One formula, four numbers.

The four constants that govern how deep the stars cut in six-, eight-, ten- and twelve-fold Islamic patterns look like four unrelated facts: 1/√3, √2 − 1, (3 − √5)/2 and 2 − √3. Three of them are the same formula. The fourth is the tradition overruling the formula, for a reason the pentagram makes plain.

The depth of a star is a single number, and for the four classical symmetries the geometry derives it rather than leaves it to taste. When the studio opens a six-, eight-, ten- or twelve-fold pattern, the star depth is already set, and it is set to an exact irrational rather than a rounded decimal. Those four values are usually presented as a table to be memorised. They are better understood as one theorem, one exception, and the reason for the exception.

What the depth is

Every star pattern in this tradition rests on a net of polygons in contact, and every polygon carries its star by the same rule: from the midpoint of each edge, two lines leave into the polygon at a fixed angle to that edge, and the star is what those lines make when they meet. Call the angle θ. A small θ gives lines that hug the edges and a fat, shallow star; a large θ gives lines that dive for the centre and a thin, deep one. The engine records this choice as the depth, which is simply tan θ, the slope of the line relative to the edge. Depth 1 means the lines leave at forty-five degrees. Depth 0.414 means they leave at twenty-two and a half.

So the question "how deep should the eight-fold star be" is the question "at what angle should the lines leave the edge", and the tradition has an answer that is not a matter of taste.

The theorem: draw the star polygon

Take a regular polygon with an even number of sides, n, and join every vertex to the vertex n/2 − 1 steps away. On the hexagon that joins each vertex to the one two steps on, and draws the hexagram. On the octagon it joins each vertex to the one three steps on, and draws the {8/3} star, the khatam. On the dodecagon it draws the {12/5}. The chords cross one another, and the crossings nearest the centre lie at a distance of exactly tan(π/n) from it, taking the circumradius as one.

The proof is two lines of trigonometry. A chord that skips n/2 − 1 vertices subtends an angle of π − 2π/n at the centre, so it passes the centre at a distance of sin(π/n). Two neighbouring chords are related by a turn of 2π/n, so their crossing sits on the line that bisects them, at an angle of π/n from the foot of either chord. A point at angle π/n along a line that passes at distance sin(π/n) is at distance sin(π/n) / cos(π/n) from the centre. That is tan(π/n), and it is the depth.

Now the table collapses. tan(π/6) = 1/√3. tan(π/8) = √2 − 1. tan(π/12) = 2 − √3. Three of the four constants are one formula evaluated three times, and the engine does not even store them: it draws the polygon, joins the vertices, intersects the chords and reads the number off the construction. The proof is the drawing.

Why the hexagram gives a third

The six-fold case is worth a moment on its own, because it is the one where the number is easiest to see. The inner hexagon of a hexagram has a circumradius of 1/√3 of the outer one, so its area is exactly one third. The star of David cuts its host into a central hexagon and six triangles, three parts to one. This is the first of the trisections that keep appearing across the engine, and it is the same number wearing a different name.

The exception: ten-fold and the pentagram

Run the formula for n = 10 and you get tan(π/10) ≈ 0.325. The engine does not use it. For ten-fold work it uses (3 − √5)/2 ≈ 0.382, which is 1/φ², the reciprocal of the golden ratio squared, and it does so because the tradition did. The reason is in the figure. Joining every vertex of the decagon to the one four steps on draws the {10/4} star, and the {10/4} star is not one polygon but two pentagrams laid over each other. Its chords therefore cross in two ways: neighbouring chords, which belong to different pentagrams, cross at tan(π/10); and chords two apart, which belong to the same pentagram, cross where a pentagram's own diagonals cross, at 1/φ².

The classical ten-fold rosette takes the pentagram's crossing, not the compound's. That is the deeper star, and it is the one on the walls of Isfahan and the pages of the Topkapi Scroll. The engine's ten-fold default is set to it, derived by intersecting the pentagram's diagonals so that the golden ratio appears as a length in the construction rather than as a number typed in. The formula was not wrong; the tradition simply chose the other crossing in the same figure, and for five-fold geometry that crossing is the golden one.

Why a derived number matters

A pattern generator that lets you drag a "star sharpness" slider is not wrong, exactly; it is just not saying anything. A slider has no reason to stop at 0.414 rather than 0.4. A construction does: at √2 − 1 the eight-fold lines from neighbouring edges meet on the star polygon's chord, and every strap in the field lines up with every other, so an interlace can pass through the whole pattern without a kink. Move the depth off that value and the straps still meet, but they bend. The tradition's numbers are the straight-strap numbers, and they are the ones the geometry finds on its own. That is what "derived, not generated" means at the level of a single constant.