https://brianweinstein.github.io/ Recent content on Brian Weinstein Hugo -- gohugo.io Sat, 22 Nov 2025 00:00:00 +0000 https://brianweinstein.github.io/posts/20251122-slop-and-shop/ Sat, 22 Nov 2025 00:00:00 +0000 https://brianweinstein.github.io/posts/20251122-slop-and-shop/ Another vertical video, snap-scrolling, content consumption app. But for a cause! https://brianweinstein.github.io/posts/20250727-camera-3000/ Sun, 27 Jul 2025 00:00:00 +0000 https://brianweinstein.github.io/posts/20250727-camera-3000/ An AI-powered camera app that captures everything in the scene except the important details. https://brianweinstein.github.io/posts/20201222-defining-meaningful-metrics-for-product-teams/ Tue, 22 Dec 2020 00:00:00 +0000 https://brianweinstein.github.io/posts/20201222-defining-meaningful-metrics-for-product-teams/ A checklist for developing better product metrics & KPIs https://brianweinstein.github.io/posts/20200219-in-defense-of-nothing-interesting/ Wed, 19 Feb 2020 00:00:00 +0000 https://brianweinstein.github.io/posts/20200219-in-defense-of-nothing-interesting/ A tribute to useful, but less interesting research findings https://brianweinstein.github.io/posts/20180124-moving-beyond-the-net-promoter-score/ Wed, 24 Jan 2018 00:00:00 +0000 https://brianweinstein.github.io/posts/20180124-moving-beyond-the-net-promoter-score/ A guide to building a more meaningful metric https://brianweinstein.github.io/posts/20161003-debate-nlp/ Mon, 03 Oct 2016 00:00:00 +0000 https://brianweinstein.github.io/posts/20161003-debate-nlp/ Natural language processing on the first 2016 presidential debate https://brianweinstein.github.io/posts/20160531-froyo-nyc/ Tue, 31 May 2016 00:00:00 +0000 https://brianweinstein.github.io/posts/20160531-froyo-nyc/ I love frozen yogurt. When I first moved to New York three years ago, I lived only 1/8th of a mile from the closest froyo shop. The convenience of this 4-minute walk is something I neither appreciated nor utilized enough at the time. After moving to Harlem last year, it’s been harder than ever to satisfy my near-constant craving for this cold candy soup — I’m now a 24-minute walk to the nearest frozen yogurt. https://brianweinstein.github.io/posts/20150128-wave-equation/ Wed, 28 Jan 2015 00:00:00 +0000 https://brianweinstein.github.io/posts/20150128-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. https://brianweinstein.github.io/posts/20150120-platonic-solids/ Tue, 20 Jan 2015 00:00:00 +0000 https://brianweinstein.github.io/posts/20150120-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. https://brianweinstein.github.io/posts/20141225-lonely-runner-conjecture/ Thu, 25 Dec 2014 00:00:00 +0000 https://brianweinstein.github.io/posts/20141225-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. https://brianweinstein.github.io/posts/20141124-comet-67p/ Mon, 24 Nov 2014 00:00:00 +0000 https://brianweinstein.github.io/posts/20141124-comet-67p/ Two weeks ago, the ESA made history by landing a spacecraft on a comet. The spacecraft, named Philae, was carried to Comet 67P by a larger space probe named Rosetta. Determining where Philae would land was a big step in this mission. Many attributes about the comet, including its topography, were taken into account. https://brianweinstein.github.io/posts/20141110-harmonographs/ Mon, 10 Nov 2014 00:00:00 +0000 https://brianweinstein.github.io/posts/20141110-harmonographs/ A harmonograph is a mechanical device consisting of two or more pendulums attached to a pen. The swinging pendulums control the motion of the pen, tracing out a geometric pattern on a sheet of paper. Since the system is damped by friction, the pattern spirals in on itself as time progresses. Each of the GIFs above simulate the output of a 4-pendulum system (modeled after a harmonograph as configured in this video). https://brianweinstein.github.io/posts/20141020-curves-of-constant-width/ Mon, 20 Oct 2014 00:00:00 +0000 https://brianweinstein.github.io/posts/20141020-curves-of-constant-width/ 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 — the first four of which are shown above. Since they don’t have fixed axes of rotation, curves of constant width (except the circle) have few practical applications. https://brianweinstein.github.io/posts/20140922-atomic-models/ Mon, 22 Sep 2014 00:00:00 +0000 https://brianweinstein.github.io/posts/20140922-atomic-models/ Evidence-based theories on the structure of atoms have been around since the early 1800s. Dalton’s billiard ball model was the first on the map, and with further discoveries and experiments — like Thompson’s discovery of the electron and Rutherford’s gold foil experiment — improved models of atomic structure were introduced. The first GIF above shows Rutherford’s planetary model, which was proposed in 1911. https://brianweinstein.github.io/posts/20140902-cops-and-robbers/ Tue, 02 Sep 2014 00:00:00 +0000 https://brianweinstein.github.io/posts/20140902-cops-and-robbers/ 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. https://brianweinstein.github.io/posts/20140811-signal-collection-parabolic-reflectors/ Mon, 11 Aug 2014 00:00:00 +0000 https://brianweinstein.github.io/posts/20140811-signal-collection-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. https://brianweinstein.github.io/posts/20140730-taylor-series-approximations/ Wed, 30 Jul 2014 00:00:00 +0000 https://brianweinstein.github.io/posts/20140730-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 — like the one describing the motion of a pendulum — that are impossible to solve in terms of elementary functions. “Approximations using the first few terms of a Taylor series can make [these] otherwise unsolvable problems” solvable for a restricted area of interest [1]. https://brianweinstein.github.io/posts/20140713-rotational-stability/ Sun, 13 Jul 2014 00:00:00 +0000 https://brianweinstein.github.io/posts/20140713-rotational-stability/ Time for an experiment! Find a book and secure it shut using tape or a rubber band. Now experiment with spinning the book while tossing it into the air. You’ll notice that when the book is spun about its longest or shortest axis it rotates stably, but when spun about its intermediate-length axis it quickly wobbles out of control. https://brianweinstein.github.io/posts/20140629-painters-paradox/ Sun, 29 Jun 2014 00:00:00 +0000 https://brianweinstein.github.io/posts/20140629-painters-paradox/ Gabriel’s Horn is a three-dimensional horn shape with the counterintuitive property of having a finite volume but an infinite surface area. This fact results in the Painter’s Paradox — A painter could fill the horn with a finite quantity of paint, “and yet that paint would not be sufficient to coat [the horn’s] inner surface” [1]. If the horn’s bell had, for example, a 6-inch radius, we’d only need about a half gallon of paint to fill the horn all the way up. https://brianweinstein.github.io/posts/20140609-lagrangian-points/ Mon, 09 Jun 2014 00:00:00 +0000 https://brianweinstein.github.io/posts/20140609-lagrangian-points/ The Lagrangian points are the five locations in an orbital system where the combined gravitational force of two large masses is exactly canceled out by the centrifugal force arising from the rotating reference frame. At these five points, the net force on a third body (of negligible mass) is 0, allowing the third object to be completely stationary relative to the two other masses. That is, when placed at any of these points, the third body stays perfectly still in the rotating frame. https://brianweinstein.github.io/posts/20140528-sonic-boom-doppler-effect/ Wed, 28 May 2014 00:00:00 +0000 https://brianweinstein.github.io/posts/20140528-sonic-boom-doppler-effect/ The Doppler effect is the shift in the frequency of a wave observed when the source of the wave (or the medium through which the wave travels) is moving relative to the observer. We’re most familiar with the Doppler effect as it appears in sound waves traveling through air (i.e., pressure waves) – think of how the pitch of a siren drops as an emergency vehicle passes you. https://brianweinstein.github.io/posts/20140519-chaos-and-the-double-pendulum/ Mon, 19 May 2014 00:00:00 +0000 https://brianweinstein.github.io/posts/20140519-chaos-and-the-double-pendulum/ A 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 Lorenz, “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. https://brianweinstein.github.io/posts/20140512-three-body-problem-3d/ Mon, 12 May 2014 00:00:00 +0000 https://brianweinstein.github.io/posts/20140512-three-body-problem-3d/ A couple of weeks ago I made a post about the classical three-body problem, which involves determining the motion of three masses interacting via gravity over time. In that simulation, the three masses were restricted to a plane. Even though we live in three spatial dimensions, a two-dimensional model for celestial orbits isn’t such a bad approximation – the orbits of masses in celestial system are often within a few degrees of the same plane. https://brianweinstein.github.io/posts/20140505-non-orientable-surfaces/ Mon, 05 May 2014 00:00:00 +0000 https://brianweinstein.github.io/posts/20140505-non-orientable-surfaces/ An orientable surface is a surface on which it’s possible to make a consistent definition of direction. Most surfaces we encounter – like spheres, planes, and tori (doughnut shapes) – are orientable. When visualized in three dimensions, orientable surfaces have two distinct sides. Non-orientable surfaces, on the other hand, have only one side. From Wikipedia, “The essence of one-sidedness is that [an] ant can crawl from one side of the surface to the ‘other’ without going through the surface or flipping over an edge, but simply by crawling far enough. https://brianweinstein.github.io/posts/20140425-three-body-problem/ Fri, 25 Apr 2014 00:00:00 +0000 https://brianweinstein.github.io/posts/20140425-three-body-problem/ Given the starting positions, velocities, and masses of three objects interacting via gravity, the classical three-body problem involves determining the motions of the three particles throughout time. What’s cool about the three-body system is that it’s impossible to solve for the motions of the objects exactly. That is, we can’t write down an equation that describes the system. Instead of finding an exact solution, we solve the system numerically, which amounts to finding an accurate approximation. https://brianweinstein.github.io/posts/20140423-fourier-series/ Wed, 23 Apr 2014 00:00:00 +0000 https://brianweinstein.github.io/posts/20140423-fourier-series/ A Fourier series is a way to expand a periodic function in terms of sines and cosines. The Fourier series is named after Joseph Fourier, who introduced the series as he solved for a mathematical way to describe how heat transfers in a metal plate. The GIFs above show the 8-term Fourier series approximations of the square wave and the sawtooth wave. Mathematica code: f[t_] := SawtoothWave[t] T = 1; nmax = 18; a0 = (2/T)*Integrate[f[t], {t, -(T/2), T/2}] anlist = Table[(2/T)*Integrate[f[t]*Cos[(2*Pi*n*t)/T], {t, -(T/2), T/2}], {n, 1, nmax}] bnlist = Table[(2/T)*Integrate[f[t]*Sin[(2*Pi*n*t)/T], {t, -(T/2), T/2}], {n, 1, nmax}] fs[t_, nmax_] := a0/2 + Sum[anlist[[n]]*Cos[(2*Pi*n*t)/T] + bnlist[[n]]*Sin[(2*Pi*n*t)/T], {n, 1, nmax}] Manipulate[Column[{Plot[{f[t], fs[t, nmax0]}, {t, -1, 1}, PlotRange -> All, AxesLabel -> {"t", "f(t)"}, PlotStyle -> {{Thick, Black}, {Thick, Red}}, ImageSize -> 700, AspectRatio -> 1/2. https://brianweinstein.github.io/archives/ Mon, 01 Jan 0001 00:00:00 +0000 https://brianweinstein.github.io/archives/ Archives https://brianweinstein.github.io/posts/ Mon, 01 Jan 0001 00:00:00 +0000 https://brianweinstein.github.io/posts/ Posts https://brianweinstein.github.io/search/ Mon, 01 Jan 0001 00:00:00 +0000 https://brianweinstein.github.io/search/ search