xEuclidxSeptember 8, 2016 10:29 AM   Subscribe

Compass-and-straightedge construction (aka Euclidean construction) is a method of drawing precise geometric figures using only a compass and a straightedge (like a ruler without the markings). MathOpenRef maintains a catalog of many common constructions, each with an explanatory animation and a proof. This YouTube video demonstrates how to construct almost every polygon that can be constructed using these methods.

Useful in many decorative arts for laying out figures and patterns, the methods can also be fun to watch, often resulting in an "aha!" moment when one sees how a particular construction is done.

Although not animated, this paper [pdf] shows how geometric constructions can be used to create many common Gothic design elements, such as trefoils and arches.
posted by jedicus (20 comments total) 68 users marked this as a favorite

A couple of entertaining games that teach you a lot of this stuff are Pythagorea (iOS, Android) and Euclidea (iOS, Android).
posted by pipeski at 10:39 AM on September 8, 2016 [7 favorites]

Finally, a guidebook for how to beat ANCIENT GREEK GEOMETRY.
posted by timdiggerm at 10:39 AM on September 8, 2016 [3 favorites]

Wow, the method for building a regular 17-sided polygon in that video is intense.
posted by painquale at 10:40 AM on September 8, 2016 [1 favorite]

yeah, isn't it? I got to the end of the 17-gon and went are you fucking kidding me?! so that's my mind blown for the day.
posted by Mary Ellen Carter at 10:46 AM on September 8, 2016 [1 favorite]

Euclid mentioned previously on the blue is another awesome game to play
posted by Dr. Twist at 10:50 AM on September 8, 2016 [2 favorites]

I learned how to do this in my geometry class.

Also, A++ title.
posted by chainsofreedom at 10:54 AM on September 8, 2016 [2 favorites]

why is the title A++? I don't get it. It's the name 'Euclid' with an 'x' on either side. My 5 year old could do that!
posted by thelonius at 11:31 AM on September 8, 2016

why is the title A++?

Straightedge. Straight edge (wiki exp).
posted by btfreek at 11:49 AM on September 8, 2016 [2 favorites]

These links are gold, thank you.

I'll add, for old school visualization: Oliver Byrne's Euclid.
posted by gwint at 11:54 AM on September 8, 2016 [1 favorite]

I love these constructions, and they take me back to my drafting class in high school. There used to be a different site where I could do them on-line, but it seems impossible to find. I appreciate the Euclid and Euclidea links above, and there's also in-browser Euclidea.
posted by cardioid at 12:11 PM on September 8, 2016 [2 favorites]

No one has ever been able to trisect an angle with just a compass and straight edge. That is all I remember from geometry.
posted by Uncle Grumpy at 12:22 PM on September 8, 2016

No one has ever been able to trisect an angle with just a compass and straight edge. That is all I remember from geometry.

You can't trisect an arbitrary angle that way, but you can trisect a right angle easily. This distinction, learned after the fact, let the wind out of my sails as a middle schooler convinced he had stumbled onto something really unique!
posted by jason_steakums at 1:15 PM on September 8, 2016 [1 favorite]

What, no 257-gon?

Wow, the method for building a regular 17-sided polygon in that video is intense.

Here's a longer video describing the 17-gon construction, and the gory mathematical details.
posted by Johnny Assay at 2:20 PM on September 8, 2016

One of my go-to sites for constructions (in addition to Cut-the-knot, which is fabulous for all sorts of things) is Whistler Alley. It's got a really nice section on conics; I found it searching for instructions for how to construct a conic passing through 5 points.

(I needed to construct a conic through 5 points because...it turns out to be relevant to a research paper I'm writing. And despite being a PhD-holding mathematician, I never really learned a lot about conics. Despite the fact that the ancient greeks (e.g., Apollonius) knew really astonishing amounts about conics, and this was only extended in awesome directions by later mathematicians. But they've sort of fallen out of the curriculum these days---I don't have time to teach them in my Euclidean Geometry course, since students don't really get much euclidean geometry in high school, and they're not anywhere else in the curriculum either. Maybe a little tiny bit in a precalculus course, but there really from an algebraic, rather than geometric, point of view.)

You may be amused to learn that the construction of the 5-point conic is basically based on Pascal's theorem, which says that if you put 6 points on a conic (e.g., an ellipse) and connect them up to form any sort of hexagon (not necessarily convex), then the intersection points of pairs of opposite sides of the hexagon will all three line up on a single line. This is cool, because in general, three points in the plane are *not* collinear. (Pappus' theorem, which the paper I was working on is related to, says that given any two distinct lines in the plane, if you put three points A, B, C on the top line, and three points A', B', C' on the bottom line, and intersect the corresponding pairs of lines: AB' and BA' intersect at P, AC' and CA' intersect at Q, and BC' and CB' intersect at R, then the three points P, Q and R are collinear. The entire collection of points and lines then forms what is known as a point-line configuration of 9 points and 9 lines, where every point has 3 lines passing through it and every line has 3 (of my specially chosen) points lying on it. If you view a pair of lines as a degenerate conic (think about what happens to a hyperbola as the bends get sharper and sharper), with the understanding that you're actually living in the projective plane, so that every pair of lines intersects, then Pappus' theorem is just a special case of Pascal's theorem!)

One of the awesome things about living in the future, as we all do, is that you can do geometry as an experimental process. Dynamic geometry software is awesome. To really believe that a theorem---say, pappus' theorem---is true for *any* pair of lines in the plane, and *any* points on those lines, there's nothing better than slapping down two lines on the plane, slapping down 6 points, drawing the required lines, and then dragging everything around. Whatda ya know! The special points stay collinear! It's awesome!!

(My go-to dynamic geometry software program is The Geometer's Sketchpad, just because I've been using it a long time, but if I were starting fresh right now, I'd probably use GeoGebra, which is free, actively under development, and has a decent iPad client as well (which may be enough to get me to switch, at least for teaching if not for research).)

It is a classical (for values of classical that include the 1800s) result that only certain regular polygons can be constructed using a straightedge and compass. It's easy to construct a triangle and a square, and it's easy to bisect angles, so it's easy to construct, for example, 2^k-gons and 3*2^k-gons for any k. It was an active area of research for a while, culminating in Galois theory, to determine for which n it is possible to construct regular n-gons. (Actually, it's still open as to how many odd-sided constructible regular polygons there are, because the classification of the Fermat primes is still open.)

However: this does not mean it's impossible to construct other regular n-gons, just that you need tools other than a compass and an unmarked straightedge! In fact, there's a fairly accessible construction for a regular heptagon, as long as in addition to your compass and straightedge, you're willing to allow either a marked ruler and sliding (neusis) or an angle trisector, or using paper-folding techniques.
posted by leahwrenn at 4:06 PM on September 8, 2016 [14 favorites]

I learned this in an euclidean geometry class I took in highschool for extra credit. We did not practice a lot, it was mostly book learning and a couple whiteboard demos.

Then in college I had to do it all over again as a prerequisite for technical drawing. I loved the ritual of cleaning the desk, stretching and taping the giant sheet of paper, sharpening the pencils, refilling the drafting pens, cleaning the straight edge to optical perfection. The first pencil line on the virgin paper feels so good.

Inking is the best part. If you get distracted and make a mistake, you can lose hours and hours of work. If you go carefully and slowly and with mystic level concentration suddenly you are done, and you have brought beauty and truth from a perfect world into this messy one.

They introduced CAD the following year. It is faster and easier and way more practical, but it does not feel the same way.

A couple of times I've done this on a cement floor with sidewalk chalk and a few pieces of strings and rocks. It makes me calm and happy. I imagine it is similar to mankind sand mandalas. I recommend it to evryone.
posted by Doroteo Arango II at 9:04 PM on September 8, 2016 [3 favorites]

Heh; by complete coincidence, I bought Andrew Sutton's Ruler & Compass book in Powell's today while in grockle mode.

I do a lot of geometric constructions, mostly in Inkscape, but some in Geogebra. I'm currently trying to work out the proportions of the perfect 12-pointed rosette. This is pretty close, but not quite there.
posted by scruss at 10:28 PM on September 8, 2016 [1 favorite]

I was on the college paper and our graphics guy was a math major who used to make very elaborate and precise graphics this way, right as computer graphics was coming in and your only tools were lines and circles. The AP had recently started syndicating collegiate content and every now and then we'd get a call from an AP graphics guy going, "Hey, how'd your guy do that?"

I vividly recall the EU neglecting to ship the euro symbol for fonts in time, or even a euro symbol graphic for news stories with the press releases, when the euro came in, and him constructing a gorgeous euro symbol from scratch using circles and lines to build up a symmetrical euro symbol that matched our default font.

Anyway I'm up to rhombus in rectangle in euclidia, thanks for stealing all my upcoming free time!
posted by Eyebrows McGee at 12:27 AM on September 9, 2016

A lot of these seem to include steps where the compass is lifted off the page and used to transfer an identical length somewhere else, which I thought was against the spirit of Euclidean construction.

But looking into it, I'm delighted to learn about the Compass Equivalence Theorem, which saves time while conserving theoretical integrity.

And not only is transferring lengths fair game, but you can do all constructions with a collapsible compass alone! (plus a bit of creative visualisation in joining points).
posted by rollick at 5:30 AM on September 9, 2016 [1 favorite]

some of the more hardcore/devotional geometers I see on social media insist on using a “rusty” compass; that is, a compass fixed at an arbitrary value. They still produce beautiful results, just very slowly. A few even make their own shell gold and mull their blue pigment from Afghani lapis lazuli.

I was slightly disappointed that the otherwise good Gothic construction pamphlet didn't acknowledge the Arabic influence more. The ‘miraculous’ appearance of complex church architecture immediately after the crusades was no coincidence: the construction techniques and even the paper required to draw plans were appropriated from Middle Eastern culture. In addition to being a vile and bloody blot on Western history, the crusades were also one colossal IP grab.
posted by scruss at 7:45 AM on September 9, 2016 [3 favorites]

A lot of these seem to include steps where the compass is lifted off the page and used to transfer an identical length somewhere else, which I thought was against the spirit of Euclidean construction.

Isn't this functionally equivalent to walking the compass around on the paper to where it needs to go?
posted by sebastienbailard at 10:19 PM on September 9, 2016

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