Making Geometry: Part I

For the last few weeks, we have made various things and I want to give links and resources to help you make and build more!

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.

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

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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