In his article Simply Symmetric, Michael de Villiers observes that symmetry is a powerful but often overlooked tool for formulating proofs: Most primary geometry curricula around the world introduce the concept of line symmetry fairly early, and sometimes also that of rotational, translational and glide reflective symmetry. However, in high school and even college geometry...
How can you identify a lover of math? Casually mention a burning tent and notice if her first thought is how to minimize her path to a river and then to the tent to douse the flames. Here is a full statement of this classic geometry problem: Ah, the great outdoors. Camping, the fresh air,...
In a 2018 blog post, I presented George Gamow’s pirate treasure problem, which can neatly be solved by capitalizing on the geometry of complex numbers. There’s more treasure to be had, however, so get ready for another adventure! An island contains a giant boulder, a lighthouse, a cave, and a jail. Among a pirate’s belongings,...
A game of enclosing sheep and wolves in fences helps children to develop their conceptual understanding of polygons.
When I was introduced to radian measure in high school, I knew just one thing: How to convert between radians and degrees. Had you asked me to illustrate a radian on a circle or to explain why radian measure was useful, I would have been stumped. In this post, I’ll describe a Web Sketchpad activity...
In last month’s blog post, I described a parabola construction technique dating back to the work of Persian polymath Ibn Sina (c. 970 – 1037). After I published the post, my colleague Scott noted that my construction could be more robust to allow for parabolas that are downward facing as well as upward facing. Such...
There can never be enough conic-section construction techniques—at least that’s my philosophy, having grown up to think that conics existed purely in the realm of algebraic equations. So to continue my conic section construction series on Sine of the Times, I’ll present a parabola construction attributed to Ibn Sina (Avicenna), a Persian polymath (c. 970 – 1037)...
In my prior blog posts, I’ve presented methods for constructing ellipses using Web Sketchpad and paper folding. The other conic sections are feeling a bit left out, so let’s explore some techniques for constructing parabolas. All three Web Sketchpad models below (and here) are based on the distance definition of a parabola: The set of...
Attention Sketchpad fans: If you’re a Mac user and would like to use The Geometer’s Sketchpad, you’re in luck—you can download it for free here. Several weeks ago, Apple released MacOS Catalina and brought an end to all 32-bit apps, including Sketchpad 5.06. Luckily, Nick Jackiw, Sketchpad’s designer, has updated Mac Sketchpad so that it...
When students find the nth roots of a complex number, they use de Moivre’s Theorem and a fair bit of calculation and trigonometry. In this blog post, I’m going to approach the topic from a more visual perspective and make use of the following geometric way to think about complex number multiplication: To multiply two complex...