 of 27, so the fractions at level level are (8/9)^level and (20/27)^level. No irrational constant can be read off a sequence of that shape. (Proved, elementary.)

The classical fractal that does give pi works differently: the Wallis sieve changes its ratio at every level, so its area is a genuine infinite product,

Product_{n >= 1} (1 - 1/(2*n + 1)^2) = pi/4

recomputed to 0.785398262 at two million factors against pi/4 = 0.785398163; the identity itself is Wallis, Proved, and the truncated product is generated by mrlynum::sieve::ratio on the odd word of two million letters, pinned at nine digits by the_plane_ratio_at_two_million_factors_rounds_to_nine_digits, so that figure is Verified. The Wallis sieve page draws the sieve level by level and reads the same product, and the cube analogue's limit 0.948815486 is Proved in closed form in the discoveries. A fixed-ratio carpet has one ratio, not a product of changing ones. That is the whole reason a hunt inside one carpet fails, and it is worth logging so nobody chases it.

What the stack lights up

Stack the grid at scales k = 1, 2, 3, .... Every lattice point (a, b) factors uniquely as g * (a/g, b/g) with g = gcd(a, b), so it belongs to exactly one layer of the stack, and the points that appear in the first layer are exactly those with gcd(a, b) = 1 - equivalently, the points visible from the corner, with no nearer lattice point on the same ray. The stack is therefore a partition of the grid into scaled copies of the lit set. (Proved, elementary.) Checked numerically as sum_{k >= 1} lit(floor(N/k)) = N^2, exact at N = 10, 100, 1000, 3000; the check has no generator in lab/, so it stands as Conjecture while the partition itself is proved. The moire demo draws those layers, one design sampled at scale 1, 3, 5 and on, stacked into a field where the interference is the finer grids landing on the coarse.

The density of those lit points is the classical coprimality density,

lit / N^2 -> 1/zeta(2) = 6/pi^2 = 0.6079271018...

(Proved, classical - Dirichlet and Mertens.) Inverting it turns the picture into an estimator:

pi = sqrt( 6 / density ) = sqrt( 6 * N^2 / lit )

The constant is classical; what the stack adds is a picture, so pi is counted out of the grid rather than imposed on it. The same visibility fact anchors the base-3 story in what base 3 hides. The primes demo reads the same grid the other way, a number being prime when its stones make one rectangle, with the sieve, the divisor pairs, the pi(x) staircase against x / ln x and li(x), and a carpet stack whose layers correlate to zero exactly at the primes. The Ulam spiral demo winds the whole numbers on squares or hexagons with the primes lit, where every straight line reads a quadratic and the prime-rich ones stand out as diagonals. The snail demo gives every cell of that winding a design tile whose side is a power of the base, so the spiral widens by that factor at each new digit, built of the primes alone or of every number by the same law.

The numbers

Lit points counted by a totient sieve as 2 * sum_{k=1..N} phi(k) - 1, which is A018805, cross-checked against a brute gcd scan at N = 10, 50, 100, 200 with exact agreement.

At N = 100000 the stack gives 3.14158..., six correct figures against pi = 3.14159265.... The rows to N = 10000 are terms of A018805, Verified; the three larger rows have no generator in lab/ and stand as Conjecture. The visible lattice points page counts the lit points of a window live and hands pi back, and the dimension counted in picks which zeta value falls out.

It is a slow estimator and an honestly noisy one. The error term in the coprime count is O(N log N), so accuracy improves only like 1/N, and because that term fluctuates arithmetically the approach is not monotone: N = 20000 is worse than N = 10000 in the table above, and N = 200000 barely improves on N = 100000 (Conjecture as a table, on the rows above). That puts it in the same family as the Wallis product and the Leibniz series - correct, convergent, not fast. The point was never speed. The the famous formulas pages walk eight of that family, one page each: the Wallis product, the Leibniz series and the Basel sum closing on pi, the harmonic sum on gamma and (1 + 1/n)^n on e, the prime count against li, Goldbach's partition count of 2n and the Mertens sum against the square root of n.

The dimension picks the zeta

The same count in dim dimensions has density 1/zeta(dim) (Proved, classical, by Mobius inversion). So the stack is a geometric generator for the whole zeta family, and the dimension you stack in decides which constant falls out. Counted by sum_{k=1..N} mu(k) * floor(N/k)^dim, cross-checked against brute gcd at N = 10, 40 for dim 2 and 3:

(Conjecture as a table: recomputed, with no generator in lab/.) Two things follow. The 3D sponge stack does not give pi: it gives Apery's constant zeta(3), which is not known to be a rational multiple of any power of pi - the same parity barrier that blocks Catalan's constant (Conjecture; no impossibility proof exists, and none is expected to be easy). And every even dimension is a fresh pi route: zeta(4) = pi^4/90 recovers pi to seven figures at N = 100000, an order better than the 2D count at the same N, on the same table.

Pi to the fourth, out of one design

Every route above runs through gcd. One more pi lives in the stack and never mentions coprimality at all. Take the design with filled corners (0,0) and (1,1) - code 9, the diagonal - and truncate its parity pattern to a side x side grid: cell (i, j) is filled when i and j have the same parity. The filled fraction rho(side) = fill/side^2 tends to 1/2, and the fluctuation around that limit is exact at every side: zero at every even side, and exactly 1/(2*side^2) at every odd side - at side = 2m the fill is 2*m^2 on the nose, while at side = 2m - 1 it is m^2 + (m-1)^2 and the numerator of rho - 1/2 collapses to 1. (Proved, elementary.) Weight the fluctuations into a Dirichlet series and only the odd terms survive, each equal to 1/(2*side^(s+2)) - the fluctuation times the weight, two different things:

Z(s) = Sum_{side >= 1} (rho(side) - 1/2) / side^s = (1/2) * lambda(s+2),  Re(s) > -1

with lambda(s) = (1 - 2^(-s)) * zeta(s) the Dirichlet lambda function. At s = 2 and s = 4:

Z(2) = pi^4/192 = 0.50733901580...
Z(4) = pi^6/1920 = 0.50072353832...

Proved; lab/py/gaussian-zeta checks the fluctuation exact in rational arithmetic to side = 80, the reductions 1/192 and 1/1920 exact, both constants matched at 90 digits with two independent pi routines and two independent lambda routes agreeing to 1.3e-82 and 5.2e-87, and the series summed straight off counted fills with no closed form anywhere - gap 8.3e-11 at side <= 999, the size of the neglected tail.

One caution is load-bearing: the constant belongs to the corner set, not the symmetry class. The orbit mate - corners (0,1) and (1,0), code 6 - has fluctuation exactly -1/(2*side^2) on odd side, so its Z(2) is -pi^4/192: same shape under cube symmetry, opposite sign under truncation to side cells, the fixed-side caveat the core already records. (Verified, side = 1..40, lab/py/gaussian-zeta.) And the machine is genuinely different from the stack above - a Dirichlet series over one design's own fill fluctuation across truncations, with no visibility and no Mobius anywhere in it. A second, disjoint way for the grid to know pi.

Where the fractal versions stand

Restricting the count from the full grid to the cells of a fractal keeps the pi but changes the constant. The box-bound theorem in the coprimality spine, Theorem (above dimension one) under THE MASTER DENSITY, proves the density for every design with dim >= 2, condition (E) and fill > base, so the row below is a theorem and the conjectural tail of the family is only its fill <= base members. Each one is a rational multiple of 6/pi^2, the multiplier being the design's own bracket against the base factor, exactly as the coprimality spine derives it (Proved, exact arithmetic). The one member of the family whose sequence reached the OEIS is the only one tabulated here; the rest are counted by that page and deliberately not given their constants anywhere in this tree, because their novelty is still live.

A coincidence to name before a reader mistakes it for support. The first row's 16/(3*Pi^2) has a namesake in the other direction: 1 - 16/(3*Pi^2) = 0.4596204 is OEIS A395134, an 1891 geometric-probability constant of Zerr, and the A396934 entry carries the crossref. That is a numeric coincidence and not evidence for this row. The two constants are complements of one another by arithmetic alone - both recomputed by lab/py/half-ball-mismatch, 16/(3*Pi^2) = 0.5403796460924681 and 1 - 16/(3*Pi^2) = 0.4596203539075319 - and Zerr's problem is not this lane's problem. The row is a theorem by the box-bound theorem and gains nothing from the crossref. A395134 at source, Verified: name "Decimal expansion of the probability that the line that passes through two points selected independently and uniformly at random in a half-disk intersects the arc at two points", formula "Equals 1 - 16/(3*Pi^2)", digits 4, 5, 9, 6, 2, 0, 3, 5, 3, 9, ..., and the Zerr attribution is a link rather than a formula line - "George B. McClellan Zerr, Solution to Problem 11134, Mathematical questions and solutions from the 'Educational Times', Vol. 55 (1891), p. 161".

The coincidence has a mechanism and a fence. The reading above is unchanged - it is still a coincidence and still not evidence for this row - but the two 16/(3*Pi^2) are now known to draw their Pi^2 from different places, and no other dimension can repeat the collision. Zerr's side decomposes. Let H be the upper unit half-disk, |H| = Pi/2. The Blaschke-Petkantschin formula for two points gives P(the line through them crosses the diameter) = I_diam / (3*Area(H)^2), where I_diam is the chord-cube integral over lines meeting the flat face. Parameterising those lines by crossing point a and angle, and substituting u = cos(phi), the inner integral is Integral_{-1}^{1} (-a*u + sqrt(1 - a^2 + a^2*u^2))^3 du, and that integral is the constant 2 for every a in [-1,1]: the odd terms of the expanded cube vanish, and the even part is the exact derivative d/du [ u*R(u)^(3/2) ] with R(u) = 1 - a^2 + a^2*u^2, which evaluates at the endpoints by R(+-1) = 1. Hence I_diam = 4, a pure integer, and P = 4/(3*Pi^2/4) = 16/(3*Pi^2). Proved, elementary throughout - parity, one binomial expansion, the product rule, the fundamental theorem of calculus - with a second proof by differentiation under the integral sign in a and a 50-digit check at 50 values of a; lab/py/half-ball-mismatch regenerates the three symbolic residuals, exactly zero, and the 50-digit check. So Zerr's Pi^2 is Area(half-disk)^2 = Pi^2/4 and nothing else; the design's Pi^2 is zeta(2) = Pi^2/6 and nothing else, (8/9)/zeta(2) = (8/9)*(6/Pi^2). One number, two unrelated mechanisms.

The mismatch theorem. The correspondence cannot generalise, and that is the useful half. A design's density is delta = B(F)*(1/zeta(dim))*Prod_{p|base}(1 - p^(-dim))^(-1), for F the filled digits and B the bracket above, a rational multiple of 1/zeta(dim): for even dim that is rational/Pi^dim, for odd dim it is a rational multiple of an irrational zeta(dim) not known to be algebraic over Q(Pi). The half-ball probabilities have their Pi pinned low, d being the half-ball's own dimension - Version L gives rational/Pi^2 at even d and a pure rational at odd d, Version H gives Q + Q/Pi^2 at even d and Q + Q*Pi at odd d. At even dim >= 4, equating rational/Pi^dim with any Q-linear combination of {1, 1/Pi^2} makes Pi algebraic, contradicting Lindemann. Proved. At dim 3 against Version L, rational/zeta(3) = 3/8 forces zeta(3) rational, contradicting Apery. Proved. At dim 3 against Version H, rational/zeta(3) = a + b*Pi would make zeta(3) algebraic over Q(Pi), which no theorem forbids and every standing conjecture does. Conjecture-conditional, and it is the one gap in the theorem. At odd dim >= 5 the argument is the dim 3 one with zeta(dim) in place of zeta(3), so it is conditional on zeta(dim) irrational, which Rivoal and Zudilin give only for infinitely many odd dim and not for each. At dim 2 with d = 2 both sides live in Q(1/Pi^2) and a match is possible; the gasket's is the only one found, checked over all base-2 and base-3 designs at dim 2 against every Version L value to d = 24. Verified by lab/py/half-ball-mismatch: 11 base-2 and 502 base-3 designs, base-2 numerators 4, 16/3, 6, 8, exactly one match, 16/3 at d = 2, carried by 3 designs. The corollary worth carrying: hunting for a Euclidean body whose chord probability reproduces some other design's density is provably futile above dim 2, so that search is closed rather than merely unfinished.

Version L is the classical family and its parity law follows from Wallis recurrences, f(2) = 16/(3*Pi^2) with f(2k+2)/f(2k) = 4k(k+1)/((2k+1)(2k+3)), and f(3) = 3/8 with f(2k+3)/f(2k+1) = (2k+1)(2k+3)/((2k+2)(2k+4)); it is Verified in exact rationals to d = 11 by lab/py/half-ball-mismatch, with an independent 10^7-sample random-point check at d = 2..7. Version H is new here and is Verified at d = 2..7, derived by an unoriented-normal Blaschke-Petkantschin reduction and integrated in closed form by lab/py/half-ball-mismatch, cross-checked by 60- and 80-digit quadrature agreeing to 2.3e-62, 7.2e-64 and 1.5e-63 at d = 3, 4, 5, and by an independent 10^8-sample random-point estimate whose deviations 2.39e-6, -6.74e-6, -3.55e-6 sit inside one sigma of 3.96e-5, 2.6e-5, 1.54e-5. The closed forms at d = 6 and d = 7 are 16 - 10363195833496113250304/(65656392092180764875*Pi^2) = 0.0074784083 and 64 - 403492347953923610203877211975*Pi/19807040628566084398385987584 = 0.0021206659, so the Version H parity law rests on six terms; as a law for every d it is Conjecture.

coprime.md upgrades the restricted densities, as above. The clean result is the unrestricted one: on the full stack the density is a theorem, and pi drops out of it by counting.

A hunt for pi in one carpet's area finds a rational. Pi is one level up, in the agreement between scales - and agreement between scales is coprimality.