The "moduli space" of triangles

I just watched a documentary about Maryam. I hope your full video is still coming.

Scott

Math lives from abstraction. But abstraction cannot exist without at least two concrete cases (what to abstract from, otherwise?). Such illustrating examples are very important!

Hey grant, crossing my finger hoping that this video seriously does come out sometime soon!! Please :)

Dhomie

Hi Grant. I was just wondering if you still planned on making this into a video.

Hi Grant, this is a beatiful animation, as always. I think there is a small inaccuracy: At 2:37 your pointer representing the triangle class in the right plot goes outside the possible region defined earlier.

I like this! I have a beginner question, why does it be called 'moduli space' instead of 'parameter space' or 'modular space'? I also found interesting (as Daniel Wisehart has already mentioned) that the ratio of the acute and obtuse area: 1/2 - pi/8 : -1/4 + pi/8 . I am looking forward to watching the complete video. Thanks!

Hitoshi Yamauchi

I sure did like it; this "moduli space" is completely new to me, a non-mathematician

Oldboy

Simply beautiful. I like the skeuomorphic background as well. Between that and the music it was so peaceful that it almost felt like a meditation.

Really well explained as usual!! I loled when you split it into the obtuse and acute triangle region. So cute! And such a great way of condensing all that info

I believe there are proofs that obtuse triangles outnumber acute ones by 3/4 as in https://www.cut-the-knot.org/Probability/ObtuseTriangle.shtml

By a factor of (pi-2)/(4-pi), or about 33% more.

Ah, i always felt like they are more obtuse than acute triangles.

Jan

This is such a cool project!!

Henry Reich

I was thinking about this exact space about month ago, "moduli spaces" as you call them are quickly becoming one of favorite things in math. Just recently I discovered that the space of all lines in the Euclidean plane is a Möbius strip without its boundary (or equivalentally an infinitely "wide" Möbius strip).

It'll be a challenge to do it justice, but hopefully, it's doable.

3blue1brown

Good to know. I'm just experimenting here with something a little different, while I'm in a different context. I kind of liked the chalkboard, thinking it made it feel one step closer to having this drawn out in front of the viewer in the real world. I'm not attached, though.

3blue1brown

Outstanding question! As far as I can tell, there is some ambiguity on the metric you put on the space of triangles, so that's a choice that would need to be made explicit. What is interesting is that the moduli spaces of Riemann surfaces, which is ultimately where we'd like to build up to here, seem to have a very "natural" metric associated with them, one which makes probabilistic questions like this much less ambiguous.

3blue1brown

That could be fun, seeing it as a sort of reflection surface. I will say, this is just a simple example meant to be a stepping stone towards more complicated moduli spaces, so it probably won't make sense to dwell on it _too_ much.

3blue1brown

Reflections count, which I often think of as rotation 180 degrees about some axis in the plane. It's worth being explicit about this, though.

3blue1brown

Good point! It's worth being explicit that rotating in 3-dimensions counts in what is meant by the word "rotate".

3blue1brown

It'd be nice to point the interested reader to a proof of why points in the shaded area correspond *exactly* to equivalence classes under similarity by reflections, rotations, and scaling. On the last question, does this way of computing that probability give you a different result than if you parameterized "all triangles" differently? If so, why is this a "good" parameterization? Why shouldn't I pick parameterizing by pairs of angles, for example?

Oh heck yes! Modular arithmetic applied to complex geometrical and topological spaces is a jam I want to learn a lot more about, so this is very welcome.

This clip was fantastic - it really got me thinking about this technique as a general one, for some reason reminding me of phase space, which I remember was one you liked too. Very curious to see where this goes!

_ericBG

I too cannot wait to see where this is going. This is also a great illlustration of the use and power of equivalence classes, similarity being one example of an equivalence relationship. Personal connection: David Eisenbud was my son's Ph. D. advisor.

I also think you need to talk about reflections as well as scaling and rotation. Without it you don't have the x>y constraint. Also, and perhaps this is where the video goes, but there should be some discussion about what happens when you go off the edge of the moduli space. (A shorter edge grows to 1, then becomes the longest edge and the moduli space point slides in somewhere else.)

Bob Dowling

Wow!

I personally think the chalkboard looks pretty good, and the video was great too

That was fun! Anxious to see where this goes. I'll agree with some other folks who commented above and say 1) Not sure the chalkboard background is a winner. I like pure black backgrounds. As it is now, makes me think I need to clean my monitor. 2) "Rotate *and/or* scale." Not one or the other. Both.

Burt Humburg

All squares are similar, so their space is a single point.

Gabe

Is a triangle similar to its mirror image? And I mean in general and not just the special case of isosceles triangles. I'm asking because it is not mentioned as a criterion, but with the chosen (visual) representation triangles with flipped side lengths of the two shorter sides are represented by the same point (x,y) on the plane, since y is by definition always the larger one.

I like the video, but I would like to point out one slight inaccuracy - "What it means for two triangles to be similar is that you can rotate or scale one into the other". I think you need to add the reflection too.

Nice build up. Is the idea to generalise for moduli space or is there more about triangles - where do non-Euclidean triangles go?

A challenging project, but well worth it. This clip is a good start.

Daniel Armesto

Really, the moduli space of triangles is itself a triangle. And the moduli space of quadrilaterals is a square? Great vid as always.

love it

Also, cutting off your space at the x=1, y=1, and x=y lines kinda feels artificial. Like, I get why - you want a unique point per similarity class, and allowing the entire plane (except for the regions where x≤0, y≤0, or x+y≤1) gives you duplicates, so you have to deal with that somehow. It still feels wrong somehow. But something tells me that this artificiality is something you're going to talk about in the rest of the video. EDIT: Maybe the reason this bugs me is that, in the "Why care about topology" video, you used this sort of thing to glue together sides of a state space (the Möbius band as the set of pairs of points on a circle) and you haven't done that here

Akiva Weinberger

When you said you wanted the space of all triangles, my first thought was to do it by angles. We know the angles of a triangle add to 180, or π radians, so this is the set {(α,β,γ) | α+β+γ=π}. This is a triangle (which diffeomorphic to what you got, which feels like confirmation of some sort). Only after thinking for a bit did the "scale the longest side to 1 and record the other two in order" occur to me. So maybe mention that this solution isn't unique? Also - the line about this making "'picking a random triangle' rigorous" kinda implies that this is *the* natural probability measure to use for picking random triangles. But is that true? Like, if you use my space above, a triangle has a 1/4 chance of being acute (if I didn't make a mistake). (I've heard — but not entirely understood — the phrase "Haar measure" in related contexts. Is that relevant?) EDIT: Oh no, I need to ensure α≥β≥γ, don't I, or else almost every triangle appears six times… Also, degenerate triangles (the case where x+y=0 or α=β=0) aren't represented well on my space, though I don't know how much we care about that.

Akiva Weinberger

This is awesome. Can't wait for more!

Kaushik Mohan

Can’t wait to see her legacy brought to life. I’ve never heard of a Reimann surface so this’ll be good. 😃

Really cool! The one thing I'm not a fan of is the chalkboardy background, I feel like it tends to be a bit of a distraction, and it dilutes the purity and beauty of your animations.

great vid

You continue to demonstrate your talent for taking a difficult concept and rendering it in a clear, comprehensible manner. Much appreciated! I look forward to the series.

David B. Hill

Beautiful! <3

Ever Salazar

Nicely explained as usual. Thanks. Loved it

Thanks. That was great. Looking forward to the rest of the project.

Chris Tietjen

this is great stuff, definitely do more

I am not sure which angle you are coming from, but I will tri and figure it out.

Sound really cool! But I'm very biased because I love Riemann surfaces.

Cole