Class 7: Wrapping up Infinity

This class we finished up the ideas explored in class 6. We reviewed all of the ideas of the past few weeks and proved that there are more real numbers between 0 and 1 than there are counting numbers. Vi Hart provides a summary of the work we have done in this video. We then went back to the story of Cantor, who was the one to discover this larger infinity. His story concludes in this video (and started in this one).

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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