In the Morning

Posts tagged math

The brachistochrone

This animation is about one of the most significant problems in the history of mathematics: the brachistochrone challenge.

If a ball is to roll down a ramp which connects two points, what must be the shape of the ramp’s curve be, such that the descent time is a minimum?

Intuition says that it should be a straight line. That would minimize the distance, but the minimum time happens when the ramp curve is the one shown: a cycloid.

Johann Bernoulli posed the problem to the mathematicians of Europe in 1696, and ultimately, several found the solution. However, a new branch of mathematics, calculus of variations, had to be invented to deal with such problems. Today, calculus of variations is vital in quantum mechanics and other fields.

(Source: saulofortz, via visualizingmath)

I was lucky to have great math teachers all through middle school and high school.  My grade school was Waldorf-based so we didn’t do math the way they do in traditional schools. I found the awe and wonder in it.

The teachers who taught me were passionate about the subject, some of them even made math jokes.  That helped.

gjmueller
:

Why math fills so many of us with dread

“We teach math as if it’s about applying a prescribed formula, circling the right answer, and going on to the next problem without thinking about what it is we’re doing,” Ellenberg replied. “But that’s so not what math is. Math is a fundamentally creative enterprise, a fundamentally humanistic enterprise. It’s a lens through which we can see the world better.”

A convex polygon with n sides is drawn in a circle. Then divided into triangles. Finally circles are inscribed in the triangles. Obviously, triangulation is not unique; a 4-gon has two, a 5-gon five, a 6-gon 14 and so on, as continue the Catalan numbers. But the sum of the radii of the circles is constant, independent of how the n-gon is divided.

(via visualizingmath)

Chaos and the Double Pendulum

chaotic system is one in which infinitesimal differences in the starting conditions lead to drastically different results as the system evolves.

Summarized by mathematician Edward Lorentz, ”Chaos [is] when the present determines the future, but the approximate present does not approximately determine the future.”

There’s an important distinction to make between a chaotic system and a random system. Given the starting conditions, a chaotic system is entirely deterministic. A random system, on the other hand, is entirely non-deterministic, even when the starting conditions are known. That is, with enough information, the evolution of a “chaotic” system is entirely predictable, but in a random system there’s no amount of information that would be enough to predict the system’s evolution.

The simulations above show two slightly different initial conditions for a double pendulum — an example of a chaotic system. In the left animation both pendulums begin horizontally, and in the right animation the red pendulum begins horizontally and the blue is rotated by 0.1 radians (≈ 5.73°) above the positive x-axis. In both simulations, all of the pendulums begin from rest.

`Mathematica code posted here.`

[For more information on how to solve for the motion of a double pendulum, check out my video here.]

(via visualizingmath)

This, ladies and gentlemen and genderqueer folks, is Pascal’s tetrahedron, a three dimensional analogue of Pascal’s triangle, and it’s pretty freaking great.

I’ve never heard of this before!

Is this a college class or is this being taught in high school now?