One over root two

The gap that grows measured the largest empty circle of a whole head at a string of sizes and found what separates a rational angle from an irrational one: the rational angle’s gap grows without bound and the irrational angle’s does not. On the scale that reading was first taken on, the golden angle’s gap wandered between 0.24 and 0.92 of a spacing from one head size to the next.

Read on the interior’s own scale it does not wander at all, and the reason it does not is the start of a sharper result.

A number that does not move #

On the interior’s scale the golden head’s largest empty circle is between 0.841 and 0.844 of a spacing at every size from 150 organs to 2,000. Three eighths of a turn climbs from 2.48 to 9.38 over the same range, a factor of 3.8. 137.3° reaches 0.863.

A reading that holds to three decimals across a thirteenfold range of head size is not a statistic of the head. It is one place in the head, measured again each time, and on the golden head that place is a single triangle — organs 2, 3 and 5 — whose circle is centred at radius 0.871.

The rings, one at a time #

Take the same reading ring by ring instead: the largest circumradius among the triangles whose circle centres fall in each ring a quarter of a unit wide. On a 2,400-organ golden head the central triangle is 0.8437, at radius 0.871, and everything beyond it is smaller.

The ring maxima are not flat either. They rise and fall with radius and peak at a sequence of radii, and at each peak the value is the same number to within a couple of thousandths: 0.7071, which is 1/√2.

What happens at a peak #

Away from the centre the head is locally a lattice. At a rise of t per organ, measured in circumferences, organ offset j is a vector whose sideways component is eⱼ — how far j turns of the divergence fall short of or overshoot a whole number of turns — and whose upward component is j·t. Every cell has area t.

The Delaunay triangulation of that lattice is built on its two shortest vectors, u and v, and it changes where those two become perpendicular. The change is a flip, and it happens at

t = √(|eᵤ · eᵥ| / (u·v)).

The pairs that flip are not arbitrary. Two offsets form a basis of the lattice exactly when u·|eᵥ| + v·|eᵤ| = 1, and each flip replaces one member of the pair by the sum of the two. The pairs walk the same ladder of offsets that the counted spirals climb as a head is read outwards.

A rectangle, and the circle through its corners #

At a flip the cell spanned by u and v is a rectangle, and the two triangles it splits into share one circumcircle: the circle through the rectangle’s four corners. Its radius is half the diagonal. With the cell’s area scaled to one, a rectangle of aspect ratio x has a diagonal of √(x + 1/x), so the largest empty circle at a flip is

√(x + 1/x) / 2, with x² = u·|eᵤ| / (v·|eᵥ|),

and the basis identity does the algebra. A square cell, x = 1, gives exactly 1/√2. Any other rectangle gives more.

On the golden lattice the flip between offsets 21 and 34 falls at radius 11.272 on the head, the cell’s aspect ratio there is 0.9997, and the empty circle is 0.7071 of a spacing.

The 21/34 flip, worked through #

Every quantity in that flip can be written down. Twenty-one turns of the golden divergence overshoot a whole number of turns by e₂₁ = 0.021286; thirty-four turns fall short by e₃₄ = −0.013156. The signs are opposite, and 21 × 0.013156 + 34 × 0.021286 is 0.27628 + 0.72372, which is 1.00000 — so the two offsets are a basis of the lattice.

The flip’s rise is √(0.021286 × 0.013156 / (21 × 34)), which is 6.2626 × 10⁻⁴ per organ, and on Vogel’s disc that rise is reached at radius 1/√(4π × 6.2626 × 10⁻⁴) = 11.272. The aspect ratio squared is 21 × 0.021286 over 34 × 0.013156, or 0.44701 over 0.44730, which is 0.9994.

That rise is not new either. The rise at which a stem’s tessellation changes its third family was given there as √(−⟨mδ⟩⟨nδ⟩ / mn) for a rung (m, n), and for the rung (21, 34) it is the same 6.2626 × 10⁻⁴. The flip that makes the largest gap square and the change in which neighbours a tessellation joins are one formula.

Every flip a factor of φ further out #

The golden lattice’s flips beyond the centre fall at radii 4.304, 6.969, 11.272, 18.241, 29.513 and 47.754. Consecutive ratios are 1.6152, 1.6191, 1.6176, 1.6182, 1.6180 and 1.6181: the flips step outwards by φ.

That follows from the formula. From one flip to the next the offsets grow by a factor of φ and their residuals shrink by a factor of φ, so |eᵤ · eᵥ| / (u·v) falls by φ⁴, the rise by φ², and the radius, which goes as the rise to the minus one half, grows by φ. The defect rings a golden head carries sit at 11.274, 18.238, 29.513 and 47.758, and the flips agree with them to within 0.004 at every one.

The Lucas lattice steps by φ too, from 5.943 to 9.633, 15.576, 25.209 and 40.786. The silver lattice steps by 1.5538 a flip, and two flips make one run of its expansion: 1.5538 squared is 2.4143, which is 1 + √2, the silver ratio.

The flip at 11.272 is a defect ring #

That radius has appeared before. The cells of a golden head that are not hexagons lie on rings, located by where the third-shortest lattice vector hands over, and one of them sits at 11.274.

It is the same place reached from two sides. At a flip four organs lie on one circle and the triangulation could be drawn either way; a tessellation that changes which neighbours it joins must put fives and sevens somewhere while it does. The ring of defects and the square cell are one event. It is also why the rings are not the transitions of the counted pair: the tessellation’s own ladder sits half a rung from the count’s, and a flip is a rung of the tessellation’s ladder.

The closed form, checked by brute force #

The closed form was derived, so it is checked against something that does not know it. At each flip of five lattices — golden, Lucas, silver, 137.3° and 106.5°, thirty-eight flips in all — the lattice is rebuilt at the flip’s rise, its shortest vector is found by search, the shortest vector that completes a basis with it is found by search, and the largest circumradius of the triangles they make is computed from their three sides.

The two routes disagree by at most 4.7 × 10⁻¹³, and every pair the walk passed through satisfies the basis identity. On the 2,400-organ head, at the four flips beyond radius six, the head’s own ring maxima sit within 0.0020 of the closed form.

The head approaches the lattice from above, and the approach is visible flip by flip. Its ring maximum is 0.7121 at the 8/13 flip near radius 4.3, 0.7091 at 13/21, 0.7078 at 21/34, and 0.70707 and 0.70709 at the 34/55 and 55/89 flips near radius 18 and 29. By radius eighteen the head and the lattice agree to the fourth decimal, which is the scale at which a Vogel head stops being a patch of individual organs and becomes a lattice.

Why a noble angle’s cell comes out square #

Write the angle’s continued fraction with partial quotients a₁, a₂, and so on. At a given run of flips let ρ be the ratio of consecutive denominators and β the ratio of consecutive residuals. The residuals obey a recurrence the quotients set, and substituting it into x² gives, for the j-th flip of a run whose partial quotient is a,

x² = 1 / ((j + ρ) · (a − j + β)).

A noble angle has every partial quotient equal to one from some point on, so every run is a single flip with j = 1 and a = 1, and ρ and β both tend to 1/φ. Then x² tends to 1 / ((1 + 1/φ) · 1/φ), and 1 + 1/φ is φ, so x² tends to exactly one. The cell is square in the limit, and the gap is 1/√2 — for the golden angle, for the Lucas angle, and for every other noble number alike.

And why every other irrational’s does not #

A partial quotient larger than one makes a run of several flips, and they cannot all be square. With a quotient of three or more, the run’s first flip has x² of at most one half, so its gap is at least 0.7282. With a quotient of two the run has two flips, and the larger of their two gaps, minimised over every possible ρ and β, is 0.7114.

So an irrational angle with infinitely many partial quotients above one has a gap of at least 0.7114 at infinitely many flips, however far out the head is read. The silver angle, 149.117°, whose continued fraction is all twos, never lets both flips of a run be square: its flips alternate between 0.7071 and 0.7282 all the way out, the second value well above the least a quotient of two allows. 1/√2 is the limit of exactly the noble angles.

That is the division Hurwitz’s theorem draws between the noble numbers and the rest, and it arrives here from a different quantity: not how close two organs come, but how large a hole the lattice leaves between them.

A quotient of four, at radius 13.5 #

137.3° is irrational, and its continued fraction has a four in it. Read straight from the lattice, with no head built at all, the golden lattice’s largest gap never passes 0.7068 between radius three and fifty. 137.3°'s peaks at 0.8614, near radius 13.5.

A quotient of four makes a run of four flips, and offset 21 stays in every one of them: 21/34, 21/55, 21/76 and 21/97, at radii 11.08, 13.56, 16.62 and 22.72. Twenty-one turns of 137.3° miss a whole number of turns by only 0.009167, so offset 21 is the short vector for the whole run while its partner is replaced three times. The gaps at the four flips are 0.8399, 0.8635, 0.8377 and 0.7490 — the middle of the run is least square, as x² = 1/((j + ρ)(a − j + β)) says it must be when a is four, and the second flip is the peak.

It is the same twenty-first organ that sets 137.3°'s closest pair at 0.6201 on the interior’s scale: a near-rational offset is both the pair that crowds and the pair that leaves the hole.

Built as a head, 137.3° does what its lattice says. Its whole-head largest gap is 0.8622, and it is not at the centre — it is at radius 13.511. At the four flips of the run the head reads 0.8395, 0.8622, 0.8369 and 0.7484, against the closed form’s 0.8399, 0.8635, 0.8377 and 0.7490, and at its six flips beyond radius six it is never more than 0.0034 from the closed form.

137.0°, with a quotient of five #

The other near neighbour tells the same story one step further. 137.0° has a five in its continued fraction, and its run of five flips again keeps offset 21 throughout — 21/29, 21/50, 21/71, 21/92 and 21/113, from radius 10.64 to 28.39. The gaps climb to 0.8905 at the second flip, 0.8895 at the third, and fall away to 0.7221 at the fifth.

Its whole head reads between 0.887 and 0.890 from three hundred organs outward, decided near radius 13. That is the head the whole-head account described as looking poor because it behaves like twenty-one rows for a while, and the run of five flips is what “for a while” is: five flips, a factor of 2.7 in radius, during which offset 21 does not change.

So the gap does tell the golden angle from its neighbours #

The whole-head account concluded that the difference between the golden angle and an irrational neighbour is a theorem that cannot be seen at the head sizes a plant makes. Radius 13.5 is about 180 organs out. By then 137.3° has opened a hole 0.862 of a spacing across in one ring, where the golden head holds 0.707 in every ring beyond its centre, and a head of three hundred organs already shows it in its whole-head reading.

The difference was hard to see in a whole-head reading only because the golden head’s central triangle, at 0.8435, is nearly as large as 137.3°'s ring. Read ring by ring, the two differ by more than a fifth.

Rational angles, bounded by two rays #

A rational angle is the third class, and it has a bound of its own. Its organs lie on q rays, and no empty circle can be larger than the circle that touches two neighbouring rays with its centre still inside the head. Measured in spacings that circle has radius √(n − 1) · sin(π/q) / √π, which grows as the square root of the head.

At 2,000 organs three eighths of a turn reads 9.3750, 97.1 percent of that bound, and five thirteenths reads 5.9561, 98.7 percent of it. The growth the whole-head reading reported is this bound being approached, ray by ray.

Three classes, and the gap tells them apart #

The largest empty circle, read away from the centre, sorts divergence angles into three classes. A rational angle’s grows without bound, as the square root of the head. An irrational angle with infinitely many partial quotients above one stays bounded, and reaches at least 0.7114 at infinitely many flips. A noble angle’s tends to exactly 1/√2.

The second and third classes are separated by at least 0.0043 at every flip that matters. That is small, and it is well inside what the head resolves: at its flips the head reads the closed form to within a few thousandths, and the bound is set by a single partial quotient of two.

Two constants from one lattice #

The closest pair a noble head keeps away from its centre tends to √(2/√5), which is Hurwitz’s constant under a square root. Its largest empty circle tends to 1/√2. Both are read from the same local lattice at the same flips, both single out the noble angles as a class, and neither singles out 137.508° from within it.

What this does not establish #

The derivation is for a lattice. That a finite Vogel head reaches the lattice’s values at its flips is measured — to within 0.0020 on the golden head and 0.0034 on 137.3° — and not derived, and the centre, where a head is not a lattice, is exactly where the whole-head reading is decided.

The bound of 0.7114 for a quotient of two is a minimum found on a grid of 400 steps in each of ρ and β, not a closed-form minimum. The measurement is on Vogel’s model with equal points, and it says nothing about seeds of different sizes or about a plant. And the largest empty circle is a statement about the worst hole among given points; it is not a packing density, and nothing here packs discs.

What would withdraw it #

A noble angle whose ring maxima, far from the centre on a larger head, settle anywhere other than 1/√2. An irrational angle with quotients above one that keeps every flip below 0.7114. A flip at which the brute-force lattice and the closed form disagree by more than rounding. Each is one head or one lattice to build, and any of them would mean the closed form is not the account of the holes.

Reading the partial quotients off the holes #

The closed form runs both ways. At every flip the gap fixes x², x² fixes (j + ρ) · (a − j + β), and ρ and β are nearly fixed by the flips before it, so a head’s ring profile carries its divergence’s continued fraction in the heights of its peaks.

That is the next measurement, and a head can fail it. Take the ring profile of a head built at an angle the reading is not told, find its peaks, invert them, and recover the partial quotients. On 137.3° the peak of 0.8614 near radius 13.5 should return its four, and the silver angle’s run of equal peaks should return a string of twos. If the recovered quotients disagree with the angle’s own expansion, the holes do not encode the arithmetic the way this essay says they do.