Dodecahedral construction of the Poincaré homology sphere, part II

26 Apr

Addendum: I forgot to mention that this post was inspired by this fun New Yorker article, which describes a 120-sided die.  It’s not the 120-cell; as far as I can tell it’s an icosahedron whose faces are subdivided into 6 triangles each.  The video is pretty fun.  Related to last week, Henry Segerman also has a 30-cell puzzle inspired by how the dodecahedra chain together.  In general, his Shapeways site has lots of fun videos and visual things that I recommend.

Side note: when I told my spouse that there are exactly 5 Platonic solids he reacted with astonishment.  “Only 5?!”  I’d taken this fact for granted for a long time, but it is pretty amazing, right?!

Last week we learned about how to make the Poincaré homology sphere by identifying opposite sides of a dodecahedron through a minimal twist.  I thought I’d go a little further into the proof that $S^3/I^*\cong \partial P^4$, where the latter is the way that Kirby and Scharlemann denote the Poincaré homology sphere in their classic paper.  This post is a guided meditation through pages 12-16 of that paper, and requires some knowledge of algebraic topology and group actions and complex numbers.

Honestly, I don’t know too much about $I^*$, but I do know that it’s a double cover of I, which is the group of symmetries of the icosahedron.  For instance, if you pick a vertex, you’ll find five rotations around it, which gives you a group element of order 5 in I.  Every symmetry will be a product of rotations and reflections.

Icosahedron from wikipedia, created from Stella, software at this website.

Last time we watched this awesome video to see how you can tessellate the three sphere by 120 dodecahedra, and explained that we can think of the three sphere as {Euclidean three space plus a point at infinity} using stereographic projection.

Hey that’s great!  Because it turns out that there are 60 elements in I, which means that $I^*$ has 120 elements in it.  Let’s try to unpack how acts on the three sphere.

First, we think of how the three sphere acts on the three sphere.  By “three sphere,” I mean all the points equidistant from the origin in four space.  The complex plane is a way to think of complex numbers, and it happens to look exactly like $\mathbb{R}^2$.  So if I take the product of two copies of the complex plane, I’ll get something that has four real dimensions.  We can think of the three sphere as all points distance 1 from the origin in this $\mathbb{C}^2$ space.  So a point on the three sphere can be thought of as a pair of points (a,b) , where and are both complex numbers.  Finally, we identify this point with a matrix $\left( \begin{array}{cc} a & b \\ -\bar{b} & \bar{a} \end{array} \right)$, and then we can see how the point acts on the sphere: by matrix multiplication!  So for instance, the point (a,b) acts on the point (c,d) by $\left( c \ d \right) \left( \begin{array}{cc} a & b \\ -\bar{b} & \bar{a} \end{array} \right)= (ac - \bar{b}d, bc + \bar{a}d)$, where I abused a bit of notation to show it in coordinate form.

What does this actually do?  It rotates the three sphere in some complicated way.  But we can actually see this rotation somewhat clearly: set equal to 0, and choose a to be a point in the unit circle of its complex plane.  Because is a complex unit, this is the same as choosing an angle of rotation θ [$a=e^{i\theta}$].

Remember how we put two toruses together to make the three-sphere, earlier in the video?  Each of those toruses has a middle circle so that the torus is just a fattening around that middle circle.  Now think about those two circles living in our stereographic projection.  One is just the usual unit circle in the xy plane of $\mathbb{R^3}$, and the other is the axis plus the point at infinity.  So how does (a, 0) act on these circles?  We can choose the basis cleverly so that it rotates the xy unit circle by θ, and ‘rotates’ the axis also by θ.  That means that it translates things up the axis, but by a LOT when they’re far on the z-axis and by only a little bit when they’re small.

We rotate the blue circle by the angle, and also rotate the red circle.  That means the green points move up the z-axis, but closer to the origin they move a little and farther away they move a lot.

Side note: this makes it seem like points are moving a lot faster the farther you look from the origin, which is sort of like how the sun seems to set super fast but moves slowly at noon (the origin if we think of the path of the sun as a line in our sky + a point at infinity aka when we can’t see the sun because it’s on the other side of the Earth).

Similarly, if we don’t have b=0, we can do some fancy change of coordinate matrix multiplication and find some set of two circles that our (a,b) rotate in some way.  In either case, once we define how the point acts on these two circles we can define how it acts on the rest of the space.  Think of the space without those two circles: it’s a collection of concentric tori (these ones are hollow) whose center circle is the blue unit circle, and whose hole centers on the red axis.  If you have a point on one of those tori, we move it along that torus in a way consistent with how the blue and red circles got rotated.

This is a schematic: the blue and green circles get rotated, so the purple point on the pink torus gets rotated the way the blue circle does, and then up the way the green circle does.

What does this have to do with I?  Fun fact: the symmetries of the icosahedron are the same as the symmetries of the dodecahedron!  (Because they’re duals).  So let’s look back at that tessellation of the 120-cell by dodecahedra, and stereographically project it again so that we have one dodecahedron centered at the origin, with a flat face at (0,0,1) and (0,0,-1), and a tower of ten dodecahedra up and down the z-axis (which is a circle, remember).

The origin dodecahedron.

Now imagine rotating around the green axis by a click (a 2pi/5 rotation).  This is definitely a symmetry of the dodecahedron.  It rotates the blue circle, and by our action as we described earlier, it also rotates the green circle, taking the bottom of our dodecahedron to the top of it (because $|e^{-\pi i/5}| = |e^{\pi i/5}| =1$).  So this identifies the bottom and top faces with that minimal rotation.  We said earlier that this rotation has order 5 in I, which means that it has some corresponding group element in $I^*$ with order 10.  10 is great, because that’s the number of dodecahedra we have in our tower before we come back to the beginning: so if we keep doing this group element rotation, we end up identifying the top and bottom of every dodecahedron in our z-axis tower of 10.

This is definitely a screenshot of that youtube video above, plus a little bit of paint so I could indicate the origin dodecahedron.

Similarly, using change of coordinate matrix basis madness, we can figure out how the rotations around the centers of each of the faces acts on the 120-cell (hint: each one will identify all the dodecahedra in a tower just like our first one did).  With 120 elements in $I^*$, each element ends up identifying one of the dodecahedra in the 120 cell with our origin dodecahedron, including that little twist we had when we defined the space.

So that’s it!  That’s how you go from the tessellation of the 120-cell to the Poincare homology sphere dodecahedral space.  Huzzah!

Dodecahedral construction of the Poincaré homology sphere

19 Apr

Update: Thanks as usual to Anschel for catching my typos!

This semester some grad students put together a learning seminar on the Poincaré homology sphere, where each week a different person would present another of the 8 faces from this classic (1979) Kirby-Scharlemann paper.  It was a fantastic seminar that I recommend to any grad students interested in algebraic geometry, topology, geometric group theory, that sort of thing. I did the last description, which is actually description number 5 in the paper.  You can read this post as a definition of the Poincaré homology sphere, without me telling you why mathematicians would care (but it has properties that makes mathematicians care, I promise).

First, start with a dodecahedron: this is one of the five Platonic solids, which are three-dimensional objects that can be created by gluing together regular (all sides are the same, all angles are the same) polygons so that the same number of polygons meet at any corner.  The fast example of a Platonic solid is a cube (three squares meet at each corner), and a non-Platonic solid is a square pyramid (4 polygons meet at the top, but only three at each corner).  If you glue two square pyramids together, you do get a Platonic solid, the octahedron.

Glue two pyramids together along their squares sides, and now four triangular faces meet at each vertex and you have a Platonic solid: the octahedron.

So after all that build up, here’s a dodecahedron: 12 pentagons glued together the only way you can: start with one pentagon, glue five to it (one on each edge), glue those together into a little pentagonal cap with a toothy bottom.  If you make two of these caps, you can glue them together; the teeth fit into each other just right.  This is the first step in this AWESOME VIDEO below (seconds 30-45 or so):

To make a the Poincare dodecahedral space, let’s first review the torus.  A long time ago, we learned about how to make a torus: take a square, identify opposite edges while preserving orientation.

First we glue the green arrow edges up together and get a cylinder, then the blue edge arrows together…

I’m a torus!

If you only identify one pair of edges and flip the orientation, you get a Mobius strip.  If you do that to both pairs of edges, you get a Klein bottle, which you can’t actually make in three dimensions.

Mobius strip picture from wikipedia

This torus/Mobius/Klein side note is just to review that we know how to glue edges together.  So look at the dodecahedron.  Each pentagonal face has a pentagonal face exactly opposite it, but twisted by 1/10 of a turn (2pi/10).  So if you identify each face with the opposite one, doing just the minimal turn possible, you get the Poincare homology sphere.  We started with 12 faces in our dodecahedron, so this glued-up space will have 6 faces.  It also has 5 vertices and 10 edges (vs. 20 vertices and 30 edges pre-gluing).  I can’t draw it for you because it’s a 3-manifold.  But here is a funny video of walking through it!

If you draw out all the identifications and you know some group theory, you can find the fundamental group of the thing, and you can prove to yourself that it is a 3-manifold and nothing funky happens at edges or vertices.

The dual to the dodecahedron is the icosohedron.  “Dual” means you put a vertex into the middle of each face of the dodecahedron, and connect edges of the dual if the corresponding faces share an edge in the dodecahedron.

Image from plus.maths.org

So you can see that the dual to the cube is the octohedron , and the tetrahedron is its own dual.  That’s all five Platonic solids!

Top row: tetrahedron, cube, octahedron.  Bottom row: dodecahedron, icosohedron.

There’s more to the story than this!  Let’s think about spheres.  The 1-sphere is a circle in the plane, aka 2-space.  Equivalently, the 1-sphere is all points that are equidistant from 0 in 2-space.  Similarly, the 2-sphere is all points equidistant from 0 in 3-space.   This gives you a notion of the 3-sphere.  How can we picture the 3-sphere?  We can use stereographic projection.

Here are the examples of stereographic projection of the circle and the 2-sphere onto the line and 2-space, respectively.  You cut out a single point from the north pole of the sphere, and attach the space to the south pole as a tangent.  Given some point on the sphere, run a line from the north pole through that point: it hits the space at exactly one point, and that’s the stereographic projection of the sphere-point.  Notice that the closer you get to the north pole, the farther out your projection goes.  If we pretend there’s one extra point (infinity) added to the plane, we can identify the n-sphere with n-space plus a point at infinity.  Look at this link and buy things from it if you want!

Projecting from the sphere to the plane: the bottom hemisphere of the red sphere maps to the pink circle in the plane, the top half maps to the rest of the plane.

What do circles that go through the north pole look like?  Just like when we projected the circle to the line, they look like infinite lines.

So we can see the three sphere as 3-space, plus a point at infinity.   Similarly here, circles that go through the north pole look like infinite lines.

Our math claim is that $\mathbb S^3/I^* \cong \partial P^4$, or that if I act on the 3-sphere by the binary icosohedral group, I get this exact dodecahedral space as the quotient.  Binary icosohedral goup is just some extension of the group of symmetries of the icosohedron, which is the same as the group of symmetries of the dodecahedron.  So we want to see a way to see this action.  The awesome video up top shows us how to start.  I’ll describe the contents of the video; you should read the next paragraph and re-watch the video after/while reading it:

Start with one dodecahedron.  Stack another on top of it, lining up the pentagons so you can glue one to another (that means the one on top is a 2pi/10 turn off from the bottom one).  Now make a tower of ten dodecahedra, all glued on top of each other.  Make a second tower of ten dodecahedra, and glue it to the first one (so it’ll twist around a bit).  Glue the top and bottom of the first tower together (they’ll line up because we did a 2pi total turn); this’ll automatically glue the top and bottom of the second tower together.  Nestle six towers like this together, so the toruses created from the towers all nestle together.  Now you have a torus of 60 dodecahedra.  Make a second torus of 60 dodecahedra.  Put the second torus through the hole of the first, so you get a solid ball.  (Here’s the weird 4-dimensional part!)  That is a 3-ball!  (The first torus also goes through the hole of the second one).  So now we have tesselated the 3-ball with dodecahedra; this is called the 120-cell.

I might make a more technical second post on this topic explaining in detail the action, but suffice it to say that we have an action by a group that has 120 elements, so that if we quotient out this 120-cell by the action, we end up with just one dodecahedron with the faces identified the way we want them to (opposite faces identified by a twist).  What is this group of 120 elements?  It’s derived from the symmetries of the icosahedron, which has the same symmetries as the dodecahedron!

Final interesting notes on this: we identified opposite sides by just one turn.  If you do two turns (so a 4pi/10 turn), you get the Seifert-Weber dodecahedral space.  If you do three turns, you get real projective space.

Jeff Weeks article on shape of space, a.k.a. is the universe a Poincare homology sphere?

Thurston book on geometry and topology

Fun website: Jeff Week’s geometry games

I’m sexist (and “so is everyone” isn’t an excuse)

10 Jun

Over the weekend we hung out for a few hours with some of my husband’s coworkers and their kids.  One wife is a very pregnant stay at home mom of two toddlers, and one husband is a stay at home dad of a toddling soon-to-be older brother.  I’ve hung out with the female coworker and her husband and child more, and their child is closer in age to our baby.  I am very impressed with him for staying at home with the kid.

When I first went back to work I had baby in day care three days a week and watched him for two, hoping that’d ease my transition back.  But those two days were SO HARD- it’s constant, mundane, brain-draining, frustrating physical work that’s incredibly, ridiculously rewarding.  (See photo)

And being alone with a little one all day with no adult interaction is rough- it was hard on me and it was hard on our marriage (which is awesome I highly recommend marriage by the way).  By the afternoons I was itching to work, but when I was at the office I was aching to be with my sweet little baby.

Aside: I have no thoughts on “having it all” except that the phrase doesn’t make any sense to me.  My hormones and heart want to be with my baby ALL THE TIME, and my mind and exercising body do not, and unfortunately all these things go together so it is impossible to have all my desires met.  Probably my wants will evolve as my child ages, and as I get more children, and I get older and my career moves, but right now it is impossible for me to have all the things I want.

So, my sexism.  When I’m with the SAHM, I take it for granted that she stays at home, watching two handfuls and running the household (she does EVERYTHING in that home) and even gardening and raising chickens.  We chatted about cooking and pregnancy and adapting to our new bodies and making friends.  Whenever I talk to the SAHD, I feel in awe that he stays at home, and we talk about the frustrations of hanging out with a little one all day and strategies to not go crazy.  Thus there are two sides to my sexism:

1. I do not feel in awe that she stays at home.  I assume that she does not go crazy or feel frustrated or feel any sort of internal struggle with all the things I said above about having little ones.  This is totally unfair to her and speaks deeply of my cultural assumptions (women can stay at home and don’t feel all the things that I feel and also I am a woman so this really doesn’t make sense).  Also, she’s got two kids so she has it way harder than him.
2. I do feel in awe that he stays at home.  This is unfair to him- it implies that I think staying at home is so hard on him, and further implies that maybe I think men can’t handle it. Also, it’s the only thing we talk about vs. a wide variety of things I can talk to her about.  I’ve now put him in a box with one interesting thing to discuss (his dadhood) vs. being a full human being with other thoughts.

I realized this on our way home from their place.  What do I do to fix this?  The clear answer is that I need to treat every person as an individual of individual circumstances, and treat each person with respect.  But while that abstraction is well and good, I need more concrete action items to get better.  When I talk to SAHDs, I won’t say things like “I’m so impressed that you stay at home” and instead I’ll talk to them like human beings.  When I talk to SAHMs, I’ll try to invite commiseration on how difficult raising kids is (you have to tiptoe here depending on how close you are with a person, b/c I’m not a SAHM but I could be if I chose to so anything coming out of my mouth could be seen as judgmental).  Hopefully these actions and saying these words will eventually change my internal attitudes too.

This all reminds me of a great essay I read two years ago, which I highly recommend.  Also, if you change one letter in the dude’s wife’s name, you get my name:

“Meanwhile, Jen is always wrong. At home with the kids, she’s an anachronistic housewife; at work, she’s ditching her kids to nurture selfish professional ambitions. Somewhere, lurking at the root of this all, is the tenacious idea that men should have a career, whereas women must choose between a career and being at home.”

Other thoughts that should make their own blog post but aren’t because the next posts will be on baking (hopefully) or math (that’s ok too): I was very eager to read the NYT op-ed titled “What Makes a Woman?” but it was a bit more defensive and less full of brainstorming of collaborative solutions between cis and trans women than I was hoping.  On a similar note, my friend recently posted about her friend’s blog about being a trans*dude , which I’ve started reading and I agree with what she said “I’ve learned so much reading it!”  I still have a lot to learn (like I don’t know why that asterisk is there in trans* but I’ll find out).

I am a woman in math

10 Jul

That should be pretty clear from the fact that I put together that women in mathematics symposium a few months ago.  And from all the photos of me with my baked goods.  I hope that the fact is not necessarily obvious from the fact that this is a baking and math blog as, I’m friends with plenty of men who bake and do math. (In fact, I’ve linked to this guy and his cookies before on this blog).

I’ll probably end up writing many things about the fact that I am a woman in math, but this is just a short post about my thoughts right this moment.

I spent some part of the afternoon reading this heartbreaking blog “What is it like to be a woman in philosophy?” where women and men write in with their stories of misogyny, sexism, and battling these things.  As I read I kept thinking to myself “phew!  Glad I have my adviser!”

Since I’m not a woman in philosophy I didn’t submit this story to them, but I do want to write it somewhere, so here goes.  One day there was a marvelous talk in our geometry, topology, and dynamics seminar, and a few days later I met with the professor I’d come to UIC to work with about various math, as well as a conference coming up.  He asked who was speaking at the conference, and I said something like “that girl who spoke on Tuesday here.”  He gave me a look, and then said, “You mean the woman.  You would never refer to a man who gave a talk as that boy.  Women have it hard enough in our field, we don’t need to make it harder by demeaning them.”

I was rightly chastised, and that was also the moment when I decided that I would ask him to be my adviser- we discuss math together well and he’s brilliant, but that’s true of many professors.  He values feminism as much as I do, and can also rebuke me when I need to be.  So he’s fallen into the category of my mom, my boyfriend, and my closest friends.  Fantastic!

As for using the word “girl”- my friends say things like “girls’ night”, and this is a common phrase in our culture (there’s about 975 million more of those, these are just the first things that came up when I googled “girls night”).  But my adviser’s right that I absolutely should not refer to a female mathematician as a girl.  Perhaps it’s unprofessional language?  Or perhaps we’ve just internalized our systematic infantalization (whoa calm down there Yen you’re not a sociology student you don’t even really know what those words mean).  So no conclusions on this (my friends didn’t have them either).

On a positive note, two of my math heroes are interviewed here, over at Roots of Unity, on being women in math.  They actually talk very little about being women in math, and more on just being them and being awesome in math, which is fantastic.  They do both mention the importance of role models, and I hope they both know that they’re huge role models to many, many graduate students at their respective universities.  Because they’re mathematicians, and women, and totally cool with both of those things, whereas a lot of us grad students sort of nervously juggle the two when we meet strangers.

I could just keep writing about being a woman in math but I will stop.  Look at this hilarious stock photo (I love stock photos)

http://www.thejanedough.com/researchers-blame-biological-clock-for-lack-of-women-in-math/

Roasted asparagus, how to cut an onion, math update

17 Jun

It’s summer!  Our local farmer’s market is teensy and the prices are about quadruple what I paid when I lived in Santa Barbara at a market about three times as large, but it’s still a good place to get incredible produce and goods that I can’t get elsewhere.  On Sunday I picked up a big bag of cheese curds, some pork sausage, mizuna (YUM) and fresh asparagus.

I don’t like asparagus.  I didn’t grow up with it and it’s always sort of fibrous and stringy and makes my pee smell funny.  But holy cow this asparagus was incredible!  Melty and tender in the middle of the thick stalks, with a pleasant bit of crispiness on the skinnier ends.  And so easy to make!

I don’t normally advocate preheating but PREHEAT THIS because you don’t want overdone/limp asparagus.  I did 415, then laid out my ingredients

I’ve never been much of a gardener but I have a great interest in paper bAGriculture

I love garlic and throw it on all savory things, so these handful of asparagus (\$4) got two cloves minced up on it

I should post this with my hipster social media account and title it MINCEtagram

Then a squeeze of lemon and a generous pour of olive oil.  Roll the asparagus around in the oil on a baking tray/cookie sheet, and toss it in the oven.

Yeup, that is some good sheet (pan)

Let it roast for between 8 and 20 minutes (I did 14)- it depends on how crisp you like it.  Pull it out and stick a fork in one and see if it gives (so is tender).  Then sprinkle with salt and pepper and devour.  The garlic, lemon, olive oil, and salt play so well together.

Kick-assparagus more like it

Next, HOW TO CUT AN ONION.  It drives me nuts when people cut an onion crosswise: only do this if you are making onion rings or want grilled onion rings rather than delicious caramelized onion wedges in a pile atop your burger.  USE THE STRUCTURE OF THE ONION to aid your cutting: onions are  partially pre-cut when you get them!

If you cut onions the wrong way, you’ll end up with awkward circular pieces and will have to move your knife in a weird circle to get dice that you want.  I’m far from a precise and careful cook, but I know that if you want things to be cooked evenly, it helps to have them around the same size/shape.

First, set your onion upright, with root on bottom and pointy Alfalfa thing on top.  Halve it that way, so instead of rings you see a Georgia O’Keefe painting in the cross section.

Step 1: chop chop from bottom to top

Then peel off the paper (or do this as a first step if you want) and lay a half down flat in front of you.  You now do a Cartesian grid cut on that, and you’ll get a nice dice from the layers of onion separating themselves.  I like cutting “vertically” (parallel to the spine of the onion) first, then transverse to the layers, but either way works.  In this photo I’m doing the transverse first.

Step 2: Make a grid, it’s up to you!

The point is that this method saves you from having to make a third series of cuts in the depth direction: onions are 3D, and we do one depth cut, then make a grid of length-width cuts.  I asked the internet “how to cut an onion” and it agrees with me, and has this nice cartoon picture:

Also, a good DON’T DO THIS photo from about.com:

I was unwilling to take this photo myself so I’m glad it’s on the internet so I can show you.  DON’T DO THIS.

Finally, what am I up to this summer?  Once you’re done with prelims, your next milestone is the thesis defense, which is a long ways away.  So I’m spending the summer reading math and trying to find a thesis topic.  This is what’s on my plate so far:

Notes on Combinatorial Group Theory by Charles Miller, available online and a pleasant read-on-your-own, graduate student level document, about 100 pages.  A two week project for me while reading other things/relaxing for summer!  I’m going to ask my advisor about a few parts I didn’t understand but overall I’d recommend for a short independent study.  Lots of good exercises.

A Primer on Mapping Class Groups, by Benson Farb and Dan Margalit, also available online but illegally?  So you should just google it.  This is an actual textbook, about 250 pages, much slower reading than the notes but still much more readable than most math texts.  I’m reading this with my friends Ellie and Mike.  I’ve actually attempted to read this before with the incredibly patient Jon McCammond and didn’t make too much progress.  Second time’s the charm!

Non-Positively Curved Cube Complexes by Henry Wilton-notes from a course he taught at Cal Tech in 2011.  These are somewhere in between the first two in terms of difficulty- I’m definitely asking my advisor for help on lots of the material in this.

Sort of a random post, but that’s me!

Brainstorming, also, Cantor sets

10 Jun

So I studied for those prelims for a few months (maybe two) on my own, reading old notes, highlighting things, rewriting relevant proofs, running through my homeworks, attempting extra exercises from the textbooks, the usual extreme studying.  I don’t know about you, but I am SO MUCH BETTER with other people when doing just about anything.  Knowing someone else will eat my baked goods makes my baked goods better.  Being accountable to a colleague makes me justify my ideas more and I write down fewer false things.

Math is a creative endeavor.  People seem to think mathematicians are somewhere between human calculator, engineer, and crazy person.  But we don’t get grouped with, say, artist, writer, poet, all that often.  I’m going to quote some wikipedia here:

Plato did not believe in art as a form of creation. Asked in The Republic,[18] “Will we say, of a painter, that he makes something?”, he answers, “Certainly not, he merely imitates.”[16]

Now, as with many creative things, there are certain tools you can use to help with math.  Going on walks is a good thing.  Eating breakfast.  Naps, even.  But the thing I love, and the thing that helped me pass my prelims (whoooo), is brainstorming.  A few weeks ago I read an article that a facebook friend posted (isn’t it funny how you know what I’m talking about when I say “facebook friend” rather than “friend”?  In this case, a guy I went to high school with) about how brainstorming is basically stupid and broken.  And, given the parameters that the article offers, it’s pretty right: get in a group, generate lots of ideas, don’t be critical.  I don’t have a lot to say about brainstorming (though Jonah Lehrer for the new yorker and a guy from Stanford’s d.school do and these are both fascinating), but I do have things to say about problem solving with a group.

Brainstorming, as this three-step outline is, isn’t the way solving problems with a group should be done.  He’s on point with steps one and two, but step three should be different: be critical.  Fight for your ideas.  Fight other people’s wrong ideas.  I work with other grad students a lot, and I often don’t realize that I’m completely wrong until I’ve been talking about an idea for a few minutes.  I need them to fight me in order to learn and understand and be better.

The idea behind brainstorming is that if you’re critical of others’ ideas, they won’t want to share them and will clam up.  I absolutely felt that way for the first year or so of graduate school- I would feel that my peers shot me (personally) down, and that I didn’t have any good ideas.  What finally changed my mind and made me more combative was realizing that I do have good ideas, sometimes.  Not all the time, probably not even 50% of the time, but sometimes I’m right, I’m absolutely right and I can prove it if you-just don’t shoot me down or interrupt me or dismiss me.  These last three things still happen, and I’ve learned to fight back in a nonconfrontational manner- it’s easier in my case because everyone in the room just wants to reach a solution.  Some people have egos (I do too), but we learn to listen to each other to the extent we’re capable, and to make each other listen when we can’t extend ourselves to do so.

Basically, I agree with this guy: “Innovation Is About Arguing” is the first bit of his title.

On a totally different topic, let’s talk about the Cantor set.  It’s a pretty cool subset of numbers with lots of unintuitive properties.  Building it is fairly straightforward.  Look at a number line.  Focus only on the segment between 0 and 1.  Cut the segment into thirds, so you’ll have notches at 1/3 and 2/3.  Delete the middle third (1/3,2/3) so you end up with two segments.  Do this over and over again, deleting the middle third from each segment you have at any one point.  Look at the picture

I DID NOT MAKE THIS PICTURE it is from wikipedia.

• It contains no open intervals, since we’d cut out the middle third from any interval that showed up.
• Despite that, no points in the Cantor set are isolated: if you take a point in it and look in a teensy neighborhood (like, $\pm .00000001$ teensy), you’ll still find other points from the Cantor set.
• It’s uncountable, which means literally that you can’t count it , even if you had infinite time.  Counting numbers are countable (1,2,3…), and so are numbers that can be written as fractions (1, 2, 1/2, 3/4…), but all the numbers between 0 and 1 aren’t (this is Cantor’s diagonalization argument, which I’ll blog about some time).  You can show that the Cantor set is uncountable by making a function from it to the interval [0,1], which hits all the numbers between 0 and 1.  If it hit all the numbers between 0 and 1, we say the map is surjective or onto [0,1].  If it’s surjective, then for every number in [0,1], there’s a number in the Cantor set that maps to it, so the Cantor set is bigger than or equal to [0,1] in size.  Since [0,1] is uncountable, this means Cantor set is too.
• If you consider the set $C+C = \{x+y : x,y\in C \}$, you get an interval, which is way incredible and unexpected because it contains no intervals- it’s like putting together two sets of discrete bread crumbs and magically having a loaf.

You think about that last bullet point, and I’ll do a post later proving it.  Hint: know how we can think of any number as a decimal expansion?  Well, you can also use base 3 and do a ternary expansion, so 1/3 = 0.1, 2/3 = 0.2, 4/9 = 1/3+1/9 = 0.11, etc.  The Cantor set, by the way we defined it, is all the numbers between 0 and 1 that contain no “2”s in their ternary expansion.

Hasta later!

3 Dec

Quora asked: What do math grad students do all day?  That’s an excellent question, and here’s an excellent answer, which compares doing mathematics with learning everything you can about vacuum cleaners, without ever having seen one, and using only words with four letters or less.   A choice tidbit: “OK, your next idea: I can use a vacuum cleaner to clean cats! That’d also be super-useful. But, alas, a bit more searching in the literature reveals that someone tried that, too, but they didn’t get good results.”

This is what math grad school is like. Only math is a significantly less happy animal.

I also mentioned this article to my adviser, and he said that mathematicians are like vacuum cleaner repairmen and innovators, but that we should also know something about the fridge, for instance.

Next, what inspired this blog: some guy talking about baking and math.  An excerpt:

For my money, the best part of the baking process (aside from the delicious final act) is the careful and precise initial measurement of the ingredients.  Keeping an accurate account of the relative proportions of each piece of the recipe is a hallmark of baking, and reflects the nature of baking itself: one part art, one part science.

I read this and immediately disagreed with it.  But I’m not a very precise person and my baked goods are made with more heart than head.  Maybe doing math will make me a better baker because I’ll care more about little details like how much of each ingredient to use, or following a recipe exactly.  Just realized that I totally lied without meaning to in the fruitcake post: definitely ran out of brown sugar and used white sugar with a random splashing of molasses instead.

Here’s my deal: life is imperfect and unpredictable.  Even if I measured everything exactly, my results will never be exactly the same each time.

Ferris Bueller had impressive amounts of control and luck, but even he couldn’t save Cameron’s car.  My cookies are sometimes like that car.

Sometimes it’s humid, sometimes I open the oven door more allowing cool kitchen air to enter, sometimes my baking soda is a little old and won’t work as effectively.  I don’t mean anything extreme, as in because I don’t have absolute control I should cede all of it; I just mean I don’t think I need to take it so seriously.  Butter, flour, sugar can’t go so terribly wrong if I mess up.  The situation is in my hands as far as it is, and that’s the way it goes.  That’s how I see math too: I have no control over it, but I can wrestle with it as far as I can, and that’s the way it goes.

Finally, a kind of remarkable insight into the mind of a baking engineer, which talks about Moistness Values and Fat Content, which leads to a randomly generated recipe builder.

Just look at those equations! So shiny!

I find computers and engineers both very intriguing in an anthropological studies sort of way.  My last paragraph makes it pretty obvious that I’m not of the exacting engineering mindset, but I certainly respect and admire it.  In any case, I’m very curious to try some of those randomly generated recipes in future baking endeavors.