Ancient Greek Geometry walkthrough / answers / cheats

Ancient Greek Geometry.md

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

• Triangle, 5 moves
• Triangle, In-Origin, 6 moves
• Hexagon, In-Origin, 9 moves
• Square, In-Origin, 8 moves ; an elegant alternate 8-move solution
• Octagon, 13 moves by underwatercolor
• Octagon, In-Origin, 14 moves; Alternative by @mrflip; another Alternative by @mrflip
• Dodecagon, In-Origin, 17 moves alt
• Pentagon, In-Origin, 11 moves by John Chrysostom. Two non-in-origin solutions: by Thomas, alternative
  • Alternative, 16 moves based on this construction
• 10-Gon, In-Origin, 17 moves (16 reported possible)
• In-origin 15-gon: 22 moves by @mrflip
• In-Origin 16-gon: 24 moves by John Chrysostom (23 moves reported possible)
• 17-Gon, 45 moves by @mrflip, improving version from @Eddy119 citing H. W. Richmond — 40 moves reported possible! In-Origin, 49 moves by @Eddy119, tweaked by @mrflip
• In-origin 20-gon: 28 moves by @mrflip
• In-origin 24-gon: 30 moves by @mrflip
• In-origin 30-gon: 37 moves reported possible
• In-origin 32-gon: 40 moves reported possible
• 34-gon, 61 moves by @mrflip: 57 moves reported possible. In-origin 34-gon in 65 by @mrflip
• In-origin 40-gon: 51 moves by @mrflip, 49 moves reported possible
• In-origin 48-gon: 56 moves by @mrflip

Circle Packs

• Circles 2, 5 moves
• Circles 2, In-Origin, 7 moves
• Circle 3, 9 moves
• Circle 3, In-Origin, 10 moves by John Chrysostom
• Circle 4, In-Origin, 12 moves by John Chrysostom
• Circle 5, 22 moves by @pizzystrizzy
• Circle 5, In-Origin, 23 moves from @pizzystrizzy
• Circle 7, 13 moves by Jason
• Circle 7, In-Origin, 14 moves by @bikerusl
• Circle 15, 47 moves by @pizzystrizzy
• Circle 19, 37 moves by @ pizzystrizzy

Circumscribed Polygons

• Origin circle circumscribed triangle: 6 moves by John Chrysostom
• Origin circle circumscribed square: 10 moves
• Origin circle circumscribed hexagon: 11 moves

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):

• pseudo-2-gon in 11 moves
• pseudo-11-gon in 44 moves by @Eddy119

Octagon in 12 moves

oh mb. i thought a nonagon was impossible

I made this like a year ago and its pretty close to a 9-gon https://sciencevsmagic.net/geo/#1A0.0A1.2A0.5A0.4A1.3A0.1L5.0L4.16A15.16L2.16L21.16L22.16L27.16L15.16L19.20L4.16L25.18A8.9L41.41L42.15L42.6A40.42L66.66L9.73A41.83A73.86L87.73A91.87L85.27A118.122L121.15L122.121L20.121A122

Here’s a really good approximation for the nonagon.

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.

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

https://sciencevsmagic.net/geo/#0A1.1A0.2A0.4A0.5A1.0A8.1A6.13A3.12A3.3A2.2L32.32L31.31L30.30L33.33L2 pentagon, compass only construction

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

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