Saturday, July 9, 2016

Tess Elation

Special opportunity today, to meet and greet the author of a fun new math book.

Emily Grosvenor (twitter, website) is a "reporter, travel writer and essayist" who has gotten all the way to Mathland with her illustrated children's book Tessalation!. After a successful Kickstarter, the book is available for pre-order or as an e-book at Amazon (free for Kindle Unlimited) and direct order from Waldorf Books. I am between having received my electronic and my physical copy, and find the book just charming.

Emily was willing to answer a few questions, so here we go!

Q. Do you remember when you first noticed tessellations?
No. I first learned about them in 4th grade -- every part of my creativity seems to have it seeds in the 4th grade, it must be a seminal year for development. We did an activity in gifted class where we made tessellations. I made an uninspired tessellation of a seal jumping out of the ocean. But it was fun. And clearly it stuck, since I was still thinking about it almost three decades later. 

Q. Do you have a tessellation from someone else that you like especially? (Maybe a favorite Escher tessellation?)
I'm a big fan of Horseman But honestly, I think my favorite is just the simple hexagon tessellation. We just got bees at our home in McMinnville. I have great hopes I'll see one there, soon!  

Q. What makes tessellations worth thinking about and exploring for you?
I find patterns soothing to look at. Honestly, visual culture bothers me a lot. Usually there is so much going on, and I get distracted easily. But with tessellation you can take in the chaos and then let the eye, and the mind, settle on an individual part. Also, I am very compelled by the idea of seeing myself as a part of a greater whole. Not just with my family, but with my community. One of themes behind Tessalation! is that the world is not as chaotic as it seems, that there is an inherent beauty and order to it, and we can be a part of it. 

Q. What was your experience on the first World Tessellation Day?
It was a crazy day! My best friend was in town with her kids and husband and we threw a party at the McMinnville Public Library. We had a tessellation station, we screened the book on the wall, we had hexagon cookies, tiling turtles tessellated games, and coloring pages from the book. It was a BLAST! I was so tired. I probably should have been tweeting out tessellations all day, but there were a couple hundred people around the world who were posting images. In all, I was happy with the outcome. When I got a chance I checked in and retweeted, liked or posted what I could. People who like tessellations really love them. Also, it's a visual meme, which makes it easy to get behind.  
Emily created World Tessellation Day, and used it to launch the book. She has a fun post about the book launch here. The twitter stream for #WorldTessellationDay had a ton of fun participation from genuinely around the world.

Q. Why should it be an annual event?
Why should anything? It's fun. Fun to post, fun to make, fun to see all of the creativity happening around the world. I was most impressed by the posts coming in from Spain showing all of the tessellated mosaics available in plain view in public spaces. There are a lot of silly holidays. We share World Tessellation Day with National Flip Flop Day, for example. Who cares about flip flops? Well, someone does. If anyone cares about something there should be a day for it.
Didn't realize it was also flip flop day - despite Danica McKellar's tweet. Doh.

Q. What’s challenging for you when you are developing a tessellation?

I actually don't do a lot of designing tessellations. Notice I did not make the illustrations in the book, for example. But I did try to create the feeling of tessellating in the rhyme scheme and overall meter of the book. I wanted there to be a strong connection between how the text feels when read out loud and what you are looking at. Can words tessellate? I think I tried to do that. 
Q. Do you have a general process you follow?
The best part of this project has been how it has opened this entire world to me of math play, tessellation and visual culture. I launched this project thinking that tessellations are awesome but not really having any idea of the scope of talent out there or of the artists who are working in tessellation. I've been touched by people who have reached out from around the world to share in the excitement. But my favorite moments are when my 3-year-old, Griffin, finds them in plain sight. Just yesterday he got a new pair of Timberline sandals and said: Mama -- there's a tessellation on my foot! 

Thanks, Emily, for the book, and holiday and interview.

Oh! I should have asked how she got connected with her talented illustrator, Maima Widya Adiputri (Tumblr, FairyFrame).

Find out much more about this book from other stops on the booktour.

Find out more about tessellations from the resources on my page.

Or start immediately making your own! One of my most recent ones to play with on GeoGebra is a funky hexagon one, with a glide reflection similar to what the Horseman has.

#TessellationNation, now #tessnat, is coming at TwitterMathCamp16. Christopher Danielson was thinking it should be based on people's questions, so hop on Twitter to chip in, or share them here.

So far:
Christopher ‏@Trianglemancsd
We proposed this session as one revolving around our questions. Maybe you could share of those here before TMC?
I would like to learn more about how to categorize tessellations.
I wonder about the relationship between "tiling" and "tessellation".
I am super curious about the tilings in mosques. Are they tessellations? Why do they so rarely appear in the math analyses of tessellations I've encountered?
#tessnat There's a start on where my mind is for #TMC16. What about you, Tessellation Nation?

Malke Rosenfeld ‏@mathinyourfeet
1. Hi #tessnat. My goals: try & try again. I would like to play with diff kinds of tiles to help me ask new questions.
2. After I play I'd like to talk abt my notices/Qs and then design a tile that is simple but creates an interesting result/design #tessnat
3. I would also like to observe someone designing/creating an anthropomorphic tiling if that ends up happening. #tessnat

Megan Schmidt ‏@Veganmathbeagle
@Trianglemancsd OH!
I want to draw the things, whatever that means. #tessnat
Ok. My needs are "be in the #tessnat morning session." :)

Friday, July 8, 2016


Quick game idea dreamed up while watching tennis. I'm excited about it, but it's untested. Maybe someone will playtest with me at TMC16?

2 or 4 players (doubles, naturally)

Materials: deck of playing cards, score sheet.

Set up: deal each player 5 cards. Randomly determine who is serving first.

Tennis scoring:

  • In a game, you score love (0), 15, 30, 40, game. But games have to be won by two points. If you get to 40-40 it is deuce (tie). One point from there is Ad (advantage). If the player with ad wins, it's game. If the other player wins it's back to deuce. 
  • It takes 6 games to win a set, but you have to win by two games. 
  • If you get to 6-6, there's a tiebreaker. The server serves once, then players take turns serving twice. First to 7 points wins the tiebreaker and the set, but, everyone say it, you have to win by two points.

(Game design aside: having to win by two points is one of the best remove first player advantage mechanics ever. The tennis tiebreaker was also a great innovation.)

A match can be one set, best of three or best of five. I recommend playing one set to start.

Playing a point: players draw up to 5 cards. The server can play a card or flip over the top card. Each card played has to be higher than the card before. Players can play two cards together as a sum. If a player can't play or chooses to pass, the other player wins the point. The same player or team serves for an entire game, then the other player or team serves the next game. Continue alternating serve throughout the match. When the deck runs out of cards, shuffle the played cards to continue.

Special cards:
Ace: 1 on a serve (flipped from the deck) 11 any other time.
Face cards: count as 10, but can only be played as part of a sum. However, King beats Queen beats Jack beats 10, so Queen+2 beats 10+2.

Variation: card pairs are multiplied rather than added.

Doubles: in a doubles match, either player on a team can play a card, or play a sum, or a sum can be made with one card from each player.

Notes: The weird scoring is the downside, but kids learning tennis scoring is not a bad thing. And it's a classic for a reason. I'm really excited by the strategy here. When to give up on a point, how to get rid of low cards, which pairs to form and play.

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.

Monday, May 30, 2016



Good for sports, bad for surgeries, okay for a test, depending on your standards.

I'm very grateful to Anne for starting #MTBoS30, and the early adopters that greased the skids for me to do it to. I've blogged more this month than I did all last year!

The posts I'm the most glad I wrote:
  • Queen of Quilts - just for the appreciation of Elizabeth, but I like that GeoGebra bit a lot as well.
  • Not Subtracting - such an opportunity, to tagalong in a conversation with Dan Meyer, Marilyn Burns and the MTBoS.
  • Commentary - for a reminder to myself to think this out. My colleague Clark Wells said this is what he was talking to me about, so there's hope for more local discussion.
  • More Tessallations - for the chance to share student work and math I love. Several students have blogged about it since, and I love that kind of resonance.
Amazingly I still haven't gotten to everything I wanted to write about. A couple of new games, some lessons from last semester, some hand drawn mathart, and the 3rd grade mathart project. (And the Hamilton song parody.)

My take away from the month is that I like blogging. Why don't I do it more?  I think because I've forgotten the reason I started, which is the reason I ask my students to write and try to get my kids to write. It is good for you on the atomic level. The conversation, the curation, the community - those are excellent side dishes. Writing makes you think.

So do I make a commitment to more? 2/month, 1/week, every Monday? I don't know - I'm not very good at these commitments. After not having given up the MTBoS30 each morning, I know I should be ready to be Dread Pirate Roberts. But I'll settle for being a member of the Goon Squad.

Sunday, May 29, 2016

All the Way

Missed another #MTBoS30 post yesterday, but it was in the service of a day chock full from 6 am to midnight, so no regrets. Time with one of the bravest people I've ever met, my son doing well on his first dan (permanent blackbelt) tae kwon do test, church, dinner with family...

With a free Sunday morning, we (as a family, even) finally got to watch the Lyndon Johnson biographical movie All the Way.

It's amazing.

I'm 51 (with considerable less grace and style than this 51 year old) and this was my birth year. I don't remember it, then, but this was the backdrop of my first memories. Kennedy and King being shot, Nixon, Vietnam and Watergate was what I knew about politics and government. It was amazing to watch this movie, with its decidedly modern viewpoint. It took decades for me to move beyond childish black and white images of these people and my black and white judgment of their actions. The filmmakers are good at filming what was actually said then in a way that makes connections to today possible.

Looking back, it's so easy to identify the right side of history. To see bigotry and name it when we are free of it (we think!). My spouse is excellent at challenging us (okay, me) to see what is inequitable now.

One of the things I enjoyed most was seeing Robert Moses portrayed as a young man, working for the Student Nonviolent Coordinating Committee. That fits seemlessly into his work as a math educator. The Algebra Project, interviews (on a Selma anniversary, NPR), or his book Radical Equation. He is one of my dearest heroes.

In All the Way, Dr. Moses is portrayed as too radical to effect change. To be so convicted to principle that he can not compromise for some gains. Dr. Moses makes direct connection between the idea of civil rights and the empowerment of mathematics education. It's so complicated, it could be easy to walk away, and understandable when people do. Education cannot solve poverty, but it's such a necessary part of any solution.

Are we not able to affect change because we need an LBJ? Someone with the conviction that can see a path to equitable education and is enough of an asshole to get it done? I think we are accountable both for holding and proclaiming the principles and doing the problem solving to get to a better place. But I am on one side of it, and often in danger of fighting the LBJs who are probably on our side.

I am humbled by how generally useless academics are in society.

One of the reasons that the Math-Twitter-Blog-o-Sphere (almost as ridiculous sounding as "snick") is so encouraging to me. We are self-organizing and devoted to the education of kids independent of what the government or publishers or pundits say. Now I'd love to see the NCTM play the role of the NAACP in pursuing systemic change, too, but I'll take what I can get. And this band of teachers, working one or six classes of students at a time is getting a lot. God bless you all in your work.

We will overcome.

PS. In the Selma anniversary interview, Dr. Moses is asked what he'd like to hear President Obama say in his address. Re responds: "I'd like to hear him speak about education. We can do all we want about voting and everything else, but if we don't provide an education for every child in this country that's what they need for the 21st century then we will just be sending them to the criminal justice system. We do not have in this country an education system that is dedicated to educating every child, so I'd like to hear him speak out about that." Me, too.