500k views

Visually stunning math concepts which are easy to explain

Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the golden ratio or the dragon curve, which are easy to understand and explain but are ...
81k views

The staircase paradox, or why $\pi\ne4$

What is wrong with this? Is $\pi=4?$
2k views

Bad Fraction Reduction That Actually Works

$$\frac{16}{64}=\frac{1\rlap{/}6}{\rlap{/}64}=\frac{1}{4}$$ This is certainly not a correct technique for reducing fractions to lowest terms, but it happens to work in this case, and I believe there ...
5k views

Is Foundational Research a Dead Field?

I'm a second year mathematics major at a pretty good school. Ever since I became a math major I have been most interested in set theory and logic, which I guess can be lumped into the category of ...
4k views

Drawing approximated regular shapes on square grid

I find myself often fooling around with pen and paper, preferably squared paper. So I began looking for ways to sketch geometric figures as precisely as possible without using compass and/or ruler. In ...
439 views

How can this “illegal geometry” problem be possible? [duplicate]

Using 2 triangles each with base of 8 and height of 3, and 2 trapezoids with heights of 3 on top, 5 on bottom and height of 5, these four figures can create an area with 64 units squared. However, ...
1k views

Interesting Mathematical Fallacies [duplicate]

I recently volunteered to help with a summer math program at a local high school for which I thought would be a breeze. As it turns out, it isn't a program for those catching up (summer school) like I ...
569 views

See Lebesgue outer measure of $[0,1]\cap\mathbb{Q}$ Lebesgue measure: $$m(A) = \inf \left\{ \sum |I_n| : A \subset \bigcup I_n \right\}$$ We know that $m(\mathbb{Q} \cap [0,1]) = 0$. Proof: ...
I have seen a [picture like this] several times: featuring a "troll proof" that $\pi=4$. Obviously the construction does not yield a circle, starting from a square, but how to rigorously and ...