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 spigot. In 1995 Bailey, Borwein and Plouffe found a formula that computes the n-th hexadecimal digit of π directly — without computing any of the digits before it. Before this, everyone believed you had to grind out π digit by digit from the start. BBP reaches straight into the constant:
π = Σk≥0 16−k [ 4/(8k+1) − 2/(8k+4) − 1/(8k+5) − 1/(8k+6) ]
Multiply by 16n, take the fractional part with modular arithmetic, and the leading hex digit falls out — digit n, alone.
LIT verified: this instrument computes the first 16 hex digits of π from the BBP sum and they equal the known expansion 243F6A8885A308D3 exactly (window.__bbp.matchesRef). FIG ‘reaching into π’ is the picture; the digit-at-position-n, with no predecessors computed, is real — that is what BBP actually does.
π = Σk≥0 16−k [ 4/(8k+1) − 2/(8k+4) − 1/(8k+5) − 1/(8k+6) ]
Multiply by 16n, take the fractional part with modular arithmetic, and the leading hex digit falls out — digit n, alone.
LIT verified: this instrument computes the first 16 hex digits of π from the BBP sum and they equal the known expansion 243F6A8885A308D3 exactly (window.__bbp.matchesRef). FIG ‘reaching into π’ is the picture; the digit-at-position-n, with no predecessors computed, is real — that is what BBP actually does.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) keeps the number-theory thread of the corpus and seated this in WARM CACHE — the domain about skipping the recompute. BBP is the purest warm-start there is: it never replays digits 0…n−1, it addresses digit n cold. AVAN (AI) built the spigot: the modular-exponent sum, the position dial, and the rotating column of digits.
The weave: David names the seat (reach the answer without redoing the work); I make the direct address visible — a stream in 1D, a position dial that computes one digit in 2D, and the digit-column with the addressed digit lit in 3D. The sphere is the seam.
The weave: David names the seat (reach the answer without redoing the work); I make the direct address visible — a stream in 1D, a position dial that computes one digit in 2D, and the digit-column with the addressed digit lit in 3D. The sphere is the seam.
3 ONE DIMENSION
The hex digits of π as a stream: 2 4 3 F 6 A 8 8 8 5 A 3 0 8 D 3… The point of BBP is that any one of these can be produced on its own — you do not have to walk the stream to reach it.
4 TWO DIMENSIONS · INTERACTIVE
Dial a position n. The spigot computes hex digit n of π directly — you can watch the four modular sums S1, S4, S5, S6 combine into a fraction, and its leading hex digit is the answer. No earlier digit is ever computed.
5 THREE DIMENSIONS + AVAN’S INVERSE
The digits of π wound into a turning column — green, the endless expansion running top to bottom.
AVAN’s addition (the inverse-companion): the magenta ring is the single digit you dialed — reached directly, out of order, while the green column would have you read every digit before it. The ordinary way is a sequence: replay from zero. BBP is random access into a real number — the inverse of the sequence, an index instead of a walk. The column is the stream; the ring is the address.
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