π LAB

HOW TO USE THE INSTRUMENT

WHAT THIS IS

π Rider asks you to remember the digits of π. The lab asks a different question: what happens if you do arithmetic to them?

You build a short pipeline of operations — add the index, multiply by π, take a sine, square it, feed the previous answer back in — and it runs over every digit of π in turn. The results fly past exactly like a ride, with π's untouched digits shown alongside as a dim reference lane, so the size and shape of the change is visible rather than inferred.

Most of what you can build here is mathematically meaningless, and that is fine. The instrument is for looking. Some transforms preserve π's famous flatness, some tilt it, some collapse it into a handful of values, and some produce structure that is entirely an artefact of the operation rather than anything about π. Telling those apart by eye is the whole exercise.

QUICK START

Press ▶ PLAY in the transport bar, or tap SPACE. Digits start streaming toward the marker on the left.
Click a preset in the TRANSFORM panel — try × π. The bright lane jumps somewhere new; the dim lane does not move, because that is π itself.
Click + OP to add a row, pick an operation and an operand. Rows run top to bottom.
Type in a row's number box while it plays. Everything on screen re-computes as you type — that is the fastest way to find out what a parameter does.
Switch BASE to 2, 16 or 64. π is genuinely re-expanded in that base, and the value axis relabels itself.
Watch DISTRIBUTION on the right. π's raw digits sit almost perfectly on the dashed even-spread line; a transform that departs from it is doing something structural.

READING THE SCREEN

Bright ring — the transformed value. Its height is the result and its color is the value; both come from the same number, so neither is the only cue. This lane is the answer.

Dim ring — π's original digit at that index, sitting slightly further away. It never changes no matter what you build. It is the control.

Stem — the change itself. The line joining the two rings. Short stems mean a gentle transform, long ones a violent one, and a run of equal stems means the transform is doing the same thing to every digit regardless of its value.

Playhead. The standing vertical bar. Digits arrive from the right and cross it; the READOUT panel always describes the digit at the bar.

Probe. The diamond riding just in front of the playhead. It chases the current value, so it traces the transform's motion as a continuous line rather than as separate points.

Gap marker. A red diamond on the dead rail below the axis, where a value stopped being a real number. See GAPS.

Graticule and axis. One line per value, labelled down the left edge in the current base. Below the lanes, the receding ruler counts digit indices.

GRID VIEW

The buttons at the left of the transport bar switch between five ways of looking at the same numbers, and V steps through them from the keyboard. The ride shows you a few dozen values going past in detail. The grid shows you tens of thousands at once: one coloured cell per value, filling a row and then wrapping onto the next, exactly like text on a page.

Everything else keeps working underneath. The chain, the base and the transport are shared, playback still runs and still makes sound, and the READOUT and DISTRIBUTION panels still describe the playhead — which the grid marks with a white box. Editing the chain repaints the whole picture.

The reason to do this is that periodicity becomes geometry. A transform that repeats every n values draws a vertical stripe when the row is a multiple of n wide, and a diagonal when it is not — so sliding WIDTH until the stripes stand up straight measures the period by eye. Structure that reads as noise going past one value at a time is often obvious as a texture.

WIDTH: Cells per row, 2 to 256. The cell width is whatever makes them span the frame, so this genuinely re-wraps the sequence rather than zooming it. This is the control that finds patterns; the rest change how they look.
ROW H: Row height in pixels, and so how many rows are on screen at once. At 1 a screen holds several hundred rows and the picture is a whole territory; at 40 it is a handful of rows read like a strip of film.
GAP: Space around each cell, as a share of its size. Zero is a solid image; anything else separates the cells into a mosaic.
STRIDE: Show every n-th value only. Combined with WIDTH this samples the sequence on a lattice, which brings out slow structure that a dense picture buries.
SKEW: Extra cells each row advances by. 0 wraps exactly at the width; anything else slides every row sideways, tilting stripes into diagonals. Negative values make rows overlap, so the same values reappear shifted — the fastest way to make the picture interfere with itself.
CONTRAST: How hard the palette is pushed away from its midpoint. Below 1 everything converges on the middle colour; above it, small differences separate and the picture goes graphic.
GLOW: A bloom over the cells, the same idea as the ride's. Purely cosmetic, and worth turning off when you are reading fine detail.
SHAPE: square tiles solidly; dot and diamond leave the background showing; bar makes each cell's height the value too, so the picture stays readable even in mono. Cells only a few pixels across are always drawn square — a circle inside three pixels is a worse square.
COLOR: digit is the ride's own palette, so a base-10 picture is coloured like the game. spectrum separates neighbouring values hardest, heat and ice read as brightness, mono strips colour out entirely, and duo is two-sided — use it with a MAP that is centred on zero.
MAP: What the colour means. value is the transformed result. value − digit is the change the chain made, so an identity chain goes flat. step to step is the difference from the previous cell, which erases level and shows only movement. π's digit ignores your chain and paints π itself — the control picture.
SNAKE: Reverse every other row, so the sequence turns back on itself instead of jumping to the left margin. Values that are neighbours in the sequence stay neighbours on the page, which removes the seam down the left edge.
RULER: The index each row starts at, down the left edge.

Moving around. Scroll with the wheel, drag the picture, drag the bar on the right, or use PAGE UP / PAGE DOWN / HOME / END. FOLLOW keeps the playing row on screen; touching the scroll yourself turns it off, because you are looking at something. Click any cell to send the playhead to that digit. SAVE PNG writes the picture out as it stands, without the playhead box.

Three that are worth setting up exactly:

× i at WIDTH 20 Two solid columns appear and everything else stays noise. They are the positions where the index is a multiple of ten, because a multiple of ten times anything ends in zero. The picture has found a fact about arithmetic, not about π.

+ i with MAP set to value − digit The value view is noise — a random digit plus a shift is still random. The change view is not: it is exactly the index modulo ten. At WIDTH 10 it draws ten flat colour columns, and at 64 it shears into diagonals. Same data, and only one of the two lenses can see it.

An expression with no d in it Try (i * 3) % base. Nothing about π survives, so the picture is pure lattice — a weave whose angle is set entirely by WIDTH. Useful as a calibration: that is what total structure looks like, and π's own digits look nothing like it.

The grid draws only the digits converted so far, so a fresh base fills in from the top while the conversion streams. Nothing is hidden — the picture is simply as deep as the data.

SPIRAL VIEWS

SPIRAL and SQUARE are the same cells as the grid, threaded onto a curve that winds out from the centre instead of wrapping into rows. Index 0 is at the middle and π runs outward from there, so how far into π a digit is becomes how far out it sits. V steps through all five views, and RINGS below is a third layout on the same plan.

Unlike the grid these are plans you move around on, not pages you scroll: drag to pan, wheel to zoom, or use +, − and 0. A spiral has no rows to page through, and the thing worth looking at is nearly always either the very centre or one arm a long way out.

The round spiral has two layouts, and they are genuinely different instruments.

packed: The radius grows as √n, so every digit gets the same amount of room and the picture is one evenly filled disc. At the default PER TURN of 1.618 — the golden angle — no two digits ever line up, which is why all 100,001 of them fit into a single flower with nothing wasted. This is the view that shows the most at once.
turns: The radius grows steadily with the index instead, so the number of digits per revolution stays fixed. Every digit that is PER TURN apart lands at the same angle, so a period draws a straight ray. Far fewer digits fit on screen — that is the trade.

PER TURN: Digits per full revolution, whole part. An integer n draws exactly n rays, in either layout. This is the control that finds periods, and it is the direct equivalent of the grid's WIDTH.
FINE: The fraction of a digit to add to it, to a thousandth. It has its own slider because the whole game is landing on 10.000 exactly and then detuning it: at +0.000 the rays stand still, and a nudge winds them into slow spirals whose lean measures how far off the period you are.
SPACING: How far apart neighbouring digits sit — between turns in turns, between adjacent dots in packed. In world units, so zoom is separate from it.
DOT: Mark size as a share of SPACING, so changing the spacing does not turn the picture into either a field of gaps or one solid blob.
ROTATE: Start angle. Turns the rays without turning the frame — useful for standing one of them straight up before you save a picture.

The square spiral is the Ulam walk: right, down, left, left, up, up, and on around, one cell per digit. It is the grid's mosaic wound up rather than wrapped.

CELL: Cell pitch in world units, and GAP insets each cell inside it, exactly as in the grid.
GROW: Sides drawn before the arm gets longer. 2 is the true Ulam walk and tiles the plane solidly, with no cell used twice and none left out. Anything else leaves the coil open.
PITCH: How many cells the arm grows by each time. +1 tiles solidly; wider values leave a gap between coils, so the picture becomes a nest of separate square rings and the same digits need several times the area.

SPIN reverses the winding of either spiral, and STRIDE, CONTRAST, GLOW, SHAPE, COLOR and MAP mean exactly what they mean in GRID VIEW. FIT frames the whole thing again, FOLLOW recentres on the playhead when it winds off screen, clicking a cell sends the playhead there, and SAVE PNG writes the picture out without the playhead ring.

Three that are worth setting up exactly:

LAYOUT turns, PER TURN 10, then the × i preset One ray — the multiples of ten — goes flat blue while the rest stay noise, because ten times anything ends in zero. Look closer and the rays are graded: the ray at index r carries exactly 10 / gcd(r, 10) different colours, so the evens are half as varied as the odds and the ray at 5 has only two. The picture is reading arithmetic, not π.

…then drag FINE off zero The rays stop being rays. They lean into slow spirals whose curl is the difference between your PER TURN and the period actually in the data — the same measurement the grid makes by tilting stripes into diagonals, done by eye at a thousandth of a digit.

An expression that drifts slowly, like (i / 3000) % base Both spirals draw concentric rings that crowd together toward the edge. That is not π and not the transform: distance from the centre goes as the square root of the index in the packed disc and in the square walk alike, so each ring out holds more digits than the last. Worth seeing once, so you recognise it when it turns up under something you thought you had found.

At the fitted zoom the packed disc puts about two pixels between neighbouring digits, and the concentric rings and little whorls you can see in it are the packing interfering with the pixel grid — they come out the same for π and for a pure lattice. The colours are the data and differ everywhere; the rings are the layout. Zoom in until each digit is several pixels across before believing any large shape you find here.

RINGS VIEW

RINGS is the third layout on the same pannable plan, and the only one that is not a single curve. The digits are dealt into closed concentric circles: the innermost ring takes FIRST of them, and every ring after it takes GROW more than the ring inside it. With the defaults that is one digit in the middle, then two, then three, then four — a target rather than a coil.

The consequence worth holding onto is that the number of digits per revolution changes as you go out. In the round spiral it is either fixed or grows on its own terms; here you set it directly, and the whole character of the picture is that one choice.

FIRST: How many digits the innermost ring holds. At 1 it is a single dot at the origin. Above that the first ring is pushed out far enough that its own digits sit one SPACING apart, because a ring of several digits cannot be at the centre — they would all be the same point.
GROW: Digits added to each ring over the one inside it. This is the control that decides the density: a ring at k spacings out has a circumference of 2π·k spacings to spend, so +6.28 is the value that keeps neighbours along a ring about as far apart as the rings are from each other. Below that the rings are sparse and read as rings; above it they crowd into solid bands. It is deliberately fractional — whole numbers make the ring counts an arithmetic run, and the fractions between them drift.
SPACING: Distance between one ring and the next, in world units, so zoom stays independent of it.
DOT: Digit size as a share of SPACING — the same bargain as the spiral's DOT, so opening the rings up does not leave the picture as a field of specks.
TWIST: Slots of rotation added per ring. Counted in slots rather than degrees so it means the same thing on every ring however many digits it holds: at +0.50 each ring sits half a digit round from the one inside it, which is the offset that stops the sparse rings lining up into spokes. Anything else winds the whole stack into a slow twist.
ROTATE: Start angle for the whole picture.

SPIN, STRIDE, CONTRAST, GLOW, SHAPE, COLOR and MAP are the same controls as everywhere else, and so are FIT, FOLLOW, SAVE PNG, click-to-seek and drag/wheel to move about. The panel header counts the rings the layout came to.

Two that are worth setting up exactly:

GROW +0.00, FIRST 10, then the × i preset With nothing added per ring every ring holds the same ten digits, so a digit's angle is fixed by i mod 10 and the rings become the turns spiral cut into closed circles. The rays come back exactly as they do there — one flat ray at the multiples of ten — and this is the setting to reach for when you want to read a period off the angle without the coil sliding it round.

GROW +6.30 and DOT ×1.00 The rings stop being visible as rings. Every digit gets about the same amount of room whatever ring it is on, and the picture fills into one even disc — the same coverage the packed spiral reaches by a different route. Slide GROW down from there and watch the disc open back into circles.

The rings share the packed disc's warning. Once GROW is above zero the digits after K rings go up as K squared, so distance from the centre grows as the square root of the index — a slow drift in the data draws rings that crowd together toward the edge here for the same reason it does in both spirals, and that is the layout speaking, not π.

THE TRANSFORM CHAIN

A value starts as the digit and passes through each enabled row in order. Row 1 gets the raw digit; row 2 gets whatever row 1 produced, and so on. Order matters — + 1 then × 2 is not × 2 then + 1.

● / ○: Enable or bypass the row. Bypassed rows keep their settings and stay in place, so you can A/B a single step without rebuilding it.
↑ ↓: Move the row through the pipeline. Reordering is usually a bigger change than editing.
✕: Delete the row.
+ OP: Add an operation row: pick an operation and, for the two-sided ones, what to apply it with.
+ EXPR: Add a free-form expression row. See EXPRESSIONS.
CLEAR: Empty the chain. The lanes then sit on top of each other, which is a useful way to confirm you are looking at π and not at your own arithmetic.

The presets across the top of the panel replace the whole chain. They are starting points, not modes — pick one and then edit it.

OPERATION REFERENCE

Two-sided operations combine the running value with an operand:

+ add add the operand

− sub subtract it

× mul multiply by it

÷ div divide by it

^ pow raise to its power

% mod remainder after dividing

min keep the smaller

max keep the larger

Operands — what the operation is applied with:

number a constant you type

i index the digit's position, 0, 1, 2, 3 …

d digit the original digit, unchanged by earlier rows

p prev the finished value of the digit before this one

π 3.14159…

e 2.71828…

base the current base

One-sided operations transform the running value on their own and ignore any operand:

sin cos tan in radians

asin atan inverse, in radians

√ sqrt square root

ln natural logarithm

e^v exponential

|v| absolute value

1/v reciprocal

floor round down

round round to nearest

frac keep the fractional part only

−v negate

About p prev. It is the finished, wrapped value of the previous digit — the one actually plotted there, not an intermediate. This makes a chain a feedback loop: every value depends on the entire history before it, so one edit anywhere changes the whole run, not just one point.

WRAPPING AND QUANTIZE

After the last row, the result is folded back into the digit range 0 … base−1 by repeatedly adding or subtracting the base until it fits. In base 10, 14 becomes 4, −1 becomes 9, and 23.5 becomes 3.5.

Wrapping keeps fractions. A digit is a whole number but a transformed value usually is not, and the lab plots it at its exact height — 3.7091 sits between the 3 and 4 gridlines, and its color is blended between the 3 and 4 colors. That precision is what lets drifting patterns show up at all.

QUANTIZE in the transport bar snaps everything to whole digits for display, so the result reads as a sequence of digits in the base rather than as a continuous signal. It is purely a lens: the underlying values are untouched and switching it off restores the full precision immediately.

GAPS

Some arithmetic has no real answer. Dividing by zero, the square root or logarithm of a negative, an expression that throws — all of these produce a value that is not a number, and there is no honest height to plot it at.

Rather than silently substituting zero, the lab drops a red marker on the dead rail below the axis and leaves the lane empty at that index. The READOUT shows ∅ GAP, the audio plays a tick instead of a note, and the DISTRIBUTION panel counts them separately. Try the 1 / d preset: every place π has a zero becomes a gap, which turns the panel into a map of π's zeros.

One exception exists so a single bad digit cannot destroy everything after it: when p prev reads a gap, it sees 0. The gap is still drawn as a gap.

EXPRESSIONS

+ EXPR adds a row where you write the arithmetic yourself. It is evaluated once per digit and its result becomes the running value, so expression rows mix freely with operation rows.

Variables:

d the original digit

i the index

v value coming into this row

p previous digit's result

base the current base

PI E constants

Functions — the usual set, unprefixed:

sin cos tan asin acos atan atan2 sqrt cbrt exp log log2 log10 abs sign floor ceil round trunc min max pow hypot random

Examples:

d * PI scale by π

d + sin(i / 12) * 4 a slow wave through the run

(d * d) % 7 squares modulo 7

d < 5 ? d : base - d fold the top half down

abs(d - p) the gap between neighbours

log(d + 1) * base / 2 compress the high digits

A row that will not compile, or that returns something which is not a number, is outlined in red with the reason underneath and is skipped — the rest of the chain keeps running while you fix it.

BASES

The base buttons and the number box accept anything from 2 to 64. This is a real conversion of π, not a re-reading of its decimal digits: base 16 begins 3.243F6A8885A308D3, base 2 begins 11.001001000011111101. Changing base changes what the digits are.

The conversion runs in the background and streams in, so playback starts immediately while the deep digits are still arriving — the SOURCE panel shows the progress bar and the count. Bases you have already visited come back instantly.

How many digits a base yields depends on how much information each one carries: base 2 gets around 200,000, base 10 exactly 100,001, base 64 about 55,000 — all from the same underlying expansion of π. Above base 10 the digits are written 0–9, then A–Z, then a–z, then + and /.

Base 2 and base 3 are the only bases where π's integer part needs two digits (11 and 10), so the first two indices are both to the left of the radix point.

TRANSPORT

▶ / ❚❚: Play and pause. SPACE does the same from anywhere except inside a text box.
⏮: Jump back to index 0, the leading 3.
scrubber: Drag anywhere in the digits converted so far. Works while playing or paused.
index box: Type an exact index and press Enter. Useful for revisiting somewhere interesting — the Feynman point sits at decimal 762.
RATE: From half a digit per second up to a few hundred. Slow rates are for reading individual values; fast rates turn the lanes into a waveform, which is where periodic patterns become obvious.

Editing while paused works exactly as well as editing while playing — the whole visible window recomputes either way.

AUDIO

AUDIO arms the sound on first click, because browsers only allow audio to start from a deliberate gesture. Click it again to silence it.

Every digit the playhead crosses is voiced, with pitch rising across the base: the lowest value is A3 and the highest is two octaves above it. In base 10 with QUANTIZE on, the lab borrows the game's own ten-note ladder so a base-10 experiment sounds like the ride it came from. Gaps get a short tick rather than a pitch, since they are not a value.

Above about sixteen notes per second the notes thin out automatically — past that density it is texture rather than melody, and stacking more voices only muddies it. Hearing a run is surprisingly good at catching short cycles that the eye reads as noise.

DISTRIBUTION

A tally of every value computed so far, not just the ones on screen. It updates on every edit, so it is the fastest way to see what a transform does globally.

The dashed line marks a perfectly even spread. π's own digits sit almost exactly on it — that flatness is the well-known thing about π — so the interesting question is always whether your transform preserves it. A tilt, a comb of spikes, or a hole means the operation is imposing structure of its own.

Below base 16 the bars are subdivided finer than one per digit, because transforms that produce fractions would otherwise pile everything into two or three bars and show nothing. Bin edges always line up with digit boundaries, so a transform producing whole numbers still reads as evenly spaced spikes. ∅ counts gaps and PEAK names the most common digit.

THINGS TO TRY

× π, then look at the distribution Ten input digits can only produce ten outputs, so a flat distribution becomes a comb of ten spikes. The lane still looks random; the panel proves it is not.

+ i at a high rate Adding the index sweeps the whole range over and over, so π's digits ride a sawtooth. Every wrap is a cliff. Turn the rate up until the sawtooth is visible as a shape.

SMEAR in base 2 Averaging each digit with the previous one is a low-pass filter. In base 2 it turns two flat values into a continuous cloud that never exceeds 1 — the ceiling is a property of the filter, not of π.

1 / d A gap wherever π has a zero. The dead rail becomes a map of the zeros, and the first one is further in than most people expect.

sin then × (base−1) Sine of a digit is not a smooth wave, because the input jumps around rather than advancing steadily. A good demonstration that a familiar function does not imply a familiar shape.

An expression using p Try (p * 3 + d) % base. This is roughly how a cheap random number generator works, and it produces something that looks like noise no matter what you feed it — a caution about reading meaning into any of this.

Compare a base against itself Set base 16 with an empty chain, note how the lanes coincide, then switch to base 2. The digits change completely but the number does not.

LIMITS AND NOTES

PRECISION: Values are ordinary double-precision floats. Chains that reach enormous magnitudes — repeated ^ pow, or e^v stacked — lose all meaningful precision before wrapping, and the result is numerical noise rather than mathematics.
DEPTH: Each base is converted from the shipped 100,000-decimal expansion of π, and the last few digits of any conversion are withheld because a truncated expansion cannot support them.
SAVED: Nothing. Every visit starts from the default chain in base 10, nothing is stored and nothing is sent anywhere — an experiment is always a fresh sheet of paper.
THE GAME: The lab cannot touch your π Rider records or settings. Experiment freely.
KEYS: SPACE play/pause · V next view · ? this panel · ESC close it. In the grid, PAGE UP PAGE DOWN HOME END and the arrow keys scroll; in a spiral, + and − zoom and 0 fits.