Spirals

Coiling bore walks that turn about an axis while advancing along it, collected with the numbers that say how hard each one spirals and what each one costs to build. Every walk here splits with no elbows — every turn folds into a piece as an L, none is stranded as a single-block piece of its own — and every one passes the full gate.

The one this started from is coil_4x4_50, which has since won a category and moved to a directory of its own; the ten came out of an exhaustive search for something tighter.

Download the whole repository as a ZIP — every trumpet and every tool, not the walks alone; these pages are under spirals/. GitHub builds it from main on every push, so it is never out of date.

The rest of the build files — every instrument, generator and tool, indexed.

Built for LaserMadeMusic, where the cutting and the playing are shown.

Standardised

Every coil here is in the same orientation, so two coils differ only where they really differ:

• forward is north for all of them, and each walk opens on a north term
• counter-clockwise, seen looking along the bore from the mouth
• canonical: of the several ways a coil can satisfy all of the above, the lexicographically smallest is the one kept, so equal coils are identical strings
• whole periods only — the repeat count that lands nearest a common target
• one block in, one block out, and no partial period between them

A bore opens at both ends, so the first and last piece cannot be removed — they are bounded by the mouth and the exit rather than by a neighbour, and the notation says as much: the first term is only the way you came in. What is standardised is that every walk has exactly one of each, and the judged metrics ignore them regardless.

Fixing forward and the sense of rotation is not enough on its own. The four rotations about the forward axis all satisfy both, and so does every rotation of the cycle that opens on a north term, so taking the first representation that fits leaves the same coil able to appear more than once looking different — and it did, four times over in one case. Pinning the representation to the lexicographically smallest is what makes distinct mean anything.

Standardising collapsed 30 walks into 10: several that looked different were the same coil in another orientation, and derived.txt records which. The hand reduction of the staircase coil, the tool's reduction of it, and the extension of the hand reduction all turn out to be one coil.

The length rule, and why it is not a cap

Whole periods of different lengths cannot all reach the same total, so some rule has to choose the repeat counts. The first one tried — as many periods as fit under a limit — is the wrong shape: it truncates the long-period coils hardest, so every one of them lands below every short-period one, and the bores ran 126 to 144 blocks, a 14.3% spread.

The target is now chosen rather than assumed. tools/standardise.js searches bore lengths from 120 to 220 blocks for the one that makes the tube lengths most alike, takes the nearest whole-period count to it rather than the largest that fits, and lands on 180: bores of 177 to 193 blocks, 2832mm to 3088mm, a spread of 9.0%.

That is as close as this set can get at a plausible size. Squeezing the spread below 10% needs a seven-metre bore and below 5% needs twelve, which is not a trumpet.

So the residual is handled where it actually bites, in the scoring: box and piece count are scored per block, because both grow with tube and comparing them absolutely would hand the shorter coils an advantage they did nothing to earn. Over this set piece count correlates 0.63 with block count. Cross-section, mean plate, distinct shapes and the rates are length-independent already; touching stays an absolute count, because the requirement is none of it at any length.

Shape and cost

Envelope, box and cross-section are in blocks; a block is 16mm of centreline. Sorted by box, smallest first.

The bore's mouth and exit are not judged. Every design has them, no design chooses them, and they are the two pieces that never join the rhythm — so every column here except blocks and mm is measured on the interior, the pieces in between. It is not a cosmetic change: the ends stick out of the envelope they bracket. coil_3x8_20 needs 4 shapes rather than 6, the difference being end pieces alone. Blocks and mm still describe the whole bore, because that is what the bore is.

Cross-section is the envelope with the coil axis taken out — how fat the coil is, which is what decides whether it fits inside anything. It is not implied by the box: coil_3x3_54_2 is 3x3 in section and coil_4x9_18 is 4x9, and the box does not say so.

Distinct is how many different piece shapes the cut list holds. A coil built by repeating one period needs only a handful, however long it runs: coil_3x8_20 is 36 pieces cut from 6 shapes. The widest needs 4. That is files to check and parts to tell apart on the bench, and it does not show up in the piece count at all.

Period is the repeating unit of the walk itself, in terms and in blocks, and it sits next to the rhythm because the two together say something neither says alone.

A piece is a flat snake, so the splitter has to start a new one exactly where the bore leaves its plane. Turns that stay in-plane — the folds, hairpins and steps both — happen inside a piece and cost no boundary at all. So the rhythm is not counting turns, it is counting departures from the plane, and it comes out as

rhythm = period terms x the share of turns that leave the plane

exactly, on every coil here. That decomposition is why the period column earns its place: a rhythm of 4 could be a long period turning gently or a short one leaving the plane at every chance, and the rhythm alone cannot tell you which. In this set the 9 eight-term coils leave the plane on half their turns, while the 0 sixteen-term ones do it on three turns in four — so the gap in rhythm is two separate factors multiplying, not one.

The share itself is not scored, and was tested rather than assumed: it takes only three values across the set, 9 of the 10 coils sit on exactly one of them, and it correlates 0.89 with the rhythm, 0.85 with distinct shapes and 0.76 with box per block. Near-constant and not independent — the same double failure that kept fill density out.

Rhythm is how many pieces you lay before the cut list starts over, and how many times it recurs. It is descriptive, not scored: across coils built by repeating a period it barely varies — 10 of the 10 here are a 4-piece rhythm — and a near-constant metric in a mean only dilutes the ones that discriminate. It is worth knowing at the bench all the same, and it is not the same thing as distinct: the two rank the set alike in only 6 of 10 places. coil_3x8_20 is 4 shapes laid in a 4-piece cycle repeated 8.5 times, between the two end pieces.

Mean plate is the average bounding box a piece is cut from, in mm2 — the laser-cutting number. Fewer, larger parts means less weeding, less sorting and fewer fingers to align, and it is the size of the part in your hand rather than the count of them. coil_4x9_18 averages 2,881 mm2 against coil_4x4_50's 1,598.

Average blocks per piece was the other reading of the same idea and is not used: the block count varies by only 11.0% across the set, so blocks-per-piece is very nearly the reciprocal of the piece count and ranks the set the same way in 15 of 10 places. Plate area is not redundant: coil_3x9_18, coil_3x3_54_2, coil_4x8_18, coil_5x7_18, coil_4x9_18 all split into 30 pieces and cannot be told apart by blocks-per-piece at all, while their mean plates are 2,473, 2,601, 2,446, 2,345, 2,881 mm2.

It is a bounding box, not the cut outline — an L-shaped piece leaves its corner behind — so it measures the size of the part, not the material consumed.

Touching counts blocks that sit face to face without being joined along the bore — two runs of the tube sharing a wall. bore_split.py warns about them, and they are the version of "density" that has a consequence: where the bore passes itself, that one wall is all that separates the two passages.

Because a shared wall shows in the finished instrument, it is the heaviest input in the scoring and is penalized convexly — 1/(1+t), so no contact scores 1.000 and eight contacts 0.111. node tools/score.js --clean ranks only the coils that have none.

It is also the one packing number that is not implied by the box. coil_3x8_20 has the smallest box in the set and 15 shared walls; coil_3x3_54_2 is 2% larger and has none. If a shared wall is something to avoid, the smallest box is not the one to build.

Rotation

A coil turns about one axis while travelling down it. Drop that axis and what is left — the lateral projection — is what rotates, and on a cubic lattice it can only turn in ninety degree steps, so the winding is counted in quarter turns and multiplied up. Steps along the coil axis project to nothing and are skipped: they are advance, not rotation.

• Rotation — the whole turn end to end, in degrees.
• Blocks / 360 — tube spent on one revolution. Lower is a tighter spiral.
• Rise / 360 — how far down its axis the coil travels to come back round. The pitch.
• Deg / block — the same, per block, averaged.
• 90deg turns, turns/m — how often the air is asked to turn a corner. The tube length is nearly the same for all of these, so this is what separates them acoustically, and it is the one number here that argues against packing tighter.
• Longest straight — the longest run without a turn.

Axis is the direction the coil advances; the search was free to build about any axis, so a coil that runs east is measured against east.

What wins what

No single spiral wins, because the measures disagree.

coil_4x4_50, the coil this started from, loses the packing categories and wins the turning ones. Set beside the ten it produced, it places 9th on box per block, 10th on pieces per block and 11th on distinct shapes, against 1st on tube per revolution and 2nd on turns per metre. Its reduction, coil_5x5_50, won a category too and was promoted out beside it, so only one of the two staircase coils is measured here now.

That is not a split verdict so much as one fact seen twice: packing tighter costs bends, and bends are what a bore notices. A coil that turns economically is a coil that does not fold itself into a small box, and every category above is downstream of that choice.

A metric deliberately left out

Fill density — blocks over box volume — was tried and dropped. At a fixed tube length it is not independent of the box: blocks is near enough constant, so density is just the reciprocal of box volume rescaled. Ranking these 10 spirals by density puts them in the same order as ranking by box in 16 of 10 positions, and the two that swap are a tie at 477 broken by a one-block difference in length. It reads like a second opinion and is not one. It would earn its place only in comparing walks of genuinely different lengths.

Touching is the metric density was reaching for. It answers the question density sounds like it answers — how hard is this bore packed against itself — and unlike density it disagrees with the box often enough to change which coil you would build.

Every block load-bearing

The staircase coil was reduced by hand — blocks removed wherever one could go without introducing an elbow — and that reduction is in this set. (The names below predate standardising, which renamed every coil; derived.txt maps them.) tools/minimal.js checks the claim: a term's floor is 3 in a coil window, 2 in a hairpin, 1 in a step, and a walk whose every term sits on its floor cannot be shortened at all.

node tools/minimal.js            # every walk
node tools/minimal.js --terms    # and which terms have slack

Before it was reduced, the staircase coil had 16 terms with slack, every one a hairpin sitting at 3 where 2 would do, and the reduction left none. It was exhaustive: the tool later reproduced it exactly, and standardising then showed the two to be one coil.

It also says something about the search. Only 1 of 10 walks here are minimal: coil_3x4_79. The search enumerated periods with legs up to 4 and never asked whether a leg was longer than it had to be, so most of what it found carries slack — coil_4x9_18 could lose 48 blocks. Two of the 1 minimal walks are the two that were reduced by hand.

Removing slack is not automatically safe. The rule is local, and a shortened walk can run into itself or start touching, so what the tool reports are candidates to put back through bore_split.py.

The reduction pass, and why minimality is not an optimisation

tools/reduce.js takes the slack out and rebuilds at comparable length. It has to enumerate rather than apply, because coiling is global and the elbow rule is local. A period coils only if it closes: drifting on one axis and returning to where it started on the other two. Shorten one leg and not its opposite and the period stops closing, so the walk wanders off diagonally instead of coiling. Taken naively, one coil went from a box of 423 to 10,452 — still elbow-free, no longer a coil.

Keeping only the reductions that still close and still wind a whole number of turns, and putting the result back through the standardiser so it can be compared with what it came from, 9 of the 10 coils reduce, and three distinct walks come out of those 9 reductions:

• six coils reduce to D3 E2 U3 N1 U3 W2 D3 N1, which is not in the catalogue: coil_3x8_20, coil_3x9_18, coil_4x7_20, coil_4x8_18, coil_4x9_18, coil_5x7_18
• two coils reduce to coil_3x3_59, one of them being coil_3x3_59 itself: coil_3x3_54_2, coil_3x3_59
• coil_3x4_68 reduces to itself — the pass finds slack in it and gives back the same period

The set is not closed under reduction. Six of the 9 reductions land on a walk that is catalogued nowhere here. That is a gap in the set rather than a turn of phrase: either those walks belong in it, or the reduction that produces them does not really close and reduce.js should be rejecting it.

This page said the opposite until the claim was counted. It was written when the search held seventeen coils, and nothing re-checked it when the category winners were promoted out to siblings of their own.

That is a much cleaner result than the first attempt, and the difference is canonicalisation: before the representation was pinned, the same coil could appear several times looking different, and reductions of it looked like new designs when they were not.

Shortening a leg still does not make a better coil — it cuts how far the coil advances per turn, so the same tube buys more revolutions in a fatter package. The best box per block and the best walls-free box per block are both unchanged by the pass.

What those 9 reductions buy:

And nothing it produced beat what was already there: the smallest box stayed at 423 against the reduced field's 462, and the smallest walls-free at 477 against 522.

That is the point worth keeping. Shortening a leg reduces how far the coil advances per turn, so the same tube buys more revolutions in a fatter, shorter package — a different design, not a better one. Every block being load-bearing is a property, not a virtue.

One thing the pass did find: four results the search reported as distinct all reduce to the same walk, E2 S3 U1 S3 W2 N3 U1 N3 — one design wearing four amounts of slack. Standardising later found more of the same, and derived.txt records every merge.

The reduced walks are kept, named *_min, with their sources recorded in derived.txt. They are in the scoring like anything else, so the composite is now over a set that contains both a design and its own reduction — worth remembering when reading a rank, since the normalization is across whatever is present.

node tools/reduce.js             # the pass
node tools/reduce.js --write     # and reduced.json

Walks kept but not scored

The scoring normalizes every metric across the set, so a walk of a different length does not merely rank oddly — it rescales everyone else. Those walks live in the repository and in the tables, below a rule, and are listed in unscored.txt with the reason. Their rate metrics — blocks/360, rise/360, turns/m, touching — stay comparable and are still worth reading.

Scoring them against each other

SCORING.md combines eight of the metrics and the touching count into a single ranking under every common mean — harmonic, geometric, arithmetic, quadratic, cubic, median, midrange, contraharmonic — and reports what the choice of mean does to the answer. It does a great deal: coil_2x2_146 places 1st under one and 17th under another. coil_3x3_54_2 comes first under 8 of the 8 means.

It also covers why to rank once rather than rank, cut and re-rank: with a set-relative normalization the survivors of a cut come out in a different order than they went in, having not been measured again.

node tools/score.js          # the scoring table
node tools/iterate.js        # what cutting and re-ranking does to it
node tools/gen_scoring.js    # SCORING.md

The gate

Every walk here has been through check.py — the parts, the sections, the seams, and a voxel model of the assembled bore flooded from the outside to prove it is one sealed passage. 11,872 checks across 10 spirals, 0 failed. The per-walk output is in checks/.

node tools/run_checks.sh     # re-run the gate over every walk

check.py needs shapely, numpy and scipy, which live in the Boxes.py venv — run it with ~/Software/boxes/venv/bin/python, as README.md says. The system python3 does not have them.

How they were found

tools/search_spirals.js walks every periodic term sequence up to eight terms with legs up to four blocks, keeping those that never reverse, return to the same lateral position each period so the thing coils rather than drifts, wind a whole number of turns per period, stay self-avoiding when repeated, and satisfy the elbow-free rule.

That rule, for a window of three consecutive terms — outer A, middle m, outer C:

The bore-designs README states only the third case and calls the other two "a fold ... as tight as you like". That holds for steps and not for hairpins: U3 N1 D3 costs two elbows and U3 N2 D3 costs none. The three cases were read off bore_split.py by probing it.

tools/mknotation.js repeats a period to length and writes it in the bore notation, trimming the tail so the last block is not mid-turn — a turn in the last block has nothing after it to make it interior, so it is always its own piece. Then every candidate goes through the splitter, and only zero-elbow walks are kept. The splitter decides, not the rule.

The tools

Every file in tools/, and what each one is for -- all fourteen of them. The list is read from the directory rather than typed into this page, so it cannot fall behind it: a tool with no description, or a description with no tool, stops the build. There was no such list until 2026-09-10, and none of these was described anywhere, here or elsewhere. This file named some of them in passing but never said what they were, and a passing mention is all the orphan check asks for.

Finding and shaping walks

Measuring and ranking

Producing what is committed

The files behind the tables

Each coil is one line of notation in walks/, named for the coil, and each has a gate transcript beside it in checks/:

walks/coil_3x8_20.txt, walks/coil_3x9_18.txt, walks/coil_3x3_54_2.txt, walks/coil_3x3_59.txt, walks/coil_4x7_20.txt, walks/coil_4x8_18.txt, walks/coil_5x7_18.txt, walks/coil_4x9_18.txt, walks/coil_3x4_68.txt, walks/coil_3x4_79.txt

standardised.json records the canonical form every walk was pinned to, and reduced.json what the reduction pass produced. Both are written by their tools, not by hand.

Regenerating

tools/build.sh               # all of the below, in order

node tools/parts.js          # piece counts and distinct shapes -> parts.json
node tools/table.js          # both tables, plain text
node tools/table.js --md     # the same as markdown
node tools/gen_readme.js     # this file
python3 ../../../../../../../../lasermade-tools/md2html.py README.md index.html    # the published page

node tools/spiral_metrics.js "$(cat walks/coil_3x8_20.txt)"   # measure one walk

~/Software/boxes/venv/bin/python ../../../../../../../tools/viewer.py \
    "$(cat walks/coil_3x8_20.txt)" \
    --out pages/coil_3x8_20.html --title "coil 3x8 20"

What has not been done

No cut files. These are walks, viewers and gate results; nothing here has been split to SVG or nested. bore_split.py --write DIR will do it.

The block counts are not equal — the scored ones run 163 to 181, so their bore lengths run 2608mm to 2896mm, a spread of 11.0%. A period is repeated to about the right length and then trimmed to wherever the tail comes out elbow-free, so these are the same bore only to within a few blocks. For comparing packing that is fine; for comparing bores it is not, and the length would have to be pinned first.

Released under CC0 1.0.