{"id":420,"date":"2013-11-06T01:07:05","date_gmt":"2013-11-06T07:07:05","guid":{"rendered":"http:\/\/blogs.ams.org\/blogonmathblogs\/?p=420"},"modified":"2013-11-25T22:25:26","modified_gmt":"2013-11-26T04:25:26","slug":"see-math-see-math-run","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/blogonmathblogs\/2013\/11\/06\/see-math-see-math-run\/","title":{"rendered":"See Math, See Math Run"},"content":{"rendered":"
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\"images-2\"<\/a>

A pyramid whose volume is the same as the cone next to it.<\/p><\/div>\n

To me, the formula for the volume of a cone says \u201cDid you know that 3 copies of the same cone occupy the same space as the smallest cylinder that contains one of them?\u201d 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.<\/p>\n

I would guess that many mathematicians recall and develop equations and concepts by relying on visual cues and imagining physical manipulations.\u00a0 But I wonder how naturally this comes to the general public.\u00a0 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.\u00a0 We know that imagery makes for good advertising as is made clear by this short clip,<\/a> which I found at the blog FlowingData.<\/a> And Pascal Wallisch, a neuroscience researcher, asserts that imagery is simply more intriguing to humans than equations at his blog Pascal’s Pensee’s.<\/a><\/p>\n

Beyond piquing our interest, mathematical images can also serve as tools for solving problems or even as explanations for already conjectured ideas.\u00a0 Flowing Data champions the use of visualizations to see data in a new light.\u00a0 Nathan Yau, who posts there, quotes mathematician John Tukey:<\/p>\n

\u201cThe greatest value of a picture is when it forces us to notice what we never expected to see.\u201d<\/h2>\n<\/div>\n
\"Sum<\/a>

Sum of the first 6 odd numbers is 6^2!<\/p><\/div>\n

While fancy graphics are at our fingertips, even simple ones can be effective. In his most recent and somewhat controversial post, \u201cDeath of math\u201d<\/a>, Stuyvesant High School teacher Gary Rubinstein writes about the image to the left: \u201cIf you don\u2019t consider yourself a \u2018math person\u2019, still give yourself a chance to revel in the beauty of this image, and I hope you\u2019ll get to experience your own \u00a0\u2018aha\u2019 moment that mathematicians live for.\u201d<\/p>\n

In other words, to “See Math” is not good enough.\u00a0 We also need to “See Math Run”.\u00a0 Imagine a quarter rolling around the perimeter of another stationary quarter.\u00a0 How many times will the rolling quarter rotate about its own center by the time it has rolled once around the stationary quarter?\u00a0 What shape will a point on the perimeter of the quarter trace out as it rolls?\u00a0 Be still my beating cardioid!<\/p>\n

\"The<\/a>

The Catacaustic of a Cardioid from Greg Egan<\/p><\/div>\n

Now consider an extension of the string art that I wrote about in a past blog.\u00a0 Shine a light from a point of a curve inwards and let it bounce off the curve.\u00a0 Find the envelope of all possible paths that a ray of light could follow.\u00a0 This envelope, to which the linear paths snuggle up, is the\u00a0\u00a0 catacaustic.\u00a0 The picture at left shows that the catacaustic of the cardiod is the nephroid (yes, the etymology there is relating to the kidney).\u00a0 Researcher John Baez\u2019s latest blogging foray is Visual Insight <\/a>here at the AMS, and my favorite post of late is on the Catacaustic of the Cardioid.\u00a0 I probably should have saved that one for Valentine\u2019s day, but Baez\u2019s explanations from his webpage<\/a> as well as the pictures by Egan are just too wonderful for me to risk forgetting about until February.\u00a0 In fact, the cardioid itself is the catacaustic of a familiar curve.\u00a0 Can you guess which one?<\/p>\n

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To me, the formula for the volume of a cone says \u201cDid you know that 3 copies of the same cone occupy the same space as the smallest cylinder that contains one of them?\u201d This fact relates (see picture) to … Continue reading →<\/span><\/a><\/p>\n

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