The proof starts with two of the "string-art" lines, a line L1 (line Q'R' in the video) with parameter s and a line L2 (line QR in the video) with parameter t, and computes the intersection point P of these two lines. Keeping line L2 fixed, one gets a series of intersection points P as one varies the parameter s. One then notices that the point of intersection of line L2 and the parabola (the "touching point") is the limit of these intersection points P as s tends toward the value t. The proof finishes by showing that this limit point is just what it was hypothesised to be, namely the point along line L2 that divides it in the same proportions as its endpoints divide the control lines.I agree that some students will not recognise the crux of the proof, but will simply see something ends up being equal to something else in some identity without understanding how that relates to what was being proved.Euclid's Elements (an ancient Greek textbook on Geometry) is a good model in this regard. He first states what he is to prove. He then starts from the assumptions and finishes with what he was to prove. Each step in between is justified.
 These are great points. You'll notice that video is actually a "bonus" step which isn't required to complete the final exercise in that lesson. We are still experimenting with how to go about "bonus" steps. In other lessons you'll see they are done in a multi-step article style. Such as in Animation: https://www.khanacademy.org/partner-content/pixar/animate/pa...We also do this for the bonus step in Character Modeling (my favorite lesson!)Our plan is to continue collecting more feedback on these different styles and find out what works best.
 I just watched the first two sequences of videos and so far they have shown how to compute midpoints of lines and how to compute the coordinates of points on parabolic arcs given three arbitrary control points, including how to prove the identities involved. They then explain how this is used in practice to generate the shape of a blade of grass. Is this not the kind of technical information you were expecting to find?
 I guess it would be great if there was some sort of application that let you apply these techniques and actually see the results.Otherwise, it's like reading the theory section of documentation without getting to use the program.
 All of our lessons give you interactive tools to create and experiment with in various ways. For example this one is fun: https://www.khanacademy.org/partner-content/pixar/modeling-c...
 Yeah, I take my criticism back.I didn't go through the whole thing initially. Excellent examples.Thanks for your hard work. It is indeed an amazing series.
 Competence is certainly an important factor. Smaller economies are less able to spy competently. However, incompetent spycraft can become much more invasive, since you know it is happening and it affects you more noticeably. -----
 It's odd to see the Netherlands referred to as previously being privacy conscious.Back in the mid 2000's I recall looking in the CIA Factbook and noting that the Netherlands intercepted more telephone calls and did more internet monitoring than just about any other country, per capita (of course "Fact" was used rather loosely in the title of that book).Hackers in the Netherlands were introducing regular pulses into their daily internet activity and then using Fourier analysis to detect this signal in the server load of apparently well known Government buildings, believed to belong to the Dutch secret service (to what extent these rumours were true, I can't say -- I wasn't involved).For historical reasons, the Dutch people also developed a habit of "spying" on neighbours to "make sure they were ok". This is still included in some guides for foreigners living in the Netherlands, so they don't become concerned when their neighbours begin looking in their windows.In my stay in the Nethelands, I however learned that just about every spy agency in Europe was spying on just about everyone else in Europe. So it became clear in the end that the Dutch weren't necessarily any worse at it than everyone else. Of course, that threat seems to be greater in larger states with bigger economies. If you live anywhere else, you at least know everyone else is probably spying on you, even if your own Government is not. -----
 I wrote about this at length:http://jacquesmattheij.com/wholesale-privacy-violations-in-t...We're definitely not doing well at all on this front and it is getting worse rather than better. -----
 In my opinion, all that is happening with these "anti-privacy laws" is that very old spying practices are becoming codified in law. Nothing will actually change. The spying is already happening, and has been for a long time. -----
 Yes, the crimes are simply being retro-actively whitewashed and for the future you can expect a lot more of the same especially because it is now legal.Never mind that these laws are in conflict with just about every right that we as citizens are supposed to have and which we all learn in school should be important to us. -----
 Towards the end of the video, the officer in fact alleged she had kicked him. It's not clear to me when that happened. Possibly her foot scraped his leg in the initial scuffle to pull her out of the car. Alternatively, it may have happened in the portion that is off video. Either way, the female police officer noted evidence of the scuff, presumably on his uniform. Incidental things like these seem to be taken as evidence of "assault" all too often. -----
 The arrested woman is "lucky" she didn't get charged for the officer's dry cleaning. -----
 "Ferguson police beat a man and then charged him with 'destruction of property' for bloody uniforms" -----
 A good proportion of maths olympians go into mathematics in academia rather than into industry. Industry does need problem solvers, but maths olympians develop creativity and a sense of elegance and beauty as well. Industry requires efficient solutions to technical problems, more than elegant or beautiful ones, at least relative to what is possible with full academic freedom. -----
 As a minor nitpick, some mathematicians see elegance in efficient solutions. It's interesting to get your head around a specific problem and come up with the solution that is optimal regarding some given metric.However, I agree that industry and mathematics are not best friends. That's because mathematicians righteously demand and require a level of freedom and support the industry is not always willing to give because of the social problem it creates with other employees and because the value of the work of a mathematician can be too hard to judge.From my experience, the fight to get the working conditions you need is not worth it. My advise to fellow mathematicians is that — when you want go into the industry — to go where there are already mathematicians. -----
 Just to clarify, I meant efficient from the perspective of business, not efficient from the perspective of mathematics. In industry, often a solution which "works", but is neither mathematically "efficient" or "elegant" is "good enough". By and large, industry is trying to generate revenue, not scientific knowledge. -----
 In software engineering, elegance is prized and is often related to efficiency in several layers.At the human layer, one expression of this idea is the principle of least astonishment: "People are part of the system. The design should match the user's experience, expectations, and mental models." (Wikipedia). A system involving people is not efficient if the people are often surprised by its design, implementation, or behavior.Efficient software solutions also tend to involve elegance. Take Git for example as an improvement over other source control systems. The conceptual primitives that Git is built upon are elegant and recognizing them as the correct basis for source control (along with a strong implementation) resulted in Git's efficiency and power.Granted, software is different from mathematics, but I find the parallel interesting. I suspect that a mathematician's desire to find an elegant formulation, and appreciation of it, is very similar to the software engineer's. -----
 My anecdotal experience is that most may start with math for their first degree, but then (or during) pivot to some related but more applied area like CS, Econ, etc. -----
 You just have to click numbers, each next to the previous one. The total modulo 10 is shown. Once it reaches zero, your score is updated. You want that to happen before your time runs out, but you get a higher score if it doesn't happen too soon. When the length of the chain is a prime number, blockers are cleared. -----
 After reading the article I was left with the distinct impression that "constructor" is a name for something they suppose must exist to solve their problems and that the mere act of giving it a technical sounding name allows them to say they've solved the problem of the origin of life, since now they may say "constructor theory solves that problem". -----
 I think it's a little more subtle.A constructor is like a program. In this case, the rules of the language are physics itself. We're familiar with writing simple programs in C. We can imagine programs, like the halting problem solver. It turns out that program can't exist, but you kind of have to think about it for a while to understand why.A constructor (as i understand it) is sort of hand rolled assembly (matter) running right on physics. Perhaps there are space unicorns on the dark side of the moon. We can imagine that. It's not really clear if physics actually allows that though. My gut says no, but the rules of the game, much like the rules of the language may allow such a collection of matter to exist.I mean, quines are pretty weird. I wouldn't have thought that was even possible. -----
 And its not even an original mistake. Our ancestors also used words (like "god") to answer (i.e. stop asking) questions about the origin of life.But I for one look forward to worshiping the Great Constructor. -----
 Nice theory. Fly in the ointment: Our ancestors didn't stop asking, quite obviously, given that we ended up here. -----
 Actually a lot of ancestors did stop asking. I think that is established fact, not theory, as even today it is easy to find people who disbelieve even basic science when it contradicts a religious view.But fortunately, as you point out, not everyone stops asking questions. -----
 It didn't fly back. It bounced, as in what goes up must (unless it achieves escape velocity), come back down. The comet has gravity, just like earth. Just not much, because it is much smaller than earth -----
 If they could use southern Rhinos as surrogates, doesn't that imply the southern and northern white rhinos are the same species? So we are not talking about the extinction of a species here, only one less habitat for a species? (Not that this somehow makes it acceptable of course.) -----
 Surrogacy is simply using the womb. It is not interbreeding, so it does not necessarily imply they are the same species.There are many definitions of species. If they are able to interbreed and create viable (i.e. non-sterile) offspring, they might be considered to be the same species. However, there are cases where offspring are viable, but less fit for a variety of reasons. In this case, many biologists would still consider them separate. -----
 The definition of "species" varies based on the biological specialty you're talking about (it's the least arbitrary of the taxa, but it still is kind of arbitrary). Cladists/geneticists tend to have a view close to the one you mention ("interbreeding with viable non-sterile offspring defines a species"), but even they will say that definition gets too simplistic. Behavioral biologists generally have a more stringent definition of "species" that involves the population's ecological niche.But, either way, surrogacy is not breeding; it would be a northern egg and sperm implanted in a southern rhino. -----
 Not going to dig up a cite right now, but I'm pretty sure that females of one species have successfully gestated embryos of another species, if they are reasonably closely related.The distinction being that in surrogacy, there is no requirement to mix DNA to make viable offspring. There are likely immunological compatibility concerns at the placental boundary.Ok, here's the cite http://en.m.wikipedia.org/wiki/Interspecific_pregnancy -----
 I was looking up the differences in them a few minutes ago and there really isn't any, so much so that I'm not concerned with losing the Northern White Rhino sub-species. It's not a good thing, but we do have the required pieces to bring them back and if we don't then there is an identical animal which still has a decent sized population.I think the tolerance between species is highly variable, but "Fetuses of the Giant Panda have been grown in the womb of a cat by intercurrently inserting panda and cat embryos into the cat womb." http://en.wikipedia.org/wiki/Interspecific_pregnancyHowever, wikipedia says: "Following the phylogenetic species concept, recent research has suggested the northern white rhinoceros may be an altogether different species, rather than a subspecies of white rhinoceros, in which case the correct scientific name for the former is Ceratotherium cottoni. Distinct morphological and genetic differences suggest the two proposed species have been separated for at least a million years." -----
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