The fourth number divides the third

Raup’s model has a fourth number, held fixed until now: the shape of the generating curve, the opening the shell grows at. Every boundary located so far was located for a circle, exactly for a circle, and three essays ended on the same sentence — that the numbers would move with a differently shaped aperture by an amount nothing here measures.

The amount has now been measured, on the drawn outlines of eleven openings, and the sentence turns out to be wrong in the case that matters most.

Contact decided on the outline #

The opening is placed in the half-plane that holds the axis, with its extent along the radius running from D to 1 and its centre at height T — which is what D and T mean for any shape. The next whorl is the same opening multiplied by W about the apex.

Contact is decided the hard way. The later opening’s outline is sampled at 1,440 points and refined to its closest approach; the earlier opening’s own measure of distance — its gauge, which is below one inside it and one on its edge — is read at that point, and the whorls meet if it falls below one. The boundary is then bisected on that verdict, sixty steps, with no formula anywhere in it.

A second route to every boundary #

A measurement with one route vouches only for itself, so every opening that is its own reflection through its centre is also located a second way. Two copies of such an opening, one W times the other about the apex, meet exactly when (W − 1) times the centre lies inside W + 1 times the opening — a single inequality with no outline in it.

Across thirteen comparisons, including an ellipse turned 45° in its own plane, the outline bisection and the one-line test agree to within 1.7 × 10⁻¹⁶. The numbers below are therefore two numbers each.

An ellipse divides the translation #

An elliptical opening S times as tall as it is wide has, at every translation, exactly the boundary a circle has at the translation divided by S. Over 41 bisected boundaries — four aspects, four expansions and three translations — the worst disagreement is 3.6 × 10⁻¹⁶.

The shell in the first figure is the example: an ellipse twice as tall as it is wide at W = 2.4 and T = 0.5 meets its successor at D = 0.391257, and a circle at T = 0.25 meets at 0.391257. The reason is one line. Stretching the axis by 1/S turns the ellipse into a circle and T into T/S, and a stretch along the axis commutes with a multiplication about a point on it.

The drawn pair, worked by hand #

The circle’s boundary has a closed form, derived when the spire was measured: D = [(1 + W²) − (W − 1)√((W + 1)² + 4W·T²)] / 2W. At W = 2.4 and T = 0.25 the first term is 6.76, (W + 1)² is 11.56 and 4W·T² is 0.6, so the root is √12.16 = 3.48712. Then D is (6.76 − 1.4 × 3.48712)/4.8, which is (6.76 − 4.88197)/4.8 = 0.391257 — the number the outline bisection returned for the ellipse at twice that translation.

That the stretch works at all rests on the shell being a multiplication about a point on its axis, which is what growing without changing shape means. A stretch along the axis and a multiplication about a point on that axis can be done in either order and give the same shell, and nothing else in the argument is used.

Taller openings hold the boundary longer #

The threshold above which no distance from the axis puts the whorls in contact is therefore S√W/(W − 1) rather than √W/(W − 1). At W = 2.4 it is 0.5533 for an opening half as tall as it is wide, 1.1066 for a circle, 2.2131 for an ellipse twice as tall, and 3.3197 for three times as tall. The arithmetic is short: √2.4 is 1.54919, over W − 1 = 1.4 that is 1.10657, and the other three are that number halved, doubled and tripled.

The drawn outlines flip there. At 0.999 of each threshold the whorls are in contact, and at 1.001 of it they are free, for every one of the four. A tall, narrow opening is, in these terms, a spire’s worth of translation that the shell does not have to grow.

Four curves that are one curve #

Redrawn against T/S, the four curves are indistinguishable, and they are the circle’s curve. That is the plainest statement of what an ellipse’s aspect does: it relabels the translation axis and changes nothing else about the boundary.

It also says which of the earlier measurements survive for ellipses. The square law in the translation survives, with T replaced by T/S; the threshold survives, multiplied by S; and the tangency two measures of burial disagreed about is the circle’s tangency under the same relabelling.

The angle, stretched #

For a circular opening, whether the whorls touch is decided by one angle seen from the apex: the tube’s half-angle γ against (W − 1)/(W + 1). The stretch carries that rule over to ellipses unchanged except in one place. The half-angle is read with T replaced by T/S,

sin γ = (1 − D) / √((1 + D)² + 4T²/S²)

and the threshold it has to clear does not move, because W is untouched by a stretch along the axis.

So an ammonoid whorl section twice as tall as it is wide — the shape its describers call compressed — is, for the single question of whether its whorls touch, a round section with half the translation. That is a statement about the model’s geometry. Whether real compressed and depressed sections sort themselves by it is a question about shells, and it is not measured here.

With no translation the shape does nothing #

Coil the openings in a plane, with no translation, and the shape stops mattering altogether for every opening that is convex and its own mirror image across the plane of coiling. Circles, ellipses from half as tall to three times as tall, superellipses from exponent 1.5 to 8, a diamond, a square, and two keeled openings that are not centrally symmetric at all: every one meets its successor at D = 1/W, to within 1.1 × 10⁻¹⁶ over 33 boundaries at three expansions.

The argument is two lines #

Two convex sets that are both mirror-symmetric across one line meet if and only if their chords along that line meet. A shared point (x, z) brings its reflection (x, −z) with it, and by convexity the point halfway between them, (x, 0), is in both sets. So the sets meet exactly when their chords do.

With no translation the chords along the plane of coiling are [D, 1] for the earlier whorl and [WD, W] for the later one, and those meet exactly when WD is below one. The hyperbola D = 1/W is therefore the boundary for every convex opening symmetric about the plane, not only for a circle.

What that withdraws #

Every share of Raup’s cube the geometry excludes, measured in boxes with no translation, is the same share for every such opening, because the line it is measured under does not move. The hedge those numbers carried was unnecessary.

It does not withdraw everything. How deep one whorl is buried below the line depends on the shape of what is buried, so the burial figures do move with the opening’s shape, while the boundary itself does not. The boundary is a statement about chords, and burial is a statement about areas.

The tip decides how fast the boundary moves #

With translation the shape matters, and a diamond shows how much. At W = 2.4 and T = 0.3 a diamond meets its successor at D = 0.241667, where a circle at the same translation meets at 0.380276.

Measured as how far the boundary has moved from 1/W towards the axis, as a share of 1/W, the openings at that translation spread widely: the diamond 42.00 per cent, an ellipse half as tall as it is wide 33.25, a keel on the inside 27.54, a superellipse of exponent 1.5 17.89, a keel on the outside 12.60, the circle 8.73, an ellipse twice as tall 2.21, three times as tall 0.99, a superellipse of exponent 4 0.79, of exponent 8 0.01, and a square nothing at all.

The freed share, worked for two openings #

The share is the distance the boundary has moved from 1/W, divided by 1/W, which is the same as multiplying the distance by W. At W = 2.4, 1/W is 0.41667. The diamond’s boundary at T = 0.3 is 0.24167, so it has moved 0.17500, and 0.17500 × 2.4 is 0.42000 — the 42.00 per cent above. The circle’s is 0.38028, a move of 0.03639, and 0.03639 × 2.4 is 0.0873.

The two shares differ by a factor of 4.8 at one translation, and the ratio is not a constant: it is the ratio of a linear law to a square law, so it grows as the translation shrinks. At a hundredth of that translation the diamond would free a hundred times less and the circle ten thousand times less.

The square law belonged to the circle #

On logarithmic axes each opening’s displacement below 1/W is a straight line, and its slope is the power of the translation the boundary moves by. The diamond’s is 1.0000; a superellipse of exponent 1.5 gives 1.5000; the circle, the ellipse and both keels give 2.0000 to within 0.0025; exponent 4 gives 3.9998 and exponent 8 gives 7.9999.

The power is set by the tips of the two openings where they face each other: a tip sharper than any circle moves the boundary faster than a square law, a tip flatter than any circle more slowly, and a corner linearly. At the smallest translation measured, each displacement is within 0.01 per cent of the leading term derived for its shape, coefficient included. The square law reported for a spire is the law of a round tip.

Which tip faces which #

The whorls do not face each other with the same tip. The earlier whorl’s outer tip, the venter, faces the later whorl’s inner tip, the dorsum, which is W times larger. So the displacement at small T goes as

(W − 1) T² / (κ_venter + W · κ_dorsum)

with κ each tip’s radius of curvature over the opening’s half-width — which for a circle is the square law T²(W − 1)/(W + 1) already known. It follows from the fact that the radii of curvature of the two shapes add where the later opening is pressed against the earlier one.

The measurement confirms the one prediction that could embarrass it: a keel on the inside frees W times the ground a keel on the outside frees. The ratio is 1.6002 at W = 1.6, 2.4008 at 2.4 and 4.0048 at 4.

The curvature law, checked at one translation #

At W = 2.4 and T = 0.01 the law gives (1.4 × 10⁻⁴)/(1 + 2.4) = 4.1176 × 10⁻⁵ for the circle; 1.4 × 10⁻⁴ over 2.4, or 5.8333 × 10⁻⁵, for a keel on the outside, whose tip has no radius of curvature; and 1.4 × 10⁻⁴ over 1, or 1.4000 × 10⁻⁴, for a keel on the inside. Bisected on the outlines the three read 4.1176 × 10⁻⁵, 5.8333 × 10⁻⁵ and 1.4005 × 10⁻⁴.

The last is off in its fourth figure, and the difference is what a leading-order law leaves out rather than an error. It grows with the translation, as a next-order term must: the inside keel’s displacement is 1.00008 times the prediction at T = 0.005 and 1.00543 times it at 0.04. A law that matched to the last digit at every translation would be a stronger claim than a first term can make.

A turned opening, with no translation #

Turn an elliptical opening in its own plane and it is no longer its own mirror image across the plane of coiling, so the chord argument no longer applies, and the boundary moves with no translation at all. For an ellipse three times as tall as it is wide at W = 2.4, the boundary falls from 0.4167 upright to 0.3769 turned 10°, 0.2968 at 20°, 0.2278 at 30° and 0.1860 at 45°, then returns symmetrically to 0.4167 lying flat. An ellipse twice as tall reaches 0.3204 at 45°.

Every one of those sits exactly where ((W + 1) − g(W − 1))/((W + 1) + g(W − 1)) puts it, with g the opening’s radial half-extent over its half-chord along the plane.

The turned ellipse at 45°, worked #

For semi-axes 1 and S turned through an angle α, g is √((cos²α + S² sin²α)(cos²α + sin²α/S²)). At 45° both squares are one half, and for S = 3 the product is (0.5 + 4.5) × (0.5 + 0.0556) = 5 × 0.5556 = 2.7778, whose root is 1.66667 — exactly five thirds.

Then the boundary is (3.4 − 1.66667 × 1.4)/(3.4 + 1.66667 × 1.4) = 1.06667/5.73333 = 0.18605, the bisected 0.186047 to the fifth figure. The same arithmetic at S = 2 gives g = 1.25 and a boundary of 0.32039. A ratio of five thirds between how far an opening reaches and where its chord sits is what moves a boundary by more than half its distance from the axis, with no translation at all.

Freed in the plan and not in the chord #

That result has to be stated carefully, because two honest readings of it disagree. Raup’s D is defined by the generating curve’s extent along the radius, which is what a plan of the shell shows, and in that D a turned opening frees more than half the ground below 1/W — 55.35 per cent at 45°.

Measured instead along the chord the plane of coiling cuts through the opening, the same boundary is 1/W at every angle, to the last digit. A tilt changes how far the opening reaches, not where its chord sits, and the whorls meet on the chord.

A square holds, then falls #

A square opening is the extreme of a flat tip, and its boundary has two thresholds instead of a curve. It stays at exactly 1/W = 0.4167 until the translation reaches 0.7083, which is (W + 1)/2W, where the faces of successive whorls along the axis first meet. It then falls in a straight line and reaches nothing at 1.2143.

The circle falls from the start and empties sooner, at 1.1066. At a translation of 0.9 the square’s boundary is still 0.2588 and the circle’s is 0.1258, so the square keeps twice the contact region at that height after giving up nothing at all below 0.7.

Both thresholds are arithmetic. (W + 1)/2W is 3.4/4.8 = 0.70833, and the straight fall reaches zero at (W + 1)/(2(W − 1)) = 3.4/2.8 = 1.21429. Between them the boundary is 1 − 2(W − 1)T/(W + 1), which at T = 0.9 is 1 − 2.52/3.4 = 0.25882 — the bisected value to the last printed digit.

The sentence, rewritten #

The hedge the three essays carried can now be stated as a result. With no translation, the shape of the opening does nothing for any convex opening symmetric about the plane of coiling. With translation, an ellipse’s height divides the translation and its tips set the power of the translation the boundary moves by. And any asymmetry across the plane of coiling, of which a tilt is the simplest, moves the boundary as a plan reads it even with no translation.

What this does not establish #

It says nothing about openings that are not convex, and those include the one that matters most for real involute shells: an opening indented where the previous whorl lies against it. The chord argument needs convexity, so the result at no translation does not reach them.

It says nothing about which shapes shells actually have. And the curvature law for the two tips is a leading-order result: it is the first term of the displacement, measured against a bisection, not an exact boundary.

The two routes to each number are also not wholly independent, and that should be said plainly. The outline bisection and the one-line test read the same description of each opening’s shape, so an error in how a shape is described would reach both and they would agree about it. The chord argument at no translation does not share that description — it is a proof — and the eleven bisections are its check rather than its evidence.

What would withdraw it #

A convex opening symmetric about the plane of coiling whose bisected boundary with no translation is not 1/W. An ellipse whose boundary at T differs from the circle’s at T/S by more than arithmetic. A tip whose measured power of the translation is not the one its curvature predicts. Each of those is checked every time the measurement runs, and each would take one opening to find.

An opening indented by the whorl before it #

The next opening to measure is the one the chord argument cannot reach. A real involute shell’s aperture is the circle with the previous whorl’s section taken out of it, so its shape depends on D and W themselves, and it is not convex. The test is concrete: build that self-consistent indented opening, coil it with no translation, and bisect its boundary. If it is not 1/W, indentation is the one change of shape that moves the flat boundary, and the essays on Raup’s cube that were right for convex openings will need a sentence about the openings real involute shells have.