Yippee, my first animations in Mathematica!

Let a circle roll around a circle twice as big. The shape traced by a point on the outer circle is a cardioid. Now consider a third circle rolling around the second one as well (again half as big, and at the same speed); its trace is already less familiar. The more circles, the more fractal-ish the resulting curve will be. In the limit, the traced curve can be described with this parametric formula:

$\dpi{120} \begin{cases} x=\displaystyle\sum_{i=0}^\infty\dfrac{\cos(2^i\,\theta)}{2^i}\\[6mm] y=\displaystyle\sum_{i=0}^\infty\dfrac{\sin(2^i\,\theta)}{2^i} \end{cases}$

(Source of inspiration: http://www.mathrecreation.com/2013/12/brain-curve.html)

More with the MoKai Ninja Star.

Mesmer v. Fluency

Today’s meeting doodle - an #edcomic on the eternal struggle in math ed.

Today’s warmup sketch. Reference

Detail from Portrait of Fra Luca Pacioli, mathematician.
Jacopo Barbari 1495 Perspective, light and shadow of rhombicuboctahedron.

I seem to be mildly obsessed with tangent circles.

Putnam Problem 1996:A1

I liked this old Putnam problem I saw on Reddit/learnmath. The OP was looking for help understanding the solution.

Find the least number A such that for any two squares of combined area 1, a rectangle of area A exists such that the two squares can be packed in the rectangle (without interior overlap). You may assume that the sides of the squares are parallel to the sides of the rectangle.

Note: I changed it to sum of areas = 100 to make it a little more accessible. Visualization is on GGBtube.

Airport Voronoi

David Cox, teacher extraordinaire, posted this image today from Jason Davies. It’s an interactive Voronoi diagram of the worlds airports, answering which airport is closest to you. David’s comment: “Instead of having to teach things like perpendicular bisectors and systems of equations, I just wish we could do things like this.

Circles in a Circle, 1923. Vasily Kandinsky (1866 - 1944). Oil on canvas

Kandinsky GEOMETRIC!? Lovely.

From Shouldn’t We Teach GEOMETRY?, Branko Grunbaum, The Two-Year College Mathematics Journal, Vol. 12, No. 4 (Sep., 1981), pp. 232-238

I will read anything by Grunbaum.