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movies and math, animated

[mathani]

I like this rigid-body problem because it requires thinking beyond the point-mass model.

Irodov 1.271 A uniform cube with edge a rests on a horizontal plane whose friction coefficient equals k. The cube is set in motion with an initial velocity, travels some distance over the plane and comes to a stand-still. Explain the disappearance of the angular momentum of the cube relative to the axis lying in the plane at right angles to the cube's motion direction. Find the distance between the resultants of gravitational forces and the reaction forces exerted by the supporting plane.

Imagine giving the cube an initial push and then observing it decelerate. The frictional forces lie in the plane of the axis, which makes their torque zero. So what exactly reduces the angular momentum? It's the displacement of the normal force, allowing its torque to overpower the torque of the weight. Deceleration effectively shifts the normal force slightly forward, indicated here with a pressure gradient within the contact layer.

[mathani]

Normals through three points on a parabola are concurrent iff the sum of their abscissas vanishes.

[mathani]

Curious geometry of Project Euler 965.

[mathani]

A square grid in polar (r, φ) coordinates stretches into a curved grid in the xy-plane. Patches farther from the r = 0 line stretch more, while those close to the line can contract. This is exactly what is captured by the Jacobian, and why it appears when one changes variables in a double integral.

[mathani]

Every chord has a parallel tangent along the arc, per Cauchy's mean-value theorem.

[mathani]

Rotating about the origin by an angle of 2α has the same effect as composing two reflections in lines through the origin that meet at an angle α. The four angles between the four line pairs above sum to π.

[mathani]

A point in a rod that moves along perpendicular lines traces an ellipse.

[mathani]

A sister to the golden rectangle is a 72-72-36-degree isosceles triangle. Bisecting a base angle reveals a smaller similar triangle.

[mathani]

Two equal half-circles on a bigger half-circle. Every line through the cusp bisects the perimeter.

[mathani]

A circular wavefront emanating from one focus of the ellipse reflects off the boundary and refocuses as a circular wavefront at the other focus.

[mathani]

Idea via @formandalign.

[mathani]

Let me call this a Stern-Brocot crystal after the Stern-Brocot tree: a scheme for enumerating rational numbers that has many beautiful properties and applications. Beginning with "infinite" and "zero" fractions 1/0 and 0/1, adjacent fractions a/b and c/d generate the child a+c/b+d. Each new generation of children branches outward, forming a growing lattice of dendrites.

[mathani]

The area of a cycloid is three times the area of the generating circle.

[mathani]

An arc looks twice as large from the center of a circle as it does from the circumference. Note the switch of the minor and major arcs.

[mathani]

The areas of the colored intersections stay constant—when n is even in such an amalgamation of regular n-gons.

[mathani]

The Ham Sandwich Theorem says that two stacked pancake blobs can be cut exactly in half (by area) with a single straight slice. A way to find this cut uses nested bisections: the outer bisection shifts the line to halve the first pancake, while the inner bisection rotates it until the second pancake is halved as well. Here are some examples.

[mathani]

The circle of Apollonius is the set of points that are some fixed amount closer to one point than the other. If the distances are equal, it turns into the vertical bisector. The ancient geometer Apollonius is best known for studying conic sections. He's the one who came up with the names ellipse, parabola, and hyperbola.