## Wave Equation

The wave equation is a partial differential equation that describes the propagation of various types of waves. The equation appears throughout many fields in physics, including acoustics, fluid dynamics, electromagnetism, and quantum mechanics. With some modifications, it can even describe the spread of traffic jams on busy highways! The one-dimensional equation was first discovered by d’Alembert in 1746 as he studied how vibrations propagated through a string, and the two- and three-dimensional equations were solved soon after by Euler during his study of acoustics. The simulations above show the propagation of a disturbance on a two-dimensional surface for two different sets of boundary conditions  . Mathematica code posted here. ...

Jan 28, 2015 · Brian Weinstein

## Platonic Solids

A Platonic solid is a polyhedron where (1) each face is the same regular polygon, and (2) each vertex joins the same number of faces. The Platonic solids are highly symmetrical, and, in three dimensions, only five such solids can exist: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. This was first proven in Euclid’s Elements around 300 B.C., and has since been more rigorously proven using the Euler characteristic. The proofs are relatively easy to follow, and if you’re interested you can check them out both here and here. Mathematica code: pSolids={"Tetrahedron","Cube","Octahedron","Dodecahedron","Icosahedron"} Manipulate[Graphics3D[ {Opacity[0.8],Rotate[PolyhedronData[pSolids[[n]],"Faces"],th,{0,0,1}], Opacity,Circumsphere[PolyhedronData[pSolids[[n]], "VertexCoordinates"][[1;;4]]]}, Boxed->False,SphericalRegion->True],{n,1,5,1},{th,0,2\[Pi]}] ...

Jan 20, 2015 · Brian Weinstein

## Lonely Runner Conjecture

Imagine n runners on a circular track of length 1. The runners start from the same spot at the same time, and each has a distinct, constant speed. A runner is considered “lonely” whenever it is a distance of at least 1/n from every other runner. The Lonely Runner Conjecture (LRC) states that each runner will eventually, at some point in time, be lonely. Said differently, the LRC states that for each runner, the spacing around it will eventually be greater than or equal to the spacing it would experience if the all of the runners were equally distributed around the track. The conjecture has been proven to be true for 7 or fewer runners, but, interestingly enough, has never been proven to work for all cases of 8 or more runners. [In my 8-runner simulation above, I’ve only shown that it works for a specific set of runner speeds — I haven’t proven that it works for all sets of speeds.] In the GIFs above, an arc appears around a runner whenever the runner is lonely, and the color of a runner fades after it’s been lonely at least once. Mathematica code posted here. Additional sources not linked above:    ...

Dec 25, 2014 · Brian Weinstein