Infinite Gold.

The Futility Closet shares a classic Martin Gardner puzzle.

I think this is a little more subtle than the more common triangle paradox. What do you think?

Here’s the sketch at GeoGebraTube. And another gif if you’re greedy or trying to figure it out.

Square Roots

Playing around with square roots this morning, inspired by a nice illustration I reblogged this morning. The reason why the construction works can be seen as that odd similarity property of right triangles that two similar can compose a third. That took me off for a while into the border spiraland and rep-tiling, but I’m back.

Sketch 1 and sketch 2 on the ‘Tube.

I'm so confused by Aristotles wheel! How does it work?!

It’s a map or a function from the small wheel to the segment, but it doesn’t preserve distances. In real life there would be a kind of continuous slippage of the wheel or the string (if a string is what you’re imagining). Paradoxes like this are how mathematicians - one named Cantor in particular - figured out some weird things about infinity.

Square roots 1 through 10 constructed with a compass and straightedge

Nice all in one picture. Could we dynamicize…?

Pedal Curve Construction. Saving to look at later.

Magic Squares.

Been playing around with magic squares and Google spreadsheets. Inspired by Ramanujan, here’s one with my birthday as the top row. (I’m old.) The spreadsheet has a sheet for messing around with 4x4’s, for beginners to start with a 3x3, Ramanujan’s 4x4 and Ben Franklin’s ASTOUNDING 16x16 monster. Check it out.

Lieber Illustrations.

I was random reading the math education stackexchange, which led over to the mathexchange, which led to Lillian Lieber.

She was a mathematician and popular writer in the mid 20th century, and her husband Hugh Lieber illustrated her books. The three above are from her book on relativity. She was at Long Island University and something that seems to no longer exist, sadly, the Galois Institute for Mathematics and Art in Brooklyn.

More info on the Liebers; source for the first 3 pictures (illustration blog); source for the 2nd 3 pictures (vintage store!); T C Mitts at Amazon, which I’ve ordered. Check out her amazing book on infinity. (Which seems to be on Google books in its entirety, page limit still applies.)

Pat Bellew has a daily blog with excellent math history: On This Day in Math. Today’s has a great gif of the Haberdasher’s Puzzle by Dudeney. Pat writes:

b. 4-10-1857 Henry Ernest Dudeney (pronounced with a long “u” and a strong accent on the ﬁrst syllable, as in “scrutiny”). He was England’s greatest maker of puzzles of mathematical interest, publishing six books of puzzles. His ﬁrst work appears under the pseudonym “sphinx.” Although he never met Sam Loyd, they were in frequent correspondence and informally exchanged ideas. … Some of Dudeney’s most famous innovations were his 1903 success at solving the Haberdasher’s Puzzle (Cut an equilateral triangle into four pieces that can be rearranged to make a square) and publishing the first known crossnumber puzzle, in 1926. In addition, he has been credited with inventing verbal arithmetic and discovering new applications of digital roots.

Pat points out that the Kindle edition of the Canterbury Puzzles is free!

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.