Tag Archives: manifold

The (2,3,7) Triangle Group

8 Nov

Hi!  Today we’re going to use some stuff we learned about a long time ago (non-Euclidean geometry, manifolds, and groups) and put it together to explore a particular group.  This is based on a talk given by my dear friend and co-organizer Michelle Chu.  “Co-organizer?  Of what?” you ask.  Well thanks for asking!  Last weekend we did held the first Texas Women in Mathematics Symposium – we had over 60 people come, lots of talks, lots of networking, and lots of food.  By the end of the day I got to add “annual” to that description, and it seems like a lot of schools were interested in hosting it in future years.  Maybe some time I’ll write a post about how to found an annual conference (this is my second!).

Anyways, let’s first talk about tilings by triangles.  We first choose some integers p, q, r and set the three angles of a base triangle equal to \frac{\pi}{p}, \frac{\pi}{q}, \frac{\pi}{r}.  Now reflect over each of the three sides to start tiling your space.  Turns out this tiling will lead to a group.  Here’s an example with p=q=4 and r=2 (so we have a right isosceles triangle):

firstimgs

Start with the pink triangle, and reflect it over each of the three sides to make the colored triangles as shown.

secondimgs

Now do the reflections again.  I kept the pink base triangle and grayed out the first images.  Note that I colored the bottom left image yellow, for reflecting over the vertical side of the bottom orange triangle, but I also could color it orange, for reflecting over the horizontal side of the left yellow triangle.  This means that yellow+orange = orange+yellow in the group.

thirdimgs

A third iteration of the same process; there are more relations here (that I didn’t color)

I picked a particularly good example, so that my triangles could tile the Euclidean plain.  We learned some time ago about non-Euclidean geometries: the space is spherical if the sum of triangle angles is more than \pi, and hyperbolic if triangles are thin and their sum of angles is less than \pi.  So based on how I choose my p, q, and r, I’ll find myself tiling different spaces.  Here’s an example of one iteration on a sphere for p=q=2 and r=5:

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It’s pretty easy to find the integer solutions for p, q, r to tile each space.  The only triangles that tile the flat plane are when (p,q,r) = (2,3,6), (2,4,4), and (3,3,3).  We already saw (2,4,4), and I’ll just tell you that (3,3,3) is when you use equilateral triangles (so there are 6 around each vertex), and (2,3,6) are those 30-60-90 triangles we learned about back in trigonometry class: here’s the picture from wikipedia:

Similarly there are only a few (relatively) that tile the sphere: (2,3,3), (2,3,4), (2,3,5), and (2,2, n), where is some number bigger than 1.  Of course this forms an infinite family of tilings, since you can choose n.  In the example above I chose n=5, and if is bigger the base pink triangle just gets skinnier.

But I say there’s only a few that tile the sphere because everything else tiles the hyperbolic plane.  That’s a lot of choices!  It might also make you think, “hm, maybe the hyperbolic plane is interesting.”

Let’s bring us back to groups.  How does a tiling of a space lead to a group?  Well, let the reflections over the (extended) sides of the base triangle be the generators of our group.  If I name these a, b, and c, I immediately get the relators a^2=b^2=c^2=1.  Next we have to figure out the rest of the relators.  I hinted at them in the pictures above.  They are (ab)^p=(bc)^r=(ca)^q.  Now we have a group presentation [link for definition] R\triangle(p,q,r)\cong \langle a, b, c \ | a^2=b^2=c^2=(ab)^p=(bc)^r=(ca)^q=1\rangle.

Also, fun coincidence: if you create the dual tree to the tiling by putting a vertex inside each triangle and connecting two vertices by a line if one triangle is the image of another under one of the reflections, you get something that looks a lot like the Cayley graph of the reflection triangle group.  The only difference is that each edge needs to be two edges (like a little loop) to reflect that each generator has order 2.

So what’s special about the (2,3,7) triangle group?  We know from above that it tiles the hyperbolic plane.  Check out this great picture from wikipedia of the tiling:

h2checkers_237

Maybe we’ll take a second here to point out that you can see the p, q, r values in the tilings, both in this picture and the other wikipedia picture above for (2,3,6): pick your favorite triangle, and look at its three vertices.  Each vertex is adjacent to other triangles, and since there are 2\pi angle around any vertex, we can figure out that p,q,r are just \frac{n}{2}.

Also, of all the integers you can pick for p, q, r, it turns out that 2, 3, and 7 maximize the sum \frac{\pi}{p}+\frac{\pi}{q}+\frac{\pi}{r} while staying less than \pi.  [It ends up giving you \frac{41\pi}{42} for those of you keeping track at home.]

So we maximize something with the numbers 2, 3, 7.  Well it turns out we do more than that- we also minimize the volume of the resulting quotient-we touched on and defined those in this post about manifolds.  And this is unique (up to conjugation/fiddling), and makes the smallest possible quotient.  Huzzah!

On a personal note, I’ve had a demonic cold keeping me in bed for the past two weeks, so forgive me if I messed up (pretty sure I did on the reflections; I’ll try to fix those soon).  Also, hope you voted today!  I voted a week and a half ago.

What is a manifold? What is not a manifold?

29 Mar

I just went to a talk and there was one tiny example the speaker gave to explain when something is not a manifold.  I liked it so much I thought I’d dedicate an entire blog post to what was half a line on a board.

I defined manifolds a long time ago, but here’s a refresher: an n-manifold is a space that locally looks like \mathbb{R}^n.  By locally I mean if you stand at any point in the manifold and draw a little bubble around yourself, you can look in the bubble and think you’re just in Euclidean space.  Here are examples and nonexamples of 1-manifolds:

1manis

Red and orange are manifolds: locally everything looks like a line.  But yellow and green are not.

At any point on the orange circle or red line, if we look locally we just see a line.  But the yellow and green both have bad points: at the yellow bad point there are 2 lines crossing, which doesn’t happen in \mathbb{R}, and in the green bad point there’s a corner.

1manisbad

I messed up a little on the orange one but imagine that that is a smooth little curve, not a kink.

We call 2-manifolds surfaces, and we’ve played with them a bunch (curves on surfaces, curve complex, etc. etc.).  Other manifolds don’t have fun names.  In general, low-dimensional topology is interested in 4 or less; once you get to 5-manifolds somehow everything gets boring/collapses.  It’s sort of like how if you have a circle in the plane, there’s something interesting there (fundamental group), but if you put that circle into 3-space you can shrink it down to a point by climbing a half-sphere to the North Pole.

circle

Empty pink circle in the plane can change size, but not topology (will always have a hole).  In 3-space, it can contract to a point.

The other thing we’ll want to think about are group actions.  Remember, a group acts on a set X if there’s a homomorphism that sends a group element g to a map \phi_g:X\to X such that the identity group element maps to the identity map, and group multiplication leads to composition of functions: gh \mapsto \phi_g \circ \phi_h.  That is, each group element makes something happen on the set.  We defined group actions in this old post.  Here’s an example of the integers acting on the circle:

rotations

Each integer rotates the circle by pi/2 times the integer. Looks like circle is getting a little sick of the action…

So far we’ve seen groups and manifolds as two different things: groups are these abstract structures with certain rules, and manifolds are these concrete spaces with certain conditions.  There’s an entire class of things that can be seen as both: Lie Groups.  A Lie group is defined as a group that is also a differentiable manifold.  Yes, I didn’t define differentiable here, and no, I’m not going to.  We’re building intuitions on this blog; we might go into more details on differentiability in a later post.  You can think of it as something smooth-ish without kinks (the actual word mathematicians use is smooth).

diffeomo

Top is smooth and differentiable. Bottom isn’t; there are weird kinks in its frown

So what are examples of Lie groups?  Well, think about the real numbers without zero, and multiplication as the group operation.  This is a manifold-at any point you can make a little interval around yourself, which looks like \mathbb{R}.  How is it a group?  Well, we have an identity element 1, every element has an inverse 1/x, multiplication is associative, and the reals are closed under multiplication.

Here’s another one: think about the unit circle lying in the complex plane.  I don’t think we’ve actually talked about complex numbers (numbers in the form x + iy, where is the imaginary square root of -1) on this blog, so I’ll do another post on them some time.  If you don’t know about them, take it on faith that the unit circle in the complex plane is a Lie group under multiplication.  Multiplying by any number on the unit circle gives you a rotation, which preserves the circle, again 1 is the identity, elements have inverses, and multiplication is associative.  Circles, as we said before, are 1-manifolds.

more

Examples of Lie Groups: the real line minus a point, and the unit circle in the complex plane

If you have a group action on a set, you can form a quotient of the set by identifying two points in the set if any group element identifies them: that is, and become one point if there’s a group element so that g.x=y.  For instance, in the action of the integers on the circle above, every point gets identified with three other points (so 12, 3, 6, and 9 o’clock on a clock all get identified under the quotient).  Your quotient ends up being a circle as well.  We denote a quotient of a group G acting on a set X by X/G.

quo

So here’s a question: when is a quotient a manifold?  If you have a Lie group acting on a manifold, is the resulting quotient always a manifold?  Answer: No!  Here’s the counterexample from the talk:

Consider the real numbers minus zero using multiplication as the group operation (this is the Lie group \mathbb{R}^{\times}) acting on the real line \mathbb{R} (this is a manifold).  What’s the quotient?  For any two non-zero numbers a, b on the real line, multiplication by a/b sends to a, so we identify them in the quotient.  So every non-zero number gets identified to a point in the quotient.  What about zero?  Any number times zero is zero, so zero isn’t identified with anything else.  Then the quotient \mathbb{R}/\mathbb{R}^{\times} is two points: one for zero, and one for all other numbers.

If the two points are “far apart” from each other, this could still be a 0-manifold (locally, everything looks like a point).  But any open set that contains the all-other-numbers point must contain the 0-point, since we can find real numbers that are arbitrarily close to 0.  That is, 0 is in the closure of the non-zero point.  So we have two points such that one is contained in the closure of the other, and we don’t have a manifold.  In fact our space isn’t Hausdorff, a word I mentioned a while back so I should probably define in case we run into it again.  Hausdorff is a serious way of explaining “far apart.”  A space is Hausdorff (adjective) if for any two points in the space, there exist disjoint neighborhoods of the two spaces.  So the real line is Hausdorff, because even if you take two points that look super close, like 2.000000000001 and  2.000000000002, you can find infinitely many numbers between them, like 2.0000000000015.

hausdorff

Any two points on the real line, if you zoom in enough, have space between them.  So the real line is Hausdorff.

If you’re curious as to when the quotient of a smooth Manifold by a Lie Group is a manifold, you should go take a class to fully appreciate the answer (the Quotient Manifold theorem). The phrasing of the Quotient Manifold Theorem below is from a book by John Lee, called Introduction to Smooth Manifolds (the version from wikipedia gets rid of one of the conditions but also gets rid of much of the conclusion).  Briefly: a smooth action means the function on M is smooth (see the picture above; we didn’t do an in-depth definition of smooth), a free action means there aren’t any fixed points, and a proper action has to do with preimages of certain types of sets.

Theorem 21.10. Suppose G is a Lie group acting smoothly, freely, and properly on a smooth manifold M. Then the orbit space M/G is a topological manifold of dimension equal to dimMdimG, and has a unique smooth structure with the property that the quotient map π:MM/G is a smooth submersion.

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