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.
- 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
- 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
- Origin circle circumscribed triangle: 6 moves by John Chrysostom
- Origin circle circumscribed square: 10 moves
- Origin circle circumscribed hexagon: 11 moves
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
With Carlyle circles, you can actually pick which vertex you want to construct (e.g. to star trick from); so, you can construct any of k in (2cos(k*2π/17)) in the same number of moves, which is 19 moves in origin, 17 moves 2x circle (so far).
Vertices (1, 13, 16, 4); (9, 15, 8, 2); (3, 5, 14, 12); (10, 11, 7, 6)
This is all from this 1991 paper by DeTemple titled Carlyle Circles and the Lemoine Simplicity of Polygon Constructions.
Feel free to try to find ways to reduce the number of steps, the ones above are in-origin, so 2x unit circle would be 17 moves, but maybe this can be simplified further; and then basically I bisect the (2cos(k*2π/17)), but maybe there's a trick to reduce the 3 moves involved to bisect, maybe there's a way to get both of the roots at once without having to bisect for each (e.g. 3 and 5), and maybe one of them makes it faster to star, like what I did yesterday.
Basically, to solve for sum of roots = S, product of roots = P, you need to solve the quadratic x^2-Sx+P=0, and a Carlyle circle is when you draw a circle with center (S/2, (1+P)/2) and edge (x=0,y=1), and the circle intersects at (x=roots, y=0). See https://en.wikipedia.org/wiki/Carlyle_circle
As for the 51-gon, you just have to use the vertical -1/2 line (left side of triangle) to cut between the 5th and 6th vertex, like this.
It's pretty easy, you can probably use the
3, 5, 14, 1210, 11, 7, 6 link to build it quickly.UPDATE: This is what I'm talking about, you can finish the 51 sides.UPDATE 2: Use this, uses 6th vertex.I'm wondering, @ILoveMath62, for approximations of polygons that can be exactly constructed with a trisector, do you have a specific angle trisection approximation technique, or do you just do different things for every polygon?