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

Here's a 22-gon made by bisecting the 11-gon:
https://sciencevsmagic.net/geo/#0A1.1L0.0L2.1A0.3A0.0L3.3L9.9A3.9L11.11A9.11L14.14L4.15A1.1A15.31L32.1A35.0A54.54A1.71L72.0A75.N.0A24.2A112.0L10.126A0.137L138.1L54.54A35.117L174.174L192.192L162.N.205A2.2A205.241L242.2A257.234A269.205A257.218A291.279A234.315L279.279L234.234L269.269L2.2L257.257L205.205L291.291L218.218L305.241L256.256L1.305A218.305L579.315A279.315L623.623A315.623L663.663A623.663L705.256A1.1A256.766L1.256L745.745A256.866L745.866L705.766A1.952L766.952A766.1019L952.1019A952.1019L1080.1080L579

Hmm... is my construction of the 17-gon the fastest (least amount of moves) for drawing it inside the origin circle?

Yes it still is! Will update. You can save a lot of moves by doing the star trick -- instead of drawing all the scalloped circles and then the lines, get two edges to intersect at the points of the 17-sided star, and then walk them around the shape.

I guess you're not going to add any more approximations? Is there an easy way to make approximations for an n-gon?

If you check the Wikipedia article on the 17-gon, it says that another more recent construction is given by Callagy, and the source is here: https://www.jstor.org/stable/3617271 (if you need to go over the paywall I have a copy from my university account)
It finds the central angle of the 17-gon directly, but I'm not sure if it's faster...
Maybe this is faster?

Here it is. https://i.imgur.com/g7yLseA.png

https://sciencevsmagic.net/geo/#0A1.1A0.0L1.1L5.0L4.2L3.4A1.1A4.12L13.6A0.0A6.24L23.28A0.28L18.52A0.0A52.69L70.79A28.70L93.93A92.92A93.141L142.155A28.17A174.189A17.228L229.243A17.108A0.0A108.335L334.352A0.0A277.108A420.N.452A0.13L490.490A0.0A490.547L548.570A18.571A18.597A571.584A570.640A584.656A640.672A656.0A1.686L672.672L656.656L640.640L584.584L570.570L18.18L571.571L597.597L616.616A597.616L938.938A616.938L992.992A938.992L1046.1046A992.1046L1100.1100A1046.1100L1158.686A672.1219L686.1219A686.1219L1266.1266L1158 here's that version: I tried to star it but I'm not sure how to do it so

If you look at my versions you’ll see what I mean: after doing the two three-node hops using the tangent line to N (total 9 hops), I mark a circle all the way across, giving two edges that converge at the tightest angle (largest star), and draw the giant circle at the star’s diameter. Since the 8 hop is relatively prime to 17, you can then use symmetry to walk the edges around without marking anything else off. Most of the other shapes in the catalog end up with some shape in the construction giving you a reflection for “free”, like the pentagon one does, but I haven’t found one here. Also, using G’ adds a step; constructing N on G gives you a tangent annoyingly close to the perp thru M but works and saves a step. Actually both tangents to N clip hit at vertices, Might be worth seeing if that can save any steps. One note: when you make the giant star the script can get super confused and crashy and evening sometimes adds three steps for extending a line for free. it seems to not like super long line segments, which is why you’ll see in some of those that I extended them in parts. Once you have the star set up it should only be 15 steps to the end.

…

On Thu, May 7, 2020 at 2:27 AM Eddy119 ***@***.***> wrote: ***@***.**** commented on this gist. ------------------------------ https://sciencevsmagic.net/geo/#0A1.1A0.0L1.1L5.0L4.2L3.4A1.1A4.12L13.6A0.0A6.24L23.28A0.28L18.52A0.0A52.69L70.79A28.70L93.93A92.92A93.141L142.155A28.17A174.189A17.228L229.243A17.108A0.0A108.335L334.352A0.0A277.108A420.N.452A0.13L490.490A0.0A490.547L548.570A18.571A18.597A571.584A570.640A584.656A640.672A656.0A1.686L672.672L656.656L640.640L584.584L570.570L18.18L571.571L597.597L616.616A597.616L938.938A616.938L992.992A938.992L1046.1046A992.1046L1100.1100A1046.1100L1158.686A672.1219L686.1219A686.1219L1266.1266L1158 here's that version: I tried to star it but I'm not sure how to do it so — You are receiving this because you authored the thread. Reply to this email directly, view it on GitHub <https://gist.github.com/a973b1c60f4a38fc3277ddd57ce65b28#gistcomment-3293871>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/AAABP4CYZVD6ALPVH6RGQFLRQJPEVANCNFSM4MX3P6PQ> .

I guess the next challenge is the 51-gon!
I gave up on the 13-gon... Still looking for a nice simple construction that tricks the code... Please send here if anyone finds one
Also, anyone know how small the error has to be to trick it?

Well, to construct a 13-gon, you have to trisect a certain angle... check http://web.archive.org/web/20151219180208/http://apollonius.math.nthu.edu.tw/d1/ne01/jyt/linkjstor/regular/7.pdf out for 3θ (3 times the desired angle) or https://commons.wikimedia.org/wiki/File:01-Dreizehneck-N%C3%A4herung.svg for a very good approximation... I tried to do both but I just couldn't...

Once we know how to approximately trisect an angle well enough to trick the website, we can pretty much make all of the polygons up to 40 except 23, 25, 29, 31

41, 43, 47, 53, 59, 67, 71... basically all the prime numbers that are neither fermat primes nor pierpoint primes

51-gon should be constructible as it is 17*3 and (2^n)*, 3* or 5* (7, 9, 11, 13, 19 , 37) should be approximately constructible as 7, 9, 11 is already done and 13, 19, 37 are pierpoint primes

Should be recognised by website except Bold:
2 (...)
3 (fermat no.)
4 (2^2)
5 (fermat no.)
6 (3*2)
7 (approx. constructible, pierpoint prime)
8 (2^3)
9 (3^2, approx. const.)
10 (5*2)
11 (approx. constructible as shown by me)
12 (3*2^2)
13 (should be approx. const. , please share, pierpoint)
14 (7*2, approx. const.)
15 (7*5)
16 (2^4)
17 (fermat prime, next fermat is 257)
18 (9*2, approx. )
19 (pierpoint, should be approx. const., please share)
20 (5*2^2)
21 (7*3, should be approx. please share)
22 (11*2)
23 (not pierpoint, if someone can approximate this, congrats, please share) 2025 update: @Doomslug682 did it below! copying for redundancy
24 (3*2^3)
25 (5*5, not pierpoint, if someone can approximate this, congrats, please share) 2025 update: also by Doomslug682 above, backup
26 (13*2, please share 13)
27 (9*3, should be approx. const., please share)
28 (7*2^2)
29 (not pierpoint, please share)
30 (5*2*3, please share)
31 (not pierpoint, please share)
32 (2^5)
33 (11*3, approx, please share)
34 (17*2)
35 (7*5, approx const because one is pierpoint and other is fermat)
36 (2*2*3*3, approx)
37 (pierpoint, approx)
38 (19*2, please share)
39(13*3)
40 (5*2^3) ...
41 (not pierpoint, please share)
42 (7*2*3, please share)
Italic involves trisecting an angle and Bold can't be made either by bisecting nor trisecting

I. Have. Done. The. 13. Gon. (13-gon , tridecagon , triskaidecagon , approximate)
https://sciencevsmagic.net/geo/#1L0.0A1.1A0.0L2.2A0.2L8.8A2.8L11.11A8.11L14.14A11.14L17.17A14.17L20.20A17.20L23.23A20.25L24.1L5.5A0.0A14.30A0.5L30.35L36.5A1.40L41.5A45.38A43.59A38.66L67.44A5.56L79.39A44.114L113.56A39.79L138.138L5.5A138.172A27.27A172.203L202.219A27.35L242.59A43.43A44.38L296.39A72.296L359.359A0.359L394.394A366.366A394.453L454.66L476.43A5.496L495.495L511.511A476.595L56.599L242.N.242L5.599A503.727A599.749L599.749A599.825L727.0A714.727L956.956A955.955A956.1043L1042.1043L0.0A1.1077A1.1119A1077.1144A1119.1168A1144.1186A1168.1202A1186.1A1077.1226A1.1253A1226.1275A1253.1299A1275.1317L1214.1214L1202.1202L1186.1186L1168.1168L1144.1144L1119.1119L1077.1077L1.1L1226.1226L1253.1253L1275.1275L1299.1299L1317
Source: https://n-polygon.blogspot.com/2012/01/13-gon-at-116e-11.html

19-gon (not in origin):
https://sciencevsmagic.net/geo/#1A0.0L1.1L2.2A1.2L5.5A2.5L8.8A5.8L11.11A8.11L14.14A11.14L17.17A14.17L20.20A17.21L22.15L16.0A1.0L28.27A0.1L27.27L32.32L0.33A0.32L41.38L27.41A33.38L52.41L51.51A33.33A34.41L67.41L68.79A41.85L84.95A78.24A25.21A17.17L21.21L139.139L20.139L144.0A144.20L151.151L117.151A117.117A151.235L234.8A255.277A255.255A277.283L8.283L293.25L290.290A25.0L393.393A25.25A393.N.290A25.393A25.25A393.388A393.462A25.556A462.388L598.556L508.508A388.598A556.393L720.676L462.676A508.25L817.817A676.393L901.598A388.598L976.976L901.720A393.720L1053.817L1053.556A508.556L1153.1153A556.1153L1201.1201L388.462A676.676A462.1299L462.676L1309.1299A462.1299L1414.1414L508.25A817.1309L1535.1535L25
Source: https://n-polygon.blogspot.com/2012/01/19-gon-at-198e-12very-simple-improved.html

19-gon in origin:
https://sciencevsmagic.net/geo/#1A0.0L1.1L2.2A1.2L5.5A2.5L8.8A5.8L11.11A8.11L14.14A11.14L17.17A14.17L20.20A17.21L22.15L16.0A1.0L28.27A0.1L27.27L32.32L0.33A0.32L41.38L27.41A33.38L52.41L51.51A33.33A34.41L67.41L68.79A41.85L84.95A78.24A25.21A17.17L21.21L139.139L20.139L144.0A144.20L151.151L117.151A117.117A151.235L234.8A255.277A255.255A277.283L8.283L293.25L290.0A356.290L359.359L412.412A356.356A412.471L472.472L506.506A505.505A506.582L583.0L34.N.621A34.34A621.677A34.709A677.738A709.646A621.784A738.874A784.916A874.960A916.677L916.709L833.833A709.833L1160.1160L916.709L960.960A709.960L1335.1335L646.646L738.738L1013.1013A738.1013L1624.1624L621.621L784.784A621.784L1809.874L34.34A874.1809L1920.1920L34.677A916.677L2061.874A34.874L2126.2126L2061

21-gon (7*3)
https://sciencevsmagic.net/geo/#1A0.1L0.0L2.2A0.0A2.1L5.0L6.6A0.2A1.6L11.11A16.5L21.2L3.16A11.3L24.24A21.1A34.34A1.55L54.58L0.0L59.58A0.100L58.100L118.118L59.1A118.100L150.58L157.157L59.59L179.150L179.12L13.100A12.12A42.13A41.179A13.267L179.179L13.13L258.250L12.12L100.100L237.250A12.250L162.162A250.162L484.258A13.201L258.201A258.201L571.571L484.237A100.237L656.656A237.656L58.58A656.58L760.760A58.760L815.815L0.0A815.267A179.267L936.936A267.936L59.59A936.59L1043.1043L916.916L0

25-gon didn't work: https://sciencevsmagic.net/geo/#0A1.1L0.0L2.1A0.3A0.1L3.3L9.9L4.9L0.0A10.10A0.26L25.30A2.30A1.25L51.10L45.73A64.64A79.79A64.73L99.14A0.120L121.144A0.0A144.181L180.180L209.209A191.209L217.217A191.191A217.223L230.230L235.235L99.181L206.N.196L235.196A257.257A196.312L313.343A344.344A343.400L401.0A1.313L344.1L432.432A1.485L432.485A432.485L539.1A432.1L606.606A1.606L660.660A606.706L660.706A660.706L750.433A2.2L433.433L801.539A485.539L852.852A539.852L11.11A852.1002L11.2A433.2L1076.801A433.801L1120.1076A2.1160L1076.1160A1076.1209L1160.1209A1160.1209L1265.1265L1002.1120A801.1120L1376.1376A1120.1376L1425.1425A1376.1425L1479.1479L750
Source:
https://n-polygon.blogspot.com/2011/12/25-gon-construction-at-88e-10very.html

Thanks Eddy! I'll batch these up at some point. I found a new (for me) trick when stellating a polygon to fill out the edges: a diameter of the circle goes through the points of the star, so if your construction can naturally have two diameters with enough of an angle spread you can avoid drawing a circle to reflect the vertices off. You can see that in the 20-gon solution I just put up, where it saves one move.

I think the problem with the 25-gon is that it places the point to the right of the bottommost point of the 25-gon somewhere else to where it should be... if someone wants to research this problem here's how I found out
https://sciencevsmagic.net/geo/#0A1.1L0.0L2.1A0.3A0.1L3.3L9.9L4.9L0.0A10.10A0.26L25.30A2.30A1.25L51.10L45.73A10.45L92.92A10.10A92.115L116.N.14A0.141L142.171A0.0A171.218L217.217L246.246A0.246L288.288A287.287A288.332L331.N.0A1.234L233.233L353.381A233.233A381.464L0.14A493.493A14.538L493.493L14.14L518.518A14.518L659.538A493.538L720.720A538.772L720.772A720.830L772.659A518.659L885.885A659.885L949.949A885.949L1016.1016A949.1084L1016.1084A1016.1152L1084.1152A1084.1212L1152.1212A1152.1212L1264.830A772.830L1317.1317A830.1317L1369.1369A1317.1369L1409.1409A1369.1409L1453.1453A1409.1499A1453.1499L1453.1499L15.15A1499.1264A1212.15L494.494L1264

I'm not sure how floating point math works but I guess it results in some less approximate constructions being recognised and some more approximate constructions not being recognised.

Could you use the spiral of Theodorus to create any-sided* shape?

spiral example https://sciencevsmagic.net/geo/#N.1A0.0A1.2A0.3L2.2L7.5L6.6L4.4L7.7L5.5L2.6L12.6A2.23L5.N.5L2.2L6.6L23.23L5.5L6.6L4.23A6.40L39.39A23.45L44.40A23.65L64.65L44.64L45.64L98.82A23.40L119.119L136.136L23.23L151.136L5.N.2L6.6L23.23L136.136L5.5L6.23L5.5L2.136A23.136L183.183A136.182L167.206A136.183L220.220L176.176L136.136L174.176L5.176A65.176L270.270A176.276L277.303A176.270L320.320L339.339L176.339L5.N.5L339.339L176.176L136.136L23.23L6.6L2.2L5.5L6.23L5.136L5.176L5.339A176.339L395.395A339.402L403.435A339.395L450.450L473.473L339.473A339.473L5.473L544.544A473.551L552.591A473.544L611.611L637.637L473.637L5.N.5L2.2L6.6L23.23L136.136L176.176L339.339L473.473L637.637L5.5L473.339L5.176L5.136L5.23L5.6L5.637A473.637L717.717A637.724L725.768A637.717L785.785L816.816L637.816L5.816A637.816L908.908A816.915L916.967A816.908L990.990L1023.1023L816.1023L5.1023A816.1023L1130.1130A1023.1137L1136.1193A1023.1130L1220.1220L1258.1258L1023.1258L5.1258A1023.1258L1328.1371A1258.1379L1378.1442A1258.1371L1465.1465L1507.1507L1258.1507L5.1507A1258.1507L1635.1635A1507.1643L1642.1710A1507.1635L1734.1734L1780.1780L1507.1780L5.1780A1507.1780L1914.1914A1780.1921L1920.1989A1780.1914L2014.2014L2064.2064L1780.2064L5.2064A1780.2064L2207.2207A2064.2215L2214.2292A2064.2207L2317.2317L2369.2369L2064.2369L5.N.2369L2064.2064L1780.1780L1507.1507L1258.1258L1023.1023L816.816L637.637L473.473L339.339L176.176L136.136L23.23L6.6L2.2369L5.2064L5.1780L5.1507L5.1258L5.5L1023.5L816.637L5.473L5.339L5.176L5.136L5.23L5.6L5.2L5

7-gon (made from 14-gon)
7-gon

23-gon (made by trigonometry [taking the tangent])
23-gon

25-gon (trig)
25-gon

26-gon (half of 13-gon)
26-gon

27-gon (trig)
27-gon

28-gon (from 14-gon)
28-gon

30-gon (from 5-gon)
30-gon

32-gon (from 16-gon)
32-gon

33-gon (from 11-gon)
32-gon

nice!
seems like 27-gon isn't recognized tho?
(I'm alive, i don't even remember what i did 5 years ago fully)

Next challenge is 23-gon and 25-gon in origin 😈

33-gon (from 11-gon) 32-gon

Wrong link? link is also 32-gon

edit: 33-gon is interesting because it's not a constructible polygon; 11 is already an approximation and trisecting that is another approximation, so hopefully @Doomslug682 comes back and gives us his 33-gon

edit 2: i just realized once you trisect a side of a 11-gon (line trisection can be done exactly), that's a 33-gon

edit 3: no it does not, if trisecting a side of a triangle gives you a nonagon, a nonagon (9-gon) would be constructible but it isn't... Doomslug682, please give us your 33-gon, even better would be explanations...

edit 4: new hypothesis: if you can approximate a non-constructible n-gon closely, you can also approximate a non-constructible n⋅m-gon, where m itself is constructible. edit/hypothesis 4.2: the n⋅m-gon will have the same amount of error as the n-gon. edit 4.1: you can overlay 2 fermat prime polygons together to get F_1 * F_2 gon, e.g. 5 * 3 for 15-gon but not 3 * 3 or 5 * 5 tho. (edit 4.1 is proven knowledge, Gauss-Wantzel Theorem) edit 4.1.1: for distinct Fermat primes p,q one can subtract integer multiples of 2π/p from 2π/q and bisect suitably to obtain 2π/(pq), so the pq-gon is constructible.