A Complicated Construction of Something Simple

[sdi]

The 3-4-5 triangle is my desert-island geometric fact: if you have a triangle with a sides of length 3, 4, and 5, it's a right triangle. This is great because it's super easy to mark a rope into 3+4+5=12 equal lengths, and this means it's super easy to make yourself a right triangle. I've used this before to lay out an orchard, making sure all the rows were nice and parallel, and it worked beautifully.

Because it's so easy to make a 3-4-5 triangle directly, it seems pretty silly to go to much greater lengths to make one using a pair of compasses, but let's not let mere uselessness stop us! After all, there's the Taoist saying: "When purpose has been used to achieve purposelessness, the thing has been grasped." ;)

I'm feeling playful, so in honor of the sovereign Sun whose day it is, and his dutiful son Pythagoras, let's pick up our compasses and hop to it:

Given points A and B, draw

1. circle AB,
2. circle BA intersecting circle AB at points C and D,
3. circle CD intersecting circle AB at point E≠D,
4. circle EB intersecting circle BA at points F and G,
5. circle FA,
6. circle GA intersecting circle FA at point H≠A,
7. circle HA intersecting circle EB at point I;

then triangle HIE is a 3-4-5 triangle. I'm not going to write up a full proof right now, but a quick sketch runs like this: let's define AB=2. It can be shown that EAB is collinear, therefore EB=EI=EA+AB=2×AB=4. Suppose line FG intersects line AB at X: it can be shown that AX=3/2. H is the reflection of A about line FG and FG is perpendicular to AB, therefore EABH is collinear and AH=HI=AX+XH=2×AX=3. Finally, EH=EA+AH=AB+AH=5.

I believe this to be the simplest possible construction (that is, using the fewest circles) of such a triangle using only compasses. (There are much easier ways to draw a right angle, though!)

Comments

[temporaryreality]

I'm enjoying this series, even if I'm not able to play along on paper (still working on much more basic constructions!). I can see, too, that I need to actually put time into Euclid...

[sdi]

I just picked up a copy of Euclid a couple weeks ago, myself—I've only ever gone through abbreviated versions of it, and never the whole thing. (It'll be a while before I do go through the whole thing, but...) I spent a little while flipping through it and what surprised me most is that so little of it is what we, today, call geometry! Most of it is concerned with arithmetic and number theory, and only a few of the books are about geometric constructions proper!

[temporaryreality]

In some ways I feel so innumerate - I hardly know what's going on in the latter parts of the book! If I reframe it kindly, "beginner's mind" at work here :D

[sdi]

Don't be hard on yourself—he's really tricky! Everything Euclid does is for a very specific reason, but without someone to guide you, it's like a puzzle to sort it all out, and most of the time you can't possibly hope to guess what he's aiming at until you've gotten a few propositions on.

I was hoping to take some time to start poking through it in little bits since I've been too ill and brain-foggy lately to get through a chapter of Plotinus, but no, Euclid is no easier! So I content myself with my own doodling while I wait to get healthy again :)

[temporaryreality]

I was a little more serious about working through portions of it some months ago and I took it in very small bites, basically working through each sentence individually and slowly enough that I could envision what he was talking about (without putting it on paper) before moving on to the next.

That I'm no longer (currently) working on it in that way says only that I'm way too flighty (though the practice was an effort to curb mental dissipation). My unending Sisyphean rock-rolling in a nutshell.

....

Might I add prayers for your health to my morning devotions for a bit?

[sdi]

I'm impressed you could follow it all in your head! I always have to draw things out.

And yes, I would welcome that—thank you very much.

[temporaryreality]

Well, step by step by step, just repeating things until I got them... and only for the first several propositions. Don't be too impressed! :)

and you're welcome. I hope for your improved health.

[boccaderlupo]

Fascinating and splendid!

Axé...