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.

Curvy

Not too much time today, so I'll just post the GeoGebra thing I've been working on.

I have a Tumblr account as well as this blog, it's a fun mathematical space, and the reblogging is an interesting spin on curation. Here's a post I wrote for a 'my favorites' at Twitter Math Camp 16 to help #MTBoS folk get started on Tumblr.

One of my favorite ever mathart sources was @tilman's http://geometrydaily.tumblr.com/.  There's a site that has a bit of a similar flavor, http://www.dailyminimal.com/.

The other day, they had this:

Nice, right? Rotating ellipses... what I always try to think of is how can this be generalized? Rotating conic sections (definitely sometime will do!)... what if it didn't just rotate around a point? What if it followed a curve? One idea I really liked from the dailyminimal was having a start and stop point. So ...

el bigote

You can choose how many ellipses, make your own parametric curve or hit the random button, design your own ellipse, use the red and green to pick starting and stopping point... hopefully lots of flexibility.

On GeoGebraTube, so go play. If you make something cool, let me know!

Polygonal Spiral

I have a fascination with spirals. Exhibit 1, my GeoGebraTube materials, searched for spirals:

That's some of them...

By the way, if you haven't been following Megan Schmidt's spiral adventures, you're missing out.

I've been interested in polygonal spirals for a while, but then my student Andrew's tessellation got me thinking again.

It's Archimedean since the spirals have a kind of constant width. At first I thought it was triangular, but it's clearly hexagonal. Interesting that two of them fit together to fill the space... that's something I need to think about more.

To build them in GeoGebra I made a list of N directional unit vectors, and then a scaling sum to get a spiral of points, like 1*v_1, 1*v_1+2*v_2, 1*v_1+2*v_2+3*v_3, ... with a modular function to reuse vectors in order.

Then I connected up to points 1/Nth of the way to corresponding vertices to make trapezoids. Then I rotated them N times, around a center I located by intersecting the perpendicular bisectors of points I wanted to correspond.

GeoGebra geek paragraph: The colors are the hardest thing to get, because in GeoGebra you can't set the color of different elements of a list. My current workaround was suggest by someone on the GeoGebra forums a few years ago, and I keep reworking it.

Execute[Sequence["Delete[R_{"+i+"}]", i, 1, oldN]]
Execute[Sequence["Delete[R_{"+i+"}]", i, 1, oldN]]
SetValue[oldN,N]
Execute[Sequence["R_{"+i+"}=Element[list8, "+i+"]", i, 1, N]]
Execute[Sequence["SetDynamicColor[R_{"+i+"},  "+i/8+","+(.5+(i*(-1)^i)/(2*N))+","+1-i/8+",.75]", i, 1, N]]

Essentially you constantly create and destroy objects from the elements in your list, and then set their color.

Usually I have a list of color names and run through them with the SetColor, but in this one I wanted a higher opacity, so made up a way to set R, G and B values in the SetDynamicColor command. 

Here's the result! It was hard to think about how else to dynamicize it, since it's a pretty rigid structure. Any ideas?

On GeoGebraTube, too.

Hexatiles

Daniel Mentrard is one of my GeoGebra heroes. The guy can seemingly do anything, and he is generous with help for his lessors.

Recently he posted this:

Don't stare too long - this is hypnotoad territory.

I misunderstood it. I thought it was a single triangular tile, sometimes flipped and sometimes rotated. So I got playing around with it, trying to generalize to other cases than the hexagonal. I made a simple triangle, with two triangles in it. The red has vertices on the border, and the blue has vertices in the region. (The commands are Point[<object>] for the first, and PointIn[<region>] for the second.) The other command is Sequence. The basic structure is usually something like Sequence[Rotate[thing, i*360^o/n, centerpt],i,1,n].

GeoGebraGeek paragraph: The fine point in this one is I wanted it do do something differently for 2, 4, 6 than for 8, 10, 12. So I defined a Boolean variable, a = n >7. Booleans read as both true/false and 1/0. The numeric is handy for dynamic coloring (eg. turn something green if it's where you want it) or for goofy stuff like this; Sequence[Rotate[thing, i*(1+a)*360^o/n, centerpt],i,1,n/(1+a)]. That makes it skip every other side for the values where they'd overlap.

Then some motions and - voilà - you get a kind of tiling.

I like the hexagonal and the square symmetries the best. Though that dodecagonal could be neat with a complementary triangle tile for the gaps.

With World Tessellation Day coming up, I've been thinking about tessellations a lot. I've mostly concentrated on Escher style, but I think I want to look more into decorating plainer tiles and then seeing the results of the symmetries, too.

The sketch is on tube.geogebra.org for play, of course. (Some of my other tessellations are in this geogebrabook.)

Now I'm thinking one of the decorations should have been a quadrilateral. Next time!

A gif of using the sketch; A decagonal symmetry.

Square symmetry and dodecagonal for the same decorations.

Queen of Quilts

Actually, Quilts wouldn't make a bad 5th suit...

Elizabeth Clark is a friend and mentor to all of us in my family in many ways, but this post is about the specific inspiration of her quilts to me.

We were over today and she showed this amazing quilt: a Sierpinski gasket stitched over in lissajous and something like the Mystery Curve.

Wow.

For some reason this gave me a flash for how to do a similar thing in GeoGebra. Have a square where you set the pattern, and then use translations and dilations to propagate it. When you click a square it toggles a true/false value, which controls the color. Then in some of the patterns, I use the same Boolean value to determine of the translation happens.

I'm pretty happy with the results. I tried one generation further of the fractal, but it didn't show up well and increased computing time a lot. I know other people have made better and slicker versions of this with programming, but it's fun to make for yourself, too.

Of course, the interactive is on GeoGebraTube if you want to play for yourself.

Hopefully I'll get a chance to blog more about Elizabeth because she gave me one of my favorite problems, and she is endlessly creative.

Guess Again

Graham Fletcher was tweeting from a Marian Small presentation this morning.

I think the theme was number sense activities. These line questions caught my interest so I wanted to make a GeoGebra version. This post is about how I made it. I'm always torn about these things because so much of the good math is in making them, but I'm mostly making them for students. But learning how to make in GeoGebra is often not something that is  engaging to a majority of them...

If you want to try it first, it's on GeoGebraTube.



















First thought: Pick a random value - maybe  1 to 100? - and a smaller value, then have students guess a higher value. (Basically Marian's second photo.) Should it just be multiples of 5? I liked the idea of multiples, so I randomly pick a scale from 1 to 5, then pick a value 1 to 30 for a central value. Pick a number lower and higher, and the setup is done.

Steve Phelps was playing with GeoGebra color the other day, so I added the unnecessary frill of different colors, but it makes it a little cooler to me. Replaying it, I realized it would definitely be better if different points were unknown, so the sketch randomly selects the low, medium or high point for guessing.

A GeoGebra fine point. If you have the values randomly determined directly, then anything that makes the sketch recompute changes the value. So I have variables set randomly, but just assign those values to game values when the New button is pressed using the SetValue command. The guesses count also uses the SetValue command: SetValue[numguess, numguess+1] everytime the guess value changes.

Then it's just bells & whistles using the condition to show object on the Advance tab. A random element of {1,2,3} is picked to hide a value, which shows if guess == value. Some different text

The "Within..." is probably the sketchiest part. I want some further information, but not perfect, so it randomly assigns a number up to two times how far away you are.

There's a bit of oddness from uploading to GeoGebraTube in terms of scaling, so I had to ask users to hit the button to start. There's a command GGBOnInit that might solve this, but I don't know how to use this yet.

Pretty fun to make, and reasonably fun to play. I don't have students to play it with at the moment, so if you get a chance, let me know how it goes.

And - as always - if you have something you'd like to see made, drop me a line!