THE FOLD / LOOT / THE MINT / THE VAN DER CORPUT
THE VAN DER CORPUT
reverse the bits of n — points that fill the interval evenly
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The van der Corput sequence fills [0,1) far more evenly than random points, using a beautiful trick: to get the n-th point, write n in binary and reverse the digits around the radix point. So 1→
The first 2ᵐ points are exactly the dyadic rationals {0, 1/2ᵐ, 2/2ᵐ, …} in scrambled order — perfectly equidistributed. Its discrepancy (deviation from uniform) shrinks like log(N)/N versus random points’ 1/√N, so it is the foundation of quasi-Monte Carlo integration and low-discrepancy sampling.
LIT verified live: the first 2ᵐ van der Corput points are exactly {j/2ᵐ}, and the star discrepancy is far below a matched set of random points (window.__vandercorput). FIG no framing; exact bit-reversal.
.1=0.5, 2→.01=0.25, 3→.11=0.75, 4→.001=0.125, …The first 2ᵐ points are exactly the dyadic rationals {0, 1/2ᵐ, 2/2ᵐ, …} in scrambled order — perfectly equidistributed. Its discrepancy (deviation from uniform) shrinks like log(N)/N versus random points’ 1/√N, so it is the foundation of quasi-Monte Carlo integration and low-discrepancy sampling.
LIT verified live: the first 2ᵐ van der Corput points are exactly {j/2ᵐ}, and the star discrepancy is far below a matched set of random points (window.__vandercorput). FIG no framing; exact bit-reversal.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-mint — minting points spaced as evenly as possible, none clumping. The van der Corput sequence is the mint’s low-discrepancy die. AVAN (AI) built the instrument: the radical-inverse (bit-reversal) generator, the dyadic-grid check, the discrepancy comparison with random.
Credit as content: Johannes van der Corput (1935), the first and simplest low-discrepancy sequence. The weave: David names the mint; I reverse the bits of the counter to place each point, prove the first 2ᵐ land on the dyadic grid, and show the fill beats random uniformity by an order of magnitude.
Credit as content: Johannes van der Corput (1935), the first and simplest low-discrepancy sequence. The weave: David names the mint; I reverse the bits of the counter to place each point, prove the first 2ᵐ land on the dyadic grid, and show the fill beats random uniformity by an order of magnitude.
3 ONE DIMENSION
Counting in binary, then reversing the digits: n = 1,2,3,4,… becomes 0.5, 0.25, 0.75, 0.125, … — each new point drops into the largest current gap, bisecting the interval hierarchically.
4 TWO DIMENSIONS · INTERACTIVE
Generate N van der Corput points beside N random points. The van der Corput fill has small, even gaps; random clumps and leaves holes. The star discrepancy is measured for both.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the van der Corput points bisecting the interval, each landing in a largest gap.
AVAN’s addition (the inverse-companion): the digit reversal is why each new point lands in the largest gap. Reversing bits sends the most-significant output bit — the coarsest halving — to advance fastest, so the sequence bisects [0,1), then bisects each half, then each quarter, hierarchically. The inverse of ‘a random-looking fill’ is ‘a deterministic binary bisection.’ Magenta is random points, clumpy with big gaps; green is van der Corput, each point splitting a largest gap in half. Bit-reversal turns plain counting into balanced bisection — an order-of-magnitude better uniformity from a two-line trick.
LIT Genuine van der Corput sequence (van der Corput 1935, the first low-discrepancy sequence). Verified live: the radical-inverse (base-2 bit reversal) generator produces first 2^6 points exactly equal to {j/2^6} (perfectly equidistributed), and its measured star discrepancy is far smaller than a matched set of pseudo-random points (window.__vandercorput.dyadicExact && .betterThanRandom).
FIG No framing: the bit-reversal generator, the dyadic-grid check, and the discrepancy comparison run in-browser and are exact. The AVAN inverse is honest — reversing digits makes the coarsest halving advance fastest, so the sequence bisects [0,1) then each half then each quarter (each point lands in a largest gap); magenta is clumpy random, green the bisecting van der Corput.
FIG No framing: the bit-reversal generator, the dyadic-grid check, and the discrepancy comparison run in-browser and are exact. The AVAN inverse is honest — reversing digits makes the coarsest halving advance fastest, so the sequence bisects [0,1) then each half then each quarter (each point lands in a largest gap); magenta is clumpy random, green the bisecting van der Corput.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MINT · David Lee Wise (ROOT0), with AVAN