A pyramid whose volume is the same as the cone next to it.
To me, the formula for the volume of a cone says “Did you know that 3 copies of the same cone occupy the same space as the smallest cylinder that contains one of them?” This fact relates (see picture) to the formula for the volume of a pyramid which says “Hey, did you know that 3 copies of the same pyramid occupy the same space as the smallest rectangular prism containing one of them?” I ‘see’ the equations and interpret them as ‘doing’ something with shapes.
I would guess that many mathematicians recall and develop equations and concepts by relying on visual cues and imagining physical manipulations. But I wonder how naturally this comes to the general public. Now that it is so easy to create graphics and even three-dimensional models, we can use visualizations as tools for research, education, and even for raising public awareness concerning the beauty of mathematics. We know that imagery makes for good advertising as is made clear by this short clip, which I found at the blog FlowingData. And Pascal Wallisch, a neuroscience researcher, asserts that imagery is simply more intriguing to humans than equations at his blog Pascal’s Pensee’s.
Beyond piquing our interest, mathematical images can also serve as tools for solving problems or even as explanations for already conjectured ideas. Flowing Data champions the use of visualizations to see data in a new light. Nathan Yau, who posts there, quotes mathematician John Tukey:
“The greatest value of a picture is when it forces us to notice what we never expected to see.”
Sum of the first 6 odd numbers is 6^2!
While fancy graphics are at our fingertips, even simple ones can be effective. In his most recent and somewhat controversial post, “Death of math”, Stuyvesant High School teacher Gary Rubinstein writes about the image to the left: “If you don’t consider yourself a ‘math person’, still give yourself a chance to revel in the beauty of this image, and I hope you’ll get to experience your own ‘aha’ moment that mathematicians live for.”
In other words, to “See Math” is not good enough. We also need to “See Math Run”. Imagine a quarter rolling around the perimeter of another stationary quarter. How many times will the rolling quarter rotate about its own center by the time it has rolled once around the stationary quarter? What shape will a point on the perimeter of the quarter trace out as it rolls? Be still my beating cardioid!
The Catacaustic of a Cardioid from Greg Egan
Now consider an extension of the string art that I wrote about in a past blog. Shine a light from a point of a curve inwards and let it bounce off the curve. Find the envelope of all possible paths that a ray of light could follow. This envelope, to which the linear paths snuggle up, is the catacaustic. The picture at left shows that the catacaustic of the cardiod is the nephroid (yes, the etymology there is relating to the kidney). Researcher John Baez’s latest blogging foray is Visual Insight here at the AMS, and my favorite post of late is on the Catacaustic of the Cardioid. I probably should have saved that one for Valentine’s day, but Baez’s explanations from his webpage as well as the pictures by Egan are just too wonderful for me to risk forgetting about until February. In fact, the cardioid itself is the catacaustic of a familiar curve. Can you guess which one?
Good article