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@mrflip
Last active January 29, 2026 22:54
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Ancient Greek Geometry walkthrough / answers / cheats

Solutions for Ancient Greek Geometry (https://sciencevsmagic.net/geo)

Most solutions taken from the about thread. See the comments below for more additions since my last check-in.

Polygons

Circle Packs

Circumscribed Polygons

Non-Constructible Figures

Abuse of floating-point math can make the widget approve non-constructible polygons (polygons with edge count 7, 9, 11, 13, 14, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 35, ..., which cannot be precisely constructed using straightedge and compass):

@ILoveMath62
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Two hexagon alternatives: 1. 9 moves 2. 9 moves

@ILoveMath62
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Line segment with length π with an error of only 8.49e-8

@ILoveMath62
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@Furlii
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Furlii commented Jan 1, 2026

hi

@ILoveMath62
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Hello!

@ILoveMath62
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Have you made any polygons yet? If yes, share some.

@ILoveMath62
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Finally made the 59-gon after many, many attempts. 87 moves.

@ILoveMath62
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@Ian-Zander, the reason why my link for the 50-gon originally led to the 49-gon was pretty much the same reason as when the link that @Doomslug682 posted for the 33-gon led to the 32-gon.

@ILoveMath62
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Octagon in 12 moves

@elijahdarcydominguez-afk

@ILoveMath62
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@ILoveMath62
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ILoveMath62 commented Jan 10, 2026

Regular polygons whose edge count is a Pierpont prime are constructible with angle trisection, but the current known angle trisector doesn’t do it perfectly. “You had one job angle trisector. One job.” Let’s say that it does perfectly. Could all regular polygons whose edge count are 1 above a multiple of 3 be constructible with angle trisection? Please figure this out. If it’s true, then a regular hendecagon is constructible with angle trisection, because the icosidigon would be, which is great. If it’s false, try doing it yourself. If it still turns out as false, that’s great too, because we want extremely small errors to the polygons we construct. Floating point causes the widget to be tricked into thinking the polygons are regular. Heptagon by taking the tangent in 47 moves

@Eddy119
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Eddy119 commented Jan 11, 2026

  1. Exact Angle trisection is impossible with just a compass and unmarked straightedge
  2. It's not as simple as 3n+1, this takes some effort to explain, but check the Wikipedia article on Pierpont primes for now
  3. 11-gon is a prime number but isn't a Pierpont prime so it isn't trisector constructible (but it apparently is neusis constructible, which I'm still intrigued about)
    Anyway for approximations without a trisector/neusis I'm not sure if Pierpont/trisector constructible polygons make it simpler to approximate... Anyway that tangent trick seems to work for all polygons

@Furlii
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Furlii commented Jan 12, 2026

@Furlii
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Furlii commented Jan 12, 2026

this app doesn't let me take arbitrary points on the plane

@Furlii
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Furlii commented Jan 12, 2026

if we construct a composite sided polygon, say 15, and let V0, V1, ... ,V13, V14 be the vertices ordered counterclockwise, then there are 8 vertices of our interest, not 14
V1, V2, V4, V7, V8, V11, V13, V14

if prime p-sided polygon, then there are simply p-1 vertices of our interest, example 7 :
V1, V2, V3, V4, V5, V6
the other vertices V0 (which is constructible) is a matter of choice, ideally it is point (1,0) but it can be anywhere

ref
https://en.wikipedia.org/wiki/Euler%27s_totient_function
https://en.wikipedia.org/wiki/Cyclotomic_polynomial

@ILoveMath62
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ILoveMath62 commented Jan 13, 2026

First of all: it never lets you take arbitrary points on the plane. You have to construct them by compass (which draws circles) and ruler (which when using pencil, draws lines) Getting some point (let's say (7,4)) requires getting other points within the coordinate's x and/or y. (for example: (7,0)) You can't do it instantly.

@Furlii
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Furlii commented Jan 14, 2026

I see. I usually do geometry constructions in geogebra.

also if you're wondering why, theres a regular pentagon construction that uses only 1 circle but it requires constructing a line across an arbitrary point. (Steiner-Poncelet construction)

@Furlii
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Furlii commented Jan 14, 2026

reference for compass only pentagon construction : https://www.cut-the-knot.org/pythagoras/MascheroniPentagon.shtml (Mohr-Mascheroni construction)

@Furlii
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Furlii commented Jan 14, 2026

you can construct a regular hendecagon via two angle quintisections btw. The short answer why regular hendecagons are not constructible by angle trisector is because the field extension ℚ(ζ+ζ⁻¹)/ℚ where ζ are 11th roots of unity, has degree of 5, not 3.

@ILoveMath62
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I already know that. 11 mod 5 is 1, so it’s constructible via angle quintisection/pentasection.

@Furlii
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Furlii commented Jan 16, 2026

I wouldn't say it like that (6 mod 5 = 1, 25 mod 5 = 0).

More like ϕ(11) = 10 = 2 * 5. The modulo doesnt exactly say anything unless we're talking about prime-sided regular polygons.

@ILoveMath62
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But 11 is prime!

@ILoveMath62
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ILoveMath62 commented Jan 19, 2026

Since 11 is prime, it should count in this case, right?

I wouldn't say it like that (6 mod 5 = 1, 25 mod 5 = 0).

More like ϕ(11) = 10 = 2 * 5. The modulo doesnt exactly say anything unless we're talking about prime-sided regular polygons.

@ILoveMath62
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ILoveMath62 commented Jan 21, 2026

occ (origin circle circumscribed) decagon: 21 moves occ dodecagon: 18 moves

@ILoveMath62
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@ILoveMath62
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@elijahdarcydominguez-afk, your nonagon does count. The error is small enough to trick the widget.

@ILoveMath62
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Nonadecagon in 40 moves by taking tangent

@ILoveMath62
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Here’s a neusis construction of the hendecagon (took me a while to understand)

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