Tuesday, June 21, 2016

Book Club - Summer 16

In my math capstone class, the students can pick their own book from a list. Then we have a day for book chat. These are my notes. Links on student names go to their reviews on their own blog.

Journey through Genius, by William Dunham: Nick. Read most of this... explores a handful of the most important theorems and proofs from math history. If you're reading this, the book won't be that bad. If you're trying the proofs, it can be very difficult, and I wouldn't recommend it. Dunham claims that Archimedes is the greatest of the Greek mathematicians, the crown story. Personal stories of the mathematicians, too; for example, Cardano's tortured life. Not a lot of fun reading, but some really good explanation of why they use the methodology.

Love and Math, Edward Frenkel: Rebecca, Kourtney, Erin. "It's hard work being a teacher..." Great about the hard work being a mathematician, and the difficulties in being Jewish in an anti-semitic world. He had to literally scale the walls to get his math education. He was also moving in discussing how important collaboration is, and how important making mistakes is. The math is really high level - we could see some familiar things from abstract algebra, but there were good analogies for a lot of the ideas. "Where does love come in?" Love of math. Started in physics, but was convinced to go deeper, mentor by mentor.

The Math Book, Clifford Pickover: Anthony. Each page is like a wikipedia article about an idea, but nothing in depth. References are given so if you want to go deeper you can. "The integers came from God, and all else are hand made."  - Kronecker. It covers big ideas, inventions and famous mathematicians. Lots of fun ideas, like the birthday paradox or the infinite monkey theorem. The Johnson Theorem, riddles like the barber paradox. History of zero... I'd recommend this for teachers for the history.

Joy of X, Steven Strogatz: Jordan Drake, Nick. Easy read, a narrative. He explained negatives, but noted how we as a society avoid them (building floors, bank statements, temperatures). Strategies for finding your soulmate. What made it so easy to read was taking complicated ideas, like sine and cosine, but then gives real life examples and good visual images to support it. I went through math doing it because I could, but this gets at why these things work or are true. He gives a lot of practical applications. He gives the reasons for "why are we using this?"  Really fun to read with a lot of 'aha!' moments.

e: The Story of a Number, Eli Maor: Marty. I thought the whole book would be leading up to e, but we see it already in the third chapter. It starts with Napier, goes through logarithms, explores finance, and then calculus (Newton and Leibniz), ... It was at times interesting and boring. The most beautiful formula which makes a connection with imaginary numbers. Lots of appendices of intense proofs.

The Number Mysteries, Marcus du Sautoy:  Heather, Brianna
5 different math mysteries, got into the history of the ideas,
  • primes; the building blocks of all numbers.
  • geometry; nature is as efficient as possible.
  • tricks for games; confusing but interesting, Monopoly and more.
  • codes; Everything is a code, languages, DNA, ISBN, modular/clock arithmetic.
  • prediction; patterns are detectable, which make things predictable. Seasons lead to the calendar, etc.
"Coolest thing?" How I could relate math to all these things I had never noticed. There was just so much information.

Mathematician’s Lament, Paul Lockhart: Hannah.  K-12 math needs to be scrapped. Math is an art, it's about playing and imagination. Instead, teachers give facts and formulas for memorizing. Takes away the creativity and engagement of solving. Teachers try to relate it to life when it doesn't. It can be fun because it doesn't relate to your life. A good problem is anyone you don't know how to solve. You want students to struggle and be frustrated. Geometry is the most destructive because it destroys proofs. Instead of being charming, it's a boring list. Write a paragraph of a proof, tell the story of your thinking. Why can't 1 + 1 = 0? Even/odd + even/odd, sum of odd numbers, ... so many ways to reveal the nature of math. "As a future teacher did you find yourself agreeing or disagreeing?" The get rid of the curriculum and let every student figure out what they're working on - I disagree. But the emphasis on memorization needs to go. "Math can be fun when not related? That's really clever. Counter to the message we get."

The Calculus of Friendship, Steven Strogatz: David. Less of a math book, and ever increasing life lessons. The teacher retired and became a famous white water rafter, which is connected with limits and infinity. Irrationality, chaos theory, etc. The monk and the mountain. Will a monk who walks up the mountain and down in irregular patterns ever be at the same point at the same time. Inspiring about going into being a teacher and the effect you can have. Readable even if you don't know calculus.

Mathematical Mindsets, Jo Boaler: Michelle, Tabatha Lathrop.  Growth mindset = you can get better at how well you learn things, fixed mindset = you can learn things, but you can't change your intelligence in an area. This really affected me. The brain research is interesting; when you're making mistakes is when you're learning.  Feedback makes a big difference. Then she connects with math mindset. Most effective teaching is when learners explore the question, and then get  explanations of how and why. Using what the kids came up with is helpful, with engagement to start. Students will say they don't like math because it's too much answer time than learning time. The faster they can do math the better, kids think, when the reality is almost the opposite. Kids don't ask 'when are we going to use this?' in other subjects. I am literally a different person because I read this book.  (Others connected to Carol Dweck's Mindset in response.)  It's interesting as an adult learner, trying to think about where you're fixed or growth.

The Magic of Math, Arthur Benjamin: Andrew Meeuwsen. Topics align well with the course, but not a lot of the fun history. Lots of worse than dad jokes: mathematician dad jokes. Lots of tricks for doing specific problems. Many connections to his mathemagic show. FOIL, squares, magic of 9, magi of counting, magic of proofs... a lot of good math, but a lot of filler, too. The book has a steep slope, from arithmetic to calculus. My favorite was about infinity. It covers a lot of the subjects from undergrad mathematics.

Jerry missed the discussion, but has a review of The Calculus Gallery.

Sunday, June 19, 2016

World Tessellation Day 2016 Gallery

I got to hand draw a couple tessellation attempts for a service project Playing with the patterns later made two that I liked.

There's a classic semiregular tiling pattern with squares and equilateral triangles. I wondered if you could make one that had three triangles at each vertex but in different combinations.

Turns out maybe not. Everything I tried wound up with a spot with 6 green triangles. But I did find a new to me combo. I like that it has one 60, one 90 and one 120 degree angle at each vertex. (On MathToyBox)

The other one was built around a kite with a 90 degree vertex. I had to make that one afterward in GeoGebra. Not a lot of flexibility in design, but I liked the square & rhombus gaps.

Some of what I saw around Twitter and Facebook was just so delightful, I wanted to archive it.

Another great Cristóbal Vila video, Ars Qubica, via Daniel Ruiz Aguilera.

The founder of this here holiday sharing pics from a tessellation get together. With excellent toys.

I guess math doesn't suck!

Some excellent tiles. That's the Cairo Tessellation at right!

Now some awesome environments...

And nowsome action shots! Love the ones from Simon Gregg's class especially!

Happy World Tessellation Day! See you next year.

No better way to end than Jennifer Silverman and Steve Vai shredding!

Friday, June 17, 2016

World Tessellation Day One

Emily Grosvenor came up with the idea of a World Tessellation Day in connection with her charming children's book, Tessalation! June 17th is M. C. Escher's birthday (1898) and there could be no more fitting day.

Tessellations are definitely my favorite topic in mathematics. The intersection of history, art, geometry (shape and transformation), algebra, and even analysis... what could be better. Some of the greatest surprises in math have come from tilings (quasicrystals, pentagon 15) and some of the greatest mathart. I've seen them engage students of all ages.

For my post, I've been thinking about so many things, but that coalesced into a 'My Favorites' post:

My Favorite Tessellations

HM: pattern blocks.

From a recent class, Hannah made this neat dodecagon and octagon tiling. They remind me a lot of these from Simon Gregg and Daaniel Ruiz Aguilera.

10. Non-Euclidean Tilings

Hyperbolic, especially. Here's a beauty from John Baez's Google+ page.

9. Pythagorean Tiling

A tessellation that demonstrates the most famousest of theorems? That's saying a lot, that is.

8. Archimedean (Semi-Regular) Tilings

So what combinations are possible? Is this all of them? Could the semi-regular tilings be the first of these kind of problems?

And then you add the delicious topological feature of dual tessellation relationships...  The gif on the right is from thinking about a Sam Shah prompt on this idea. (On GeoGebraTube)

7. Pentagon 15

How deep are tessellations? They still surprise us. Every quadrilateral tessellates.

A monohedral tiling is a tiling where all the tiles are congruent. An isohedral tiling is a monohedral tiling where for any two tiles there is a symmetry of the tiling that maps one tile to the other. There are exactly three types of convex hexagon monohedral tilings. (Here's a good NRICH problem with one.) Every convex quadrilateral has a monohedral tiling. And we knew all 14 convex pentagon monohedral tilings. Several by one of my favorite mathematicians, Marjorie Rice. (Her website.)  And then they found the 15th. (GeoGebraTube

6. Pinwheel Tiling

Straight from the mind of John Conway.
5. Spiraling Polygrins

I went from fond of these to berserk when Christopher Danielson started making them. (On GeoGebraTube or TMWYK store.)

4. Rep-Tiling

When a tile can be composed to make a larger similar image of itself. Then it makes a tessellation by either deflating each tile into smaller images. Or inflating by composing larger and larger similar arrangements.

3. Penrose Tiling

These were my exposure to quasiperiodic tilings. There properties are many and wonderful. At one point I was stuck on my thesis and my advisor (Nigel Higson) gave me these to work on. My best ever Mathematica program generated them by projecting n-dimensional integral lattices onto an intersecting plane. For part my thesis I then made quasiperiodic integral operators out of them.

2. Islamic Tilings

Most recently, Daniel Ruiz Aguilera got me working on the Qarawiyyin Mosque Tiling. (GeoGebraTube) Endless riches with new work still being done. As a bonus, these are often interspersed with knotting, another favorite.

1. Escherized Tiling

Instead of mine, let me show some ooooold student work from a couple of preservice art teachers in one of my first courses taught at Grand Valley. I still keep these in my office.

Current: Self-tiling. Since Math Munch unveiled this great Lee Sallows self-tiling I've been curious. They deflate in only one way, but inflate in four ways - I can't figure out what that means about the structure. (GeoGebraTube)

So many types that didn't make the list. And despite the numbering, I'm just crushing on them all.

I hope one of these pave the way for you, or maybe showed you a new kind, or just reminded you of old favorites.

And happy first World Tessellation Day! Tile on, brothers and sisters.