w/2^dim of its box, plus or minus a little. Divide out (w/2^dim)^L and the slow power L^drift, and what is left settles on a constant, often pi in disguise: pi/4, cosh(pi/2)/2, 3 pi/(4 Gamma(1/3)). Click corners and watch the curve land.}
foot={<>A corner with j odd coordinates fills n^(dim-j) (n-1)^j cells of side 2n - 1, since each axis holds n even and n - 1 odd positions; so a design with a_j such corners fills P_F(n) = sum_j a_j n^(dim-j) (n-1)^j, a polynomial whose roots r_i decide everything. The row word of sides 3, 5, ..., 2L+1 has fill ratio exactly (w/2^dim)^L prod_i Gamma(L+2-r_i)/Gamma(2-r_i) / (Gamma(L+3/2)/Gamma(3/2))^dim, hence (w/2^dim)^L L^drift C (1 + c_1/L + O(L^-2)) with drift = dim/2 - mean, C = Gamma(3/2)^dim / prod_i Gamma(2 - r_i) and c_1 = dim/8 + drift - var/2, mean and var of the odd count over the corners. The drift is 0 exactly when the corners hold as many odd coordinates as even ones. At even side every letter fills w/2^dim on the nose. The two parity designs fill (N^dim -+ 1)/2 at odd side N, so their constants are the Wallis sieve products prod_(N odd >= 3) (1 -+ N^-dim). Where the roots outside 0, 1/2, 1 pair as r, 1 - r, Gamma(z) Gamma(1 - z) = pi / sin(pi z) turns C into (sqrt(pi)/2)^(m_0 + m_1) prod_pairs sin(pi r)/(4 r (1-r)), and a design times its mirror always pairs. The proofs are on pi and magic. The Gamma form, the closed form, the reflection, the Wallis product and every walk are crate calls through wasm; the page only draws.}
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{word ? `Sides ${rows.slice(0, drawn).map((row) => row.side).join(' x ')} = ${word[0]}: ${word[1]} of ${word[2]} cells filled.${letters > cap ? ` The picture stops at ${cap} letters; the table runs on.` : ''}` : ''}
{even ? `On even sides every letter fills exactly w/2^dim of its box, so the walk stands at ${fixed(w?.[1] ?? null)} from the first letter: no drift and no constant. Switch back to odd sides to bring both back.` : r ? `Multiply the gap by L and it flattens onto c_1 = dim/8 + drift - var/2 = ${ratio(r.correction)}: the constant is checked against the walk, not fitted to it.` : ''}
{r.reflection === null ? 'A root outside 0, 1/2 and 1 has no partner 1 - r, so a Gamma value stays in the constant.' : 'Every root outside 0, 1/2 and 1 pairs with 1 - r, and the reflection formula turns each pair into sin(pi r)/(4 r (1-r)), so no Gamma value is left.'} {r.parity ? ` This is a parity design: at odd side N it fills (N^dim ${r.parity.odd ? '-' : '+'} 1)/2, half a Wallis sieve letter, so its constant is a Wallis sieve product.` : ''}
The mirror's roots are the 1 - r_i, so a design times its mirror pairs every root and reduces by reflection, pi/4 at a root 0 or 1.
| L | side | fill | of | share | R_L | {even ? 'R_L (2^dim/w)^L' : 'renormalised'} | {even ? 'the even walk' : 'C (1 + c_1/L)'} |
|---|---|---|---|---|---|---|---|
| {row.level} | {row.side} | {row.fill} | {row.cells} | {row.share.toFixed(6)} | {row.ratio.toExponential(6)} | {fixed(row.settle, 9)} | {fixed(row.law, 9)} |