Calculus through symmetry

Find lines tangent to curves and areas under curves by appealing to symmetries of these curves and a little algebraic manipulation, but no limits.  For example, we can find lines tangent to y=1/x by reflection and stretching symmetries.

Tangents to 1/x by stretching

Tangents to 1/x by stretching

Resource: worksheet3.pdf

Wallpaper symmetry

Many of us made “snowflakes” as a kid, by folding paper in half horizontally and then vertically, then cutting a design through all the sheets.  (Nevermind that snowflakes should have 6-fold symmetry instead of 4-fold.)  Just as folding paper witnesses mirror reflection, rolling paper into a cylinder witnesses translational symmetry, rolling paper into a cone witnesses rotational symmetry, and rolling with a twist (like with Möbius band) witnesses glide reflection symmetry.  By combining these elements, all 17 Wallpaper Symmetry groups can be realized.

Below are some activities inspired by the first third of The Symmetries of Things.

Portion of Running Bond brickwork.

Portion of Running Bond brickwork.

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