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

I've been trying to find the 31-gon through brute force, but floating point keeps getting in the way.

Yeah. Makes sense. Did you think it was even possible to optimize the heptagon?

I literally joined yesterday.

31-gon by taking the tangent: https://sciencevsmagic.net/geo/#0A1.1A0.0L1.1L5.5A0.5L8.8A0.8L11.11A0.11L14.14A0.14L17.17A0.17L20.20A0.20L23.23A0.23L26.1A4.4A1.0L59.62A0.59L77.77A0.77L92.92A0.92L107.107A0.107L120.120A0.120L133.116A114.133A142.133A114.133L157.157A170.157L173.170A157.178A157.182L181.170A184.184A170.204L205.209L26.23A1.N.26A275.278L279.279L285.26A285.293L275.275L305.285L304.304L315.279L275.275L324.278L296.279L323.324L314.314L337.305L336.336L351.315L350.350L366.337L365.365L384.351L383.383L407.366L406.406L431.384L430.430L450.407L449.449L470.431L469.469L491.450L490.490L512.470L512.512L535.491L535.535L559.293L278.278L581.323L580.580L603.296L602.602L626.581L625.625L650.603L649.649L675.626L674.674L703.650L702.702L746.675L745.745L776.703L775.775L805.746L804.804L835.776L834.834L866.805L865.865L897.835L897.897L931.866L931.931L965.965L559

In your 37-gon attempt, move 26 scrambled (weird floating point stuff I don't understand). It basically places multiple lines at once

Something I've found while brute forcing is that changing the scale sometimes fixes it

I don't know where @pizzystrizzy, @bikerusl, @ralpjs, @mrflip, @Doomslug682, or @WaiLFiN went.

Hi, interesting stuff, I am here, I can't look at all of this right now, but I'll look into it I'm the near future, I'm still interested, and wondering how to neusis construct a 11-gon...

Maybe we should make a discord for all of this? It might make discussions easier...

(My discord is the same as my GitHub)

A discord sounds cool

Hint for faster possible 21-gon

My game isn't accepting it
edit: I was tweaking

Approximations to be done for more shapes:
43 (approximation used by taking the tangent, please share) Late Nov update: I finished it!
44 (11x4)
45 (5x9)
46 (23x2)
47 (use by tangent, please share) Early Dec update: finished by me!
48 (2⁴x3)
49 (use by tangent, please share) Early Dec update: Finished by me!
50 (25x2)
51 (use triangle on heptadecagon, please share)
52 (13x4)
53 (use tangent, please share) Mid-Dec update: finished by me!
54 (3³x2)
55 (5x11)
56 (7x8)
57 (3x19) Mid-Dec update: finished by me!
58 (29x2)
59 (use tan, please share) New Year's update: finished by me!
60 (3x4x5)
61 (use tan, please share)
62 (2x31)
63 (7x9)
64 (2⁶)
65 (13x5)
66 (2x3x11)
67 (use tan, please share)
68 (4x17)
69 (might be done with trisecting 23-gon, but might fail, please share)
70 (2x5x7)
71 (use tan, please share)
72 (8x9)
73 (pierpont, approx.)
74 (2x37 [please share 37]) Late November update: 37-gon is finished!
75 (3x25)
76 (4x19)
77 (7x11)
78 (2x3x13)
79 (use tan, please share)
80 (2⁴x5)
81 (3⁴ [use tan, please share])
82 (2*41 [please share 41]) Thanksgiving update: 41-gon finished!
83 (use tan, please share)
84 (3x4x7)
85 (5x17 [perfectly constructible as being a multiple of two distinct Fermat primes])
86 (2x43 [please share 43]) Late Nov update: I finished the 43-gon!
87 (3x29)
88 (8x11)
89 (use tan, please share)
90 (2x5x9)
91 (7x13)
92 (4x23)
93 (3x31)
94 (2x47 [please share 47])
95 (5x19)
96 (2⁵x3)
97 (pierpont, approx.)
98 (2x7² [please share 49])
99 (9x11)
100 (4x25)

Italic involves trisecting an angle and Bold can only be made by taking the tangent.

Ohhhhh... maybe instead of taking the tangent for the required angle, what if we take the tangent for 2 times the angle? That should give us more shapes.

I suspect approximating polygons with pierpont prime number of sides is probably not that much easier or different than approximating the non-constructible ones that aren't? Pierpont might be only relevant with neusis or other angle trisection, what's your experience? Are Pierpont polygons easier to approximate?

I guess if you had a reliable way to approximate angle trisection it could be easier here...

9-move Circle Pack 3 alternative

If you take the 11-move pentagon and remove the pentagon itself, you get the angle for the triacontagon. Using the star trick, this is the only possible 37-move triacontagon. (I don't know why it takes so long to render)

If you take the 35-move hendecagon (the one I would've shared if I'd never made the 28-move one) and bisect, you get a 46 move icosidigon. I can't share it though because it glitched. This is the link to it before it glitched. Finish the last move to make it.

From the 75-move nonadecagon, (enneadecagon) I decided to make different circles to get the points required, and actually get the points not on the x-axis, and form the parallel line needed to make the circle required, and use the star trick +several more to get 56 moves.
Edit: I managed to star with the starting 8π/19 angle to save 2 moves from the nonadecagon. New nonadecagon
Edit 2: I made the parallel line in a different way to save 7 moves.

Circle Pack 19 in 36 moves instead of 37. Circle Pack 15 in 41 moves. If the Circle Pack 15 had each inner circle r=1, the big circle would have r=1+sqrt(6+2/sqrt(5)+4sqrt(1+2/sqrt(5))), or this.

I took a different tangent for the icosiheptagon to make it in 49 moves.

Sharing decagon since the one shared here has an incorrect move. Dodecagon alternative (also 17 moves)

This probably explains why the widget sometimes glitches while rendering shapes. The 24-gon looks like this if you change line 28L33 into 28L10. This will load slowly.

I don’t know why these approximations even exist. 19-gon A 19-gon B nonagon

I removed 3 unnecessary moves from the icosanonagon to get 57 moves.

Hi, interesting stuff, I am here, I can't look at all of this right now, but I'll look into it I'm the near future, I'm still interested, and wondering how to neusis construct a 11-gon...

Maybe we should make a discord for all of this? It might make discussions easier...

(My discord is the same as my GitHub)

Eddy119, this reply you made has a grammar error. "look into it I'm the" Change I'm to in, please.

If you star with twice the starting angle of the heptadecagon, you can get 38 moves.

I managed to remove an unnecessary move on the 41-gon to get 62 moves. (I don’t know why it counts. This can be done)

13-gon by taking the tangent in 30 moves:
13-gon
49-gon by taking the tangent:
49-gon
50-gon from 25-gon:
50-gon

26-gon in 43 moves 18-gon in 33 moves 23-gon in 46 moves 11-gon in 27 moves

The 16-gon had 23 moves reported possible, finally I constructed it from the octagon.