THE FOLD / CHEAT / NOCLIP / THE BBP
THE BBP
the n-th hex digit of pi, without the digits before it
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The BBP formula (Bailey–Borwein–Plouffe) computes the n-th hexadecimal digit of π without computing any of the digits before it. π = Σk≥0 16−k[ 4/(8k+1) − 2/(8k+4) − 1/(8k+5) − 1/(8k+6) ], and multiplying by 16n and taking the fractional part isolates one digit — the key being that 16n−k mod (8k+j) can be found by fast modular exponentiation, so no giant number is ever built.
It shattered the belief that you must compute all earlier digits first: π becomes random-access.
LIT verified live: BBP’s hex digits for n=0…23 match the reference hex expansion of π (243F6A8885A308D313198A2E) exactly (window.__bbp). FIG no framing; exact digit extraction.
It shattered the belief that you must compute all earlier digits first: π becomes random-access.
LIT verified live: BBP’s hex digits for n=0…23 match the reference hex expansion of π (243F6A8885A308D313198A2E) exactly (window.__bbp). FIG no framing; exact digit extraction.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at noclip — clip straight through to the digit you want, passing through all the digits between without touching them. BBP is exactly that no-clip into π. AVAN (AI) built the instrument: the modular-exponentiation series, the fractional-part extraction, the reference cross-check.
Credit as content: David Bailey, Peter Borwein & Simon Plouffe (1995). The weave: David names the no-clip; I compute one digit deep inside π by modular arithmetic and confirm a run of them against π’s known hexadecimal digits.
Credit as content: David Bailey, Peter Borwein & Simon Plouffe (1995). The weave: David names the no-clip; I compute one digit deep inside π by modular arithmetic and confirm a run of them against π’s known hexadecimal digits.
3 ONE DIMENSION
The hexadecimal digits of π after the point. BBP can jump to any position and return that digit alone — the others are never computed.
4 TWO DIMENSIONS · INTERACTIVE
Pick a position n; BBP returns the n-th hex digit of π directly. A reference string confirms it.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a single hex digit, plucked from deep inside π.
AVAN’s addition (the inverse-companion): you can extract the n-th digit without the previous n−1. The formula isolates one digit through modular arithmetic (16n−k mod (8k+j)), so a digit’s position becomes an address — random access into an irrational. The inverse of ‘compute all digits up to n’ is ‘compute only digit n.’ Magenta is the digits skipped; green is the one digit addressed. π stops being a stream you must read from the start and becomes a table you can index. (It works in base 16 and 2, not base 10.)
LIT Genuine BBP formula (Bailey, Borwein & Plouffe 1995). Verified live: the modular-arithmetic digit extraction reproduces pi's hexadecimal digits at positions n=0..23 exactly against a reference string (window.__bbp.matchesRef); digits 243f6a8885a308d313198a2e.
FIG No framing: the series, the fractional-part extraction, and the reference cross-check run in-browser and agree exactly. The AVAN inverse is honest — a digit's position becomes an address via modular arithmetic, so digit n is computed without digits 0..n-1; magenta is the skipped digits, green the one addressed. Works in base 16/2, not base 10.
FIG No framing: the series, the fractional-part extraction, and the reference cross-check run in-browser and agree exactly. The AVAN inverse is honest — a digit's position becomes an address via modular arithmetic, so digit n is computed without digits 0..n-1; magenta is the skipped digits, green the one addressed. Works in base 16/2, not base 10.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NOCLIP · David Lee Wise (ROOT0), with AVAN