THE FOLD / GRIND / WARM CACHE / THE DIRECT DIGIT
THE DIRECT DIGIT
the n-th hex digit of pi, with no predecessors
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 A genuine BBP spigot. Verified live: the instrument computes pi's first 16 hex digits from the BBP series via modular exponentiation and they equal the known expansion 243F6A8885A308D3 exactly (window.__bbp.matchesRef === true). It reaches digit n through the fractional part of 16^n*pi with no digit before n ever computed — random access into a real number, which is exactly what BBP made possible in 1995.
FIG 'Reaching into pi' is the picture; the digit-at-position-n with no predecessors is real. Doubles keep it exact for the demo's positions; the mechanism (modular sums -> fraction -> leading hex digit) is the actual algorithm, not a lookup.
FIG 'Reaching into pi' is the picture; the digit-at-position-n with no predecessors is real. Doubles keep it exact for the demo's positions; the mechanism (modular sums -> fraction -> leading hex digit) is the actual algorithm, not a lookup.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of WARM CACHE · David Lee Wise (ROOT0), with AVAN