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
Matt Parker built a domino adder of four or five bits out of 10,000 dominos. Inspired by this, we offer a smaller scale activity that gets across some of the key points–and more.
We’ll build gates and describe their behavior with truth tables. See the handout for additional activities.
The following video shows the execution of a “p and not q” gate. Both input dominoes are pressed, and one of the domino chains prevents the other from propagating:
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.
The Borromean Rings is an arrangement of three interlinked loops. The trio cannot be separated, but, if any one loop is removed, the other two can be separated. This makes for an great classroom activity, for many audiences, and to demonstrate many different phenomena, including algebra, topology, cryptography, boolean logic, and NP-completeness.