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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. To help in this process, the ESA derived a 3D model of the comet’s surface, and the data (made up of over 30,000 measurements) was recently released. The images above use ESA data to model the surface of 67P. By adjusting lighting and orientation, actual photos taken by Rosetta (images 3 and 5) can be reproduced. You can play around with the model yourself! Either use the code below in Mathematica or on the Wolfram Programming Cloud (with a free account), or play with the model online by clicking here*. *This link will expire on Dec 23, 2014, after which you’ll have to upload the .obj file directly to the online viewer. Mathematica code: objLink = "http://sci.esa.int/science-e/www/object/doc.cfm?fobjectid=54726"; comet = Import[objLink, "OBJ"]; pts = Import[objLink, {"OBJ", "VertexData"}]; ListSurfacePlot3D[pts, MaxPlotPoints -> 20, Mesh -> All, MeshStyle -> Opacity[0.4]] Show[comet, Background -> Black, Lighting -> {{"Directional", LightGray, {-7, -10, 10}}}] Additional sources: [1] [2] [3] Images 3 & 5: ESA/Rosetta/NAVCAM, CC BY-SA IGO 3.0 ...

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). The different outputs are generated by using different pendulum length ratios in each simulation. Mathematica code posted here. Additional source not linked above. ...

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. One notable use of the Reuleaux triangle, though, is in drilling holes in the shape of a slightly rounded square (watch one of the triangle’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, “Other than a circle, what shape can you make a manhole cover such that it can’t fall through the hole it covers?” Mathematica code posted here. Additional source not linked above. ...

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. 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. It’s physically impossible for electrons to stably orbit like this, and the idea of orbiting electrons was entirely replaced by 1926. I can’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! ...

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 “Zombies and Humans”. 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] ...