So, the euler spiral! It’s the big one at the top. It’s traced out by the parametric equations written under it.

I thought something interesting would happen if I replaced the cosine and the sine with the derivatives of the parameterizations of various other curves, and then this happened.

So basically Euler Spirals on acid. Wicked.

Reblogged from Visualizing Math

Have I mentioned I have a thing for absurdity?

Seriously who thinks of these questions. Like its not hard, but the way it’s written. A “microwave cooking enthusiast” like who the fuck thinks of these questions

Left side, second row. Unf.

Reblogged from Peano
Reblogged from People call me Asauce

What.

Reblogged from Yugen

Thomae’s function, named after Carl Johannes Thomae, also known as the popcorn function, the raindrop function, the countable cloud function, the modified Dirichlet function, the ruler function,[1] the Riemann function or the Stars over Babylon (by John Horton Conway) is a modification of the Dirichlet function. This real-valued function f(x) is defined as follows:

$f(x)=\begin{cases} \frac{1}{q}\mbox{ if }x=\frac{p}{q}\mbox{ is a rational number}\\ 0\mbox{ if }x\mbox{ is irrational}. \end{cases}$

If x = 0 we take q = 1. It is assumed here that gcd(pq) = 1 and q > 0 so that the function is well-defined and non-negative.

## Discontinuities

The popcorn function is perhaps the simplest example of a function with a complicated set of discontinuitiesf is continuous at all irrational numbers and discontinuous at all rational numbers.

Informal Proof

Clearly, f is discontinuous at all rational numbers: since the irrationals are dense in the reals, for any rational x, no matter what ε we select, there is an irrational a even nearer to our x where f(a) = 0 (while f(x) is positive). In other words, f can never get “close” and “stay close” to any positive number because its domain is dense with zeroes.

To show continuity at the irrationals, assume without loss of generality that our ε is rational (for any irrational ε, we can choose a smaller rational ε and the proof is transitive). Since ε is rational, it can be expressed in lowest terms as a/b. We want to show that f(x) is continuous when x is irrational.

Note that f takes a maximum value of 1 at each whole integer, so we may limit our examination to the space between $\lfloor x \rfloor$ and $\lceil x \rceil$. Since ε has a finite denominator of b, the only values for which f may return a value greater than ε are those with a reduced denominator no larger than b. There exist only a finite number of values between two integers with denominator no larger than b, so these can be exhaustively listed. Setting δ to be smaller than the nearest distance from x to one of these values guarantees every value within δ of x has f(x) < ε.

Reblogged from fuck yeah mathematics

Found this on Tumblr a while ago. Didn’t reblog and subsequently lost it. Glad to have finally found it again!

Reblogged from Math is ∫εχy

A shape that can be dissected into smaller copies of the same shape is called a reptile or rep-tile.

Using an orange to illustrate an Euler spiral.