Ancient Greek Geometry

Ancient Greek Geometry

ArtbegottiWhat can you make using only circles and straight lines? Yes, you can cut a pie into a million slices, but when you've got to make the pie yourself using only two dots as your starting point, the challenge becomes much greater. That's the basic idea behind Ancient Greek Geometry, a webtoy by Nico Disseldorp that uses basic mathematical principles to spin basic drawing into an intriguing puzzle.

Ancient Greek GeometryYour only two tools when playing with Ancient Greek Geometry are a straightedge and a compass (the kind that looks like a giant pair of tweezers, not like a GPS's compass). Starting with two given points, you can build your design by drawing a line between two points (click and drag to connect them), or a circle that has one point as its center and another point along its circumference (click and drag to the appropriate radius). Believe it or not, you can construct a lot of shapes with these techniques, including triangles, squares, and the elusive pentagon. This webtoy gives you the choice of tackling these shape challenges within the suggested numbers of moves, or freely building your own designs. How complex you get is entirely up to you in this intriguing diversion.

Play Ancient Greek Geometry


I don't understand, earlier there was at least seven votes, now there are three.

What's going on?

Repairmanman: Some votes come from spambots. There's a script that gets rid of those votes a few times an hour.

We have had an issue with someone spamming votes using scripts. Every hour the site checks for these fraudulent votes from that scripted source and removes them.

Does anyone know how to make a square in eight moves?
I think it's the first step towards getting octagon in 15.

A really well done app for these kind of construction. I might even use it for teaching. One addition I'd like to see is colour, that could make the tricker ones easier.

Pentagon and decahedron are the trickiest. The trick behind this is:

You need to construct a golden (ratio 1+sqrt(5))/2.
You can get sqrt(5) by

Constructing a right angle triangle with width 2 and height 1. The other side will have length sqrt(5).

i was able to get a few of the designs but i'm not enough of a math nerd to figure out more than i've got. i did make something that looked like an owl or bug or whatever and colored it in photoshop for fun.

singsurf: there's decahedron? I don't see it, and it would be impossible as a 3D figure. If you're talking about dodecagon, then it's pretty easy

if you work forwards from the hexagon

the same way you got from the triangle to the hexagon.

For my part, I'm still on 32/40, working on move-minimal for four of them and all of pentagon.

This = brilliant. Just enough minimalist gamification to turn a deadly boring subject into "grr, so close...but what if I...hrmmm...".

Still wouldn't catch some people's attention, but it's hooked me.

Amusingly, a "walkthrough" for this game would be equivalent to a textbook on ancient Greek geometry, with the more rigorous proofs stripped out.

This game makes me happy!

For anyone who is interested in this stuff, "Sacred Geometry," please check out Spirit Science:

Lessons 5 and 10 deal specifically with this geometry.

@repairmanman To get the square in 8 moves I suggest

using 3 circles of the same size that intersect like a Venn diagram.

Hope that helps.

I did it:V
I finally did it.
Thanks vanchan.

Overall this game is loads of fun, but I'm not too good at making the shapes in smaller amounts of moves.
Plus it refuses to recognize some of my shapes, which kind of sucks.

Oookay...I constructed a decagon instead of a dodecagon, and the game recognized it and added "10-GON" into the list of challenges.

Are there any other hidden challenges? (Maybe I haven't unlocked them yet--only at 31/40 so far.)

I seem to be having a perpetual case of the off-by-one solution. I have a 15-move solution for Circle Pack 7, a 13-move solution for Circle Pack 4, and a 16-move solution for Pentagon. Those are the only ones I still need.

Haven't found any other hidden challenges, other than the 10-gon.

Never was very good at optimization, still having problems getting minimum moves for the square, octagon, circle pack 4, and. pentagon. Also, I don't know if this is a glitch or I'm just misinterpreting the requirements, but I've been able to get circle packs 3 and 4 centered over the origin, but haven't gotten the reward for it. Any ideas?

As for the hidden challenges, I can indeed confirm that there are more then just the 10-gon. In the spoiler below, you can find the four I've found, and some helpful hints on how to get them.

In order of difficulty to get:


Either double a pentagon, or modify the ending of the process to halve the final length of the sides.


Just double a decagon, no problems.


Start off doing the same as the decagon, but at the end do a few modifications/additions to shorten the final side length slightly.


Good luck. You'll need it.
Remember, the new layer button is your friend.

I did it 66 moves... so many circles *shudder*.

@lzhornyak: The "origin circle" ones mean that the circumscribing circle should be centered on one original dot and pass through the other.

Also, Circle Pack 7 is possible in 13 moves!

I'm pretty sure that any regular polygon will be recognized. I built a 16-gon and a 24-gon, and it recognized both of those. Then, for good measure, I built a 48-gon, and it recognized that. I think I'll give it the benefit of the doubt and not test it with a 96-gon.

@yuruigon: I figured as much, though if you're up for a real challenge try you hand at building a 17-gon. Unlike the 16-, 24-, and 48-gon, it requires a completely new construction method and cannot be simply multiplied off of a smaller polygon.

@Buttons: I really cannot figure out how to do that then. I cannot for the life of me figure out how to get the internal circles the proper size and place.

Huh, I had built a 15-gon, but it did not recognise it. Maybe because I used the 5-gon as a template?

Speaking of the arbitrary sided polygons, the non-constructible ones (7, 9, 11, 13, 14, 18, 19, 21, 22, 23, 25, ...) can be "constructed" and recognised by the game if you do a good enough approximation by square roots and integers (then divide by a integer). It's because javascript does roundings.

17-gon for those interested


Just place that code after the geo/ in the address bar.

@lzhornyak: Yeah, it can be pretty tricky. For circle packs 3 and 4, try figuring out what the radii of the inner circles should be in order to pack into the unit circle, then try constructing those lengths. For instance, for circle pack 3, that length is 2sqrt(3)-3. Once you have that length, copy it over to the origin, and you should be all set.

Another way to do it (this is how I solved the pentagon) is to make a bigger version elsewhere, then project the lengths down so they fit in a unit circle.

@Buttons: For many of the origin circle problems, I used roughly the same technique you've described for the pentagon.
I (partially) constructed the shape at any size, then projected it down to the unit circle.
While I did manage it, I found the projection quite complicated (and error prone). I used parallel lines for the projection, but doing a single parallel line (eg. find the line through point C that is parallel to line AB) took me about 10 operations each time. And that's assuming lines AB and AC already exist.
I'm wondering if you have a simpler technique for the projection?

Drawing a parallel shouldn't take as many as 10 operations. In 4 steps, assuming line AB and point C exist:

Draw a circle around A (or B), through C. This circle intersects line AB in two points, call one of them D.
Draw a circle around C, through A (or B).
Draw a circle around D, through A (or B).
These two circles intersect in A (or B), and in another point. Call this E.
Draw line CE, this line is parallel to AB.

ACDE (or BCDE) is a rhombus, which have parallel sides.

@Buttons: Ah, I figured it out. Not sure if what I was doing could be called projection or not, but it sure was fiddly.

I was using parallel lines to map points on one line to points on another, intersecting line.

But now I'm using lines that meet at a point to map points on one line to points on another, parallel line.

Either way it effectively scales the points to the desired size, but this new way is a lot easier.

@tchakkazulu: Thanks for this! It's a lot easier than what I was doing.

After trying to wrap my head around this concept, I find this to be quite fun. But I fail to see how even the simplest of shapes can be done in the minimum amount of moves. The only one I've done is the hexagon in 10 moves, and I can't get less than 6 moves on a triangle.

Given that you need three lines for a triangle, you have to create all 3 points with 2 circles, and I just don't see how that's even remotely possible.

Thanks Steve, I was so focused on it having to be inside the circle, that I didn't even realize this possibility.

I haven't found the majority of low move solutions but here's one that beats the target - a hexagon in the origin circle in 9 moves rather than 10:

To see how it's done use CTRL-Z repeatedly to rewind it and then CTRL-X to play it forward.

Now, can someone please put me out of my misery and put up a link for a square in 8 moves.

I'm baffled as to how that 8 move square is done. I see that it works, but I can't see how you'd come to the conclusion that the left and right point are in the right place, when it seems to me they should be a tad higher up. I.e. I don't understand the math behind it.

I also can't figure out how to get the 5 move Circle Pack 2.

I need 2 moves to make each of the centers for the 2 small circles, and 2 moves to draw each of them. That only leaves me 1 move to draw the origin circle and nothing else.

@ThemePark: Not sure how you're going about that circle pack but

Hint 1:

the origin circle is one of the small circles

Hint 2:

you need four circles and one line


@ThemePark: Here is a graphical representation for the math of the square. If it doesn't make sense I'll try to explain in better detail.

Thanks kailando, those hints did the trick!

kailando, thanks for that. [kicks self] I'm now wondering why I never saw it before.

Okay I give in, can someone please help me with this freakin pentagon?


Here's a pentagon in a target-beating 14. I can't really give hints.
It's a non-intuitive method which you can see by Googling for inscribed pentagon (I can't seem to include a specific link)

This is a really interesting game. Keeps the mind fresh. My only concern is the need for some instructions or a small tutorial before starting the game. Can someone please explain what the "in origin circle" means? I finished the triangle, hexagon and square both with the "in origin circle" highlighted. I assume b accident since I can't get the same result for circle pack 2 and 3. Circle pack 2 in 5 moves but the "in origin circle" is still faded and circle pack 3 in 12 moves with the same thing. Can anyone please explain or hint?

Hints for the 15-move pentagon:

Some edge and diagonal lengths of a regular decagon are easier to construct than others.

I constructed my pentagon in a circle with radius twice the orignal distance.

Very nice game, loved it!

Cheers Tazzmania for the pentagon in 14 moves, that's a VERY clever use of +1 and -1 on an elegant construction of sqrt(5). I just couldn't get below 17 moves.

And now to spend the night trying the 17-gon :D

oh my gosh. This is the kind of pointless time waster that I can wind up sacrificing hours of my life to!
I'm going to screencap my final pattern and fill it in with colors on GIMP. Just to add icing to this time-wasting cupcake. ;)

What counts as a move? I just used Tazzmania's 9-move hexagon solution, but it takes a lot more than 9 moves...yet the game registered it fine.

I'm so confused...


I understand your confusion. Here's the answer:

If you already have a short straight line, you can extend it for free without incurring an extra "move".
You can do this as many times as you want in both directions but you must have something to intersect with to form the new end point.

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