Linked Questions

1192
votes
69answers
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 ...
712
votes
21answers
81k views

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

What is wrong with this? Is $\pi=4?$
44
votes
7answers
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 ...
12
votes
3answers
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 ...
15
votes
4answers
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 ...
3
votes
3answers
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, ...
4
votes
3answers
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 ...
2
votes
2answers
569 views

Lebesgue measure paradox

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: ...
3
votes
5answers
152 views

Are geometric proofs less reliable than others?

When I submit a homework with a proof that uses a graph, ball, shape etc., most of the time the professors are not happy with them. They respond with a statement like: "The proof you made seems very ...
0
votes
2answers
278 views

Constructing a circle from a square [duplicate]

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 ...