The Mighty Heptadecagon — a compass & straightedge guide

I · The problem of antiquity

Twenty centuries of silence

The rules were set by Euclid around 300 BC, and they are brutally simple. You get two tools: a compass that draws circles and a straightedge that draws lines — an unmarked one, no ruler ticks, no protractor. Whatever shape you want must emerge purely from where circles and lines cross. The Greeks could build the equilateral triangle, the square, the pentagon, the fifteen-gon, and anything you get by doubling sides. And then — nothing. For every other prime number of sides, no one could find a way in.

For two thousand years the educated guess hardened into folklore: what the Greeks hadn't constructed probably couldn't be constructed. Nobody proved it. Nobody could. The question just sat there, polished smooth by generations of failed attempts.

Then, on the morning of March 30, 1796, a nineteen-year-old in Braunschweig, Germany woke up early and, before getting out of bed, saw the whole structure at once. His name was Carl Friedrich Gauss, and he had not yet decided whether to become a mathematician or a philologist. That morning decided it. The regular seventeen-sided polygon could be drawn with compass and straightedge alone — and he could prove it. It became the first entry in his scientific diary, and he remained proud of it for the rest of his life: he asked for a heptadecagon on his gravestone. The stonemason reportedly refused — carved that small, seventeen sides are indistinguishable from a circle — but the monument to Gauss in Braunschweig stands on a seventeen-pointed star.

The discovery wasn't a clever drawing. It was a proof that the drawing must exist — the construction itself wasn't published until years later, and the elegant version of it took another century.

~300 BC

Euclid's Elements

Triangle, square, pentagon, 15-gon — and the doubling rule. The toolkit is frozen for two millennia.

1796

Gauss, age 19

Proves the 17-gon is constructible — before he's chosen mathematics as a career. First entry in his diary.

c. 1800

Erchinger

The first worked compass-and-straightedge recipe, produced within a few years of the proof — reviewed by Gauss himself.

1837

Wantzel

Proves the other half: if n isn't Gauss's kind of number, the construction is impossible. The 7-gon dies formally.

1893

Richmond

The minimal construction — the elegant version this page walks through, step by step.

2024

This journal

Drawn by hand, in ink, after several failed attempts — photographed below, mistakes and all.

II · The Fermat prime

Why seventeen, of all numbers?

Here is the whole secret, in one sentence: a compass and straightedge can only solve quadratic equations. Every point you can construct comes from intersecting lines and circles, and algebraically each intersection costs at most one square root. Chain the moves and you can nest square roots inside square roots — but you can never escape them. No cube roots. No fifth roots. Square roots all the way down.

Building a regular n-gon means constructing the angle 360°/n — equivalently, the number cos(360°/n). So the real question is: which polygons have a cosine you can write using nothing but nested square roots? Gauss's answer: it works exactly when n − 1 factors into 2s — because then the problem splits, level by level, into a tower of quadratics. For a prime number of sides p, that means p − 1 must be a power of 2. Primes of that shape have a name — Fermat primes, of the form 2^{2^k} + 1 — and only five have ever been found:

F₀

3

triangle ✓

F₁

5

pentagon ✓

F₂

17

this page

F₃

257

constructed 1832

F₄

65,537

a 10-year manuscript

Since 17 − 1 = 16 = 2·2·2·2, the seventeen-gon unfolds into exactly four levels of quadratic equations — hard enough that antiquity never found it, tame enough that it yields to a compass. Meanwhile its little cousin the regular 7-gon is flatly impossible — 7 − 1 = 6 carries a stubborn factor of 3 — as are 9, 11, 13, 14, 18, and 19. Every heptagon ever drawn with these tools is an approximation. The heptadecagon, absurdly, is not.

79111314171819 — constructibility of n-gons, 7 through 19 (odd territory)

III · The mathematics

Four quadratics, one cosine

Gauss's proof runs on a beautiful engine. Place all 17 vertices on a circle and read them as the complex numbers ζ^k. Their sum is 0 — that's one equation binding all of them. The trick is how you split that sum.

Gauss ordered the vertices not 1, 2, 3, … but by powers of 3 — because 3 is a "primitive root" of 17: its powers 3, 9, 27, 81, … cycle through every nonzero residue mod 17 exactly once. Written in that order, the vertices can be dealt like cards into two hands of 8, each hand into two of 4, each of those into two of 2. Every deal produces two sums whose sum and product you can compute — which means each pair solves a quadratic. Four deals, four quadratics, and the last one hands you cos(2π/17) itself.

The tower on the right shows the actual numbers cascading down — each level using only the level above it, plus a square root.

The tower of quadratics

i

x² + x − 4 = 0
x₁ = (−1+√17)/2 = 1.5615528…

ii

y² − x₁y − 1 = 0
y₁ = (x₁+√(x₁²+4))/2 = 2.0494812…

iii

y² − x₂y − 1 = 0 (x₂ = (−1−√17)/2)
y₃ = (x₂+√(x₂²+4))/2 = 0.3441507…

iv

z² − y₁z + y₃ = 0
z = (y₁+√(y₁²−4y₃))/2 = 1.8649444…

∴

cos(2π/17) = z/2 = 0.9324722…

The full derivation — Gaussian periods, dealt by hand＋

The deck, shuffled by 3

The powers of 3 mod 17, in order, are:

3^k mod 17: 1, 3, 9, 10, 13, 5, 15, 11, 16, 14, 8, 7, 4, 12, 2, 6

All sixteen nonzero residues, each exactly once. Now deal them alternately into two hands — the even positions and the odd positions:

x₁ = ζ¹+ζ⁹+ζ¹³+ζ¹⁵+ζ¹⁶+ζ⁸+ζ⁴+ζ² = 1.5615528…
x₂ = ζ³+ζ¹⁰+ζ⁵+ζ¹¹+ζ¹⁴+ζ⁷+ζ¹²+ζ⁶ = −2.5615528…

Because all seventeen vertices sum to zero, x₁ + x₂ = −1. Multiplying the two sums out and counting how often each vertex appears gives x₁·x₂ = −4. Sum and product known → x₁ and x₂ are the two roots of x² + x − 4 = 0. One square root (√17) and the first level is solved.

Deal again — hands of four

Split each hand by alternating positions again (equivalently: powers of 9 mod 17):

y₁ = ζ¹+ζ¹³+ζ¹⁶+ζ⁴ = 2.0494812… y₂ = ζ⁹+ζ¹⁵+ζ⁸+ζ² = −0.4879284…
y₃ = ζ³+ζ⁵+ζ¹⁴+ζ¹² = 0.3441507… y₄ = ζ¹⁰+ζ¹¹+ζ⁷+ζ⁶ = −2.9057035…

Each pair has sum equal to its parent (y₁+y₂ = x₁, y₃+y₄ = x₂) and product −1. So y₁ solves y² − x₁y − 1 = 0, and y₃ solves y² − x₂y − 1 = 0. Two more square roots, built from the previous level.

The last deal — hands of two

z₁ = ζ¹+ζ¹⁶ = 2cos(2π/17) = 1.8649444… z₂ = ζ¹³+ζ⁴ = 2cos(8π/17) = 0.1845367…

Their sum is y₁. Their product, multiplied out, is ζ¹⁴+ζ⁵+ζ¹²+ζ³ — which is exactly y₃, computed one level up. So z² − y₁z + y₃ = 0, and the larger root is 2cos(2π/17). Fourth square root, done. The cosine of the seventeenth of a turn has been captured using nothing but quadratics — which is precisely what a compass can draw.

Why this had to work

The deals succeed because 16 halves cleanly all the way down: 16 → 8 → 4 → 2 → 1. That's what "p − 1 is a power of two" buys you. For the 7-gon, the same machine jams immediately — 6 splits into 2 × 3, and a three-way deal costs a cube root, which no compass can pay. This is the entire difference between impossible and drawable.

Unwound into a single expression

= 0.93247 22940 43558… · the number the whole page is about · as inked on the finished drawing below

IV · The construction

Every arc, every point

This is H. W. Richmond's 1893 construction — the shortest known path to the 17-gon — expanded into its every elementary move: each compass arc, each straightedge line, each point born from a crossing. Nothing is measured, ever. The geometry below is computed to machine precision, and the final vertices land on the true seventeenths of the circle with zero error.

Step through it. Watch how long the machine runs before the first vertex appears — and how fast the rest fall once it does.

Step 0

tip: ← → arrow keys step too

V · The journey — in graphite and ink

Where mistakes compound

On screen, every intersection above is exact. On paper, nothing is. A pencil line has width. A compass needle walks a fraction of a millimeter when you lean on it. And the heptadecagon is uniquely unforgiving about this, because the construction is a chain: J is built from B, E from J, F from E, K from F, the vertices from all of it. Every small error is inherited by every step downstream, and interest is charged. By the time the compass walks the seventeenth chord around the circle, the accumulated drift shows up as one visible, damning gap — the figure doesn't close.

That's why the failed attempts hang here next to the finished piece. They aren't outtakes; they're the actual cost of the figure. Geometry of this order is a discipline of attention — the margin for error is the whole challenge.

Failed heptadecagon attempt no. 2 — pencil construction drifting off true in the sketchbook

Attempt № 2 · abandoned

The drift

Somewhere in the middle levels an intersection was taken a hair wide, and every arc after it inherited the debt. Abandoned when the error became visible to the naked eye.

Working construction sheet with labeled points and the vertices P5, P7, P12, P17 marked in red

The working sheet

The construction, labeled

The full Richmond scaffolding in graphite — axes, the quarter point, the tilted circles — with the hard-won vertices flagged in red as they emerged.

Audit sheet with red tick marks checking every vertex position against true

The audit

Checking seventeen truths

Red strokes across every mark: the post-construction audit, vertex by vertex, hunting for the compounding error before committing ink.

The finished inked heptadecagon — the complete graph of all 136 chords, with the hand-lettered title, Gauss's cosine in full nested radicals, and the signature

The finished piece · ink · 2024

The Mighty Heptadecagon —
all 136 chords

Not just the 17-gon: the complete graph on its vertices — every one of the 136 possible chords, inked by hand. The title block carries Gauss's cosine in full nested-radical form, and the interior angle, 158 14⁄17°, exactly.

Shown whole, uncropped — the figure, the formula, the lettering, and the signature are one composition.