## Curves of Constant Width and Odd-Sided Reuleaux Polygons

A curve of constant width is a convex, two-dimensional shape that, when rotated inside a square, always makes contact with all four sides. A circle is the most obvious (but somewhat trivial) example. Some non-trivial examples are the odd-sided Reuleaux polygons â€” the first four of which are shown above. Since they donâ€™t have fixed axes of rotation, curves of constant width (except the circle) have few practical applications. One notable use of the Reuleaux triangle, though, is in drilling holes in the shape of a slightly rounded square (watch one of the triangleâ€™s vertices and notice the shape it traces out as it spins). On a less technical note, all curves of constant width are solutions to the brainteaser, â€śOther than a circle, what shape can you make a manhole cover such that it canâ€™t fall through the hole it covers?â€ť Mathematica code posted here. Additional source not linked above. ...

Oct 20, 2014 Â· Brian Weinstein

## Atomic Models

Evidence-based theories on the structure of atoms have been around since the early 1800s. Daltonâ€™s billiard ball model was the first on the map, and with further discoveries and experiments â€” like Thompsonâ€™s discovery of the electron and Rutherfordâ€™s gold foil experiment â€” improved models of atomic structure were introduced. The first GIF above shows Rutherfordâ€™s planetary model, which was proposed in 1911. In his model, negatively-charged electrons orbit an incredibly small, dense nucleus of positive charge. Despite being a completely incorrect model, most people still think this is what atoms really look like*. This is not an atom. Itâ€™s physically impossible for electrons to stably orbit like this, and the idea of orbiting electrons was entirely replaced by 1926. I canâ€™t say what an atom actually looks like, but the most accurate model we have today is governed by the laws of quantum mechanics. The location of an electron is determined by a probability distribution, called an atomic orbital, which tells us the probability of an electron existing in any specific region around a nucleus. The second image shows the surface around a hydrogen nucleus on which an excited electron is most likely to exist. Mathematica code posted here. *Advertisements and popular science articles incorrectly represent atoms all the time. Even the US Atomic Energy Commission and the International Atomic Energy Agency use the Rutherford model in their logos! ...

Sep 22, 2014 Â· Brian Weinstein

## Cops and Robbers (and Zombies and Humans)

Cops and Robbers is a mathematical game in which pursuers (cops) attempt to capture evaders (robbers). The game is one of many pursuit-evasion games, each of which is governed by a different set of rules. The general goal of these problems is to determine the number of pursuers required to capture a given number of evaders. The GIFs above show two versions of the game. The first is similar to the standard Cops and Robbers rendition, and the second is best described as â€śZombies and Humansâ€ť. In both versions, an evader moves in the direction that gets it furthest away from the pursuers (focusing more on the closer pursuers), and a pursuer moves in the direction that gets it closest to the evaders (focusing more on the closer evaders). In the first simulation, members of both groups have a constant speed. In the second simulation, members of a group move more quickly the closer they are to members of the opposite group, and slower when further away. Mathematica code posted here. Additional sources not linked above: [1] [2] ...

Sep 2, 2014 Â· Brian Weinstein

## Signal Collection and Parabolic Reflectors

A reflector is a type of antenna that receives and focuses various types of signals. Reflectors have numerous applications, from satellite dishes and telescopes, to long-distance microphones and car headlights. One common feature of these examples is their parabolic shape, giving them the name parabolic reflectors. It turns out that paraboloids are the perfect shape for focusing signals from distant sources. When pointed directly at the the incoming signal, a parabolic reflector (GIF 1) collects the signal to a single focal point, where a receiver, called a feed horn, is placed to collect the focused transmission. In many applications, parabolic reflectors are too costly to produce, so spherical reflectors (GIF 2) are used instead. The disadvantage of spherical reflectors is that they have multiple focal points, and therefore produce blurry results. Mathematica code posted here. This code is incredibly messy and I guarantee thereâ€™s a better way to calculate this. Please contact me if you have suggestions! ...

Aug 11, 2014 Â· Brian Weinstein

## Taylor Series Approximations

A Taylor series is a way to represent a function in terms of polynomials. Since polynomials are usually much easier to work with than complicated functions, Taylor series have numerous applications in both math and physics. There are many equations in physics â€” like the one describing the motion of a pendulum â€” that are impossible to solve in terms of elementary functions. â€śApproximations using the first few terms of a Taylor series can make [these] otherwise unsolvable problemsâ€ť solvable for a restricted area of interest [1]. The GIF above shows the five-term Taylor series approximation of a sine wave about x=0. Mathematica code: f[x_] := Sin[x] ts[x_, a_, nmax_] := Sum[(Derivative[n][f][a]/n!)*(x - a)^n, {n, 0, nmax}] Manipulate[Plot[{f[x], ts[x, 0, nmax]}, {x, -2*Pi, 2*Pi}, PlotRange -&gt; {-1.45, 1.45}, PlotStyle -&gt; {{Thick, Cyan}, {Thick, Dotted, Yellow}}, AxesStyle -&gt; LightGray, Background -&gt; Darker[Gray, 0.8]], {nmax, 1, 30, 1}] ...

Jul 30, 2014 Â· Brian Weinstein