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- What happens when we have two mirrors?
- What if the mirrors are parallel?
- What if instead they intersect? What if we make the angle between them larger or smaller? How many images do we see?
- Try the activity and followup here.

Rotations

- Rotations can give powerful effects, especially when you combine multiple rotations or introduce other visual effects. See David Roy’s work in wood and John Edmund’s “strobe animated” 3d printed sculpture.
- Playing with this app can give you some good experience with rotations and reflections.

Rosettes

- With one center of rotation and possible mirrors that intersect at the center of rotation you will finitely many symmetries in patterns called rosette patterns. Rosette patterns can either have mirror symmetry or not and are completely characterized by the order of rotation in the image.

Frieze Patterns and Wallpaper Patterns

- Translation (moving or stamping out an image) in one direction or parallel mirrors can create a frieze pattern.
- If there is translation in two directions, we get wallpaper patterns and there are only 17 of those!

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Develop a portfolio of symmetry patterns. Your portfolio should include a total of 5 patterns. The choice of symmetry type is up to you, as is the choice of the type of work and method of creation (working by hand is fine, as is working on computer, taking photographs, or etching on stone tablets). The only restriction is that __the work must be your own__. Each piece should be accompanied by a description with:

- Briefly, how the piece was created & any concept/idea behind the piece
- Briefly, a description of the symmetry types in the piece: Rotations with their order, reflections (horizontal/vertical/diagonal?), translations, and location of any glide reflections
- The type of pattern: Rosette, Frieze, Wallpaper
- The symmetry group/type of the pattern (note that a piece does not have to have “perfect” symmetry – if the symmetry is broken in some way, be sure to note that).

When I grade this project, I will look at:

Were the pieces that you created visually and/or conceptually interesting, original, or thought-provoking? Was the piece well-done, given the concept and method?__Execution /Concept:__Were all of the symmetries of the piece identified correctly?__Identification of Symmetries:__Were a wide variety of symmetry types and symmetry groups in the portfolio?__Variety of Symmetries:__

Your task is to create a 3-dimensional geometric structure/sculpture, and to write an accompanying short paper.

__Your structure should be well made.__Typically this means it will be sturdy and symmetric, although these aren’t strict requirements since those properties will vary by design and materials. The time and care that you put into both the design and the construction should be evident.__Your structure should make an impression on the viewer.__An ideal project will move beyond simply replicating a construction method. You should carefully consider your construction method, materials, design, and any other elements that will contribute to the visual impact of the final product.__Your structure does not have to be completely original work, but it should contain some original methods, colors, materials, or context.__In other words, whatever you do, you should make it your own.__Your paper should be 400-500 words.__The paper should contain:- What you have learned about the geometry of your piece, about the method of construction, or about related mathematics
- Details about your design, practice, and construction process
- You should include citations and sources for any resources that you use, all done APA style (ask a librarian if you need help with this)
- As an alternative, you could use another medium to do this “paper” including a detailed combination of text and graphics, a video, or whatever else you can imagine.

When I grade this project I will look at how well you met each of the four requirements listed above.

If neither of these options appeals to you, feel free to email me and propose something else. Your project is due on 5/8 at 11:59pm. At that time I’ll be closing the books on the course and I won’t be accepting any work after that!

[Featured Image is from James Nizam, found here.]

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Polyominoes! For our first exploration we took a look at polyominoes. There are a number of games and puzzles that make use of tetrominoes (Tetris) or multiply polynominoes (Blokus). We talked about all of the pentominoes and explored some of the ways that pentominoes can tile a plane or be used to make a box with no top.

Flexagons! There are so many kinds of flexagons, and a variety of resources for making more.

- We made a trihexaflexagon in class. This is shaped like a hexagon and has three faces. Vi Hart talks about this one in her first flexagon video.
- We also made a hexahexaflexagon class, which is also shaped like a hexagon, but has six faces. It appears in the video above near the end.
- There are loads of great flexagon resources around the internet, so make some flexagons!

Triangles! We worked to build things out of equilateral triangles.

- We found that we could build a tetrahedron (3 triangles around each vertex),
- an octahedron (4 triangles around each vertex), or
- an icosahedron (5 triangles around each vertex).
- Once we got up to 6 triangles around each vertex, the triangles lie flat and tile the plane.
- Then we were able to “level up” and put 7 triangles around a vertex which creates a hyperbolic plane (you can also put 8 or more triangles around)! This is a wrinkly surface that looks at any point at little like a Pringles potato chip, and is called a hyperbolic plane. You can make a hyperbolic plane with crochet and it is very fun to explore the weirdness of hyperbolic space.

Hyperbolic Paraboloids! We made two different hyperbolic paraboloids

- The first was made by folding. Mathematician Eric Demaine creating this folding and call this shape a “hypar.” You can build things out of it!
- The second was made by cutting, and was a model I developed. If you like this kind of creating, you can find other models, or of course you can create your own.

Snowflakes and Swirlflakes! This is really the beginning of some material about symmetry which we will continue next class. We talked about mirror symmetry and rotational symmetry and we will continue those discussions next time!

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Then we talked about one visual that is often connected with infinity, the Mobius strip.

On Monday, we will start on our next unit of material, focusing on geometry. As an exploration to kick us off, we started looking at “polyominoes,” which are like dominoes, but with more squares. Two squares put together make a domino. Notice that there is only one way to make a domino. By rotating the horizontal domino, you can see that it is identical to the vertical domino. The rule for polyominoes—shapes made of squares—is that if two shapes can fit on top of one another, either by rotation or reflection, they are not unique. Therefore **there is only one domino**.

Challenge 1: **How many trominoes? **How many ways can you arrange three squares to create what are called trominoes? (Tri-means three, like in triangle & tri + omino = tromino.) You may want to cut out each unique tromino from the grid paper. Then you can check that your trominoes are unique by rotating them and flipping them over (rotation and reflection).

Challenge 2: **How many tetrominoes? **How many ways can you arrange four squares, creating tetrominoes? (Tetra means four, and tetra + omino = tetromino.) It may help to cut out each unique tetromino from the grid paper as you find it. What video game are these from? How can you be sure you have them all?

Challenge 3: **How many pentominoes?** How many ways can you arrange five squares, creating pentominoes? (Penta here means five, like in the word pentagon, and penta + omino = pentomino.)

Bring your exploration with you to class, OR submit your results at this link!

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- We did a coloring exercise and learned that we can add up infinitely many numbers and still get only 1. You can find the exercise here.
- We talked about why 1/3=0.333333… and why 0.9999999…=1 (even though that seems nuts). We also watched this Vi Hart video about why: https://www.youtube.com/watch?v=TINfzxSnnIE.

- We reviewed ideas about one-to-one correspondence as a way of deciding if one set is bigger than another, reminded ourselves that there the same number of even numbers as all the counting numbers. Then we talked about whether if we had infinitely many piles of m&ms, each of which had an infinite number of m&ms, would that be a larger infinity that the counting numbers (the answer is no).
- To find a new infinity we need to not focus on counting numbers, we need other types of numbers — you can find a summary of the different types we discussed here: http://www.purplemath.com/modules/numtypes.htm. Remember that last week we watched Vi Hart explain a little about irrational numbers and why pythagoras was really wierd here: https://www.youtube.com/watch?v=X1E7I7_r3Cw.
- Next class, we will be showing that there are more real numbers between 0 and 1 than there are counting numbers. Do do that we first explored a game I call “virtual dodgeball.”
- You also have a new project assignment. See all of the details here.

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- We started by talking about infinity and the different images and ideas that come up when we talk about the infinite. Then we watched a Vi Hart video “Infinity Elephants” that provided an introduction to some important ideas, while looking back at fractals. Particularly, Vi mentioned that there were different sizes of infinity which is what our next few classes will explore. Also, if you like apollonian gaskets like the ones in the video and you want to know more about the math, here is a link to start with.
- Next, I talked about some of the weird things that happen when we start thinking and talking about infinity. You want re-watch the Hilbert Hotel problem we worked o in class here, and you can watch a another cute animated video about it here. You can read a New York Times Opinionator piece about it as well.
- After tackling the Hilbert Hotel, I defined what it means for two sets (a set being a collection of things) to be the
**same size**. If two sets can be paired evenly, with pairs featuring one object from the first set one from the second, with no leftovers, then the two collections are in “one-to-one correspondence.” We say that two sets are the same size if we can create a one-to-one correspondence between the sets.Note that if we try to put two collections into one-to-one correspondence, we can decide if the collections are the same size without even knowing how many objects are in each collection.Children learn to count using one-to-one correspondence and it is an important early numeracy activity. - A collection of objects (a set) is finite if it can be put in one-to-one correspondence with a set {1,2,3,4,…, n} for some n – that number n tells us how big the set is.The easy way to say this is that a set is finite if you can count it! A set is infinite if it is not finite.
- Infinite sets have this really weird feature that they can be put in one-to-one correspondence with a proper subset of themselves. A proper subset is just a part of a set that contains something but not the whole set. So given the set {1,2,3,4,5,6}, one proper subset is {1,2,3,4}. Note that we can never put those in one-to-one correspondence, but the infinite set {1,2,3,4,….} can be put into one-to-one correspondence with the set of even numbers {2,4,6,8,10, …}.
- We learned just a little about Georg Cantor, a mathematician that explored and discovered many of our modern mathematical ideas about the infinite. Cantor discovered that there are diffferent sizes of infinity. I found a better video for this that doesn’t cut people’s heads off. We will watch more of Cantor’s story as we explore his ideas about infinity.
- Finally, we talked briefly about different kinds of numbers, in particular irrational numbers, which so disturbed Pythagoras. They’ll come up more as we go in search of a larger infinity.

For next week, there’s a more concrete exploration of infinity for you to do before class. We will start class with talking about your explorations.

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