10 ms·
But why is a sphere's surface area four times its shadow? [video]
- jjcm 8y agoI'm really impressed with the level of quality of some of the educational videos on youtube. The author here clearly spent hours preparing each of these diagrams and animations, and he's not unique among the popular youtube educators for doing so. I'm envious of the resources today's youth have in the classroom.
- duaoebg 8y agoI remember when Wikipedia came online. I worried that future generations would be so smart that I would be quickly surpassed. Now I’m worried I won’t be.
- ryanchants 8y agoTo me, this a strange take. I'll never have the genius of Newton, Copernicus, Euler, etc. However I know and have access to so much more than them, just because of when I was born. So while I may be envious of their brilliance and insight, they would be envious of so much of what I take for granted. So you not being surpassed may not be as inherently negative as you think. Apologies if inferred incorrectly.
- lr4444lr 8y agoI think his point was that the initial expectation was that improved access to knowledge via the internet would unleash the potential for a higher average level of education. The correlation appears to the OP instead to be moving in the opposite direction.
- duaoebg 8y agoYup, that is what I was getting at. In many ways I lamented my late access to computers but it was probably better for me in the long run.
- sopooneo 8y agoThe availability of fantastic material like this will take the the rare lone geniuses of the world and multiply their results and possibly help them find like minded people to work with and learn from. But the vast majority of students still need a teacher they actually know to inspire them to work hard.
- wallace_f 8y agoIt would be nice if education could experiment with different approaches and see what produces the most area under the curve in terms education achievement plotted across all students. Unfortunately contributors like 3b1b dont get rewarded enough. Current education model keeps spending more money without any results(1). 1- https://fee.org/articles/government-spending-on-education-is-higher-than-ever-and-for-what/ https://fee.org/articles/government-spending-on-education-is...
- androsyang 8y agoIf I recall correctly 3blue1brown is actually a team of people, and it's founder created a python library which helps generate the animations.
- hylianwarrior 8y agoThis makes it no less impressive, IMO
- espeed 8y agoThat was a temporary experiment to see if more people enabled Grant to produce more videos, but the Mythical Man-Month rang true again. It's just Grant now, like it was when it started. https://en.wikipedia.org/wiki/Brooks's_law https://en.wikipedia.org/wiki/Brooks's_law
- too-soon-ago 8y agoI didn't realize that happened. Kind of a shame. But he did get that interactive page for visualizing quaternions built with a collaborator.
- espeed 8y agoHe's still experimenting with different approaches, trying to find the optimal flow like any startup would. His approach as of late has been to collaborate more with other channels, like he did on the Feynman video and the interative quaternion page you referenced above.
- quickthrower2 8y agoHaven't watched yet but an obvious lower bound is 2 times it's shadow (one for each side) - in fact a coin achieves this.
- theunamedguy 8y agoI don't see why this is getting downvoted. It's a perfectly valid argument that's very much in the spirit of 3b1b.
- lightedman 8y agoBecause a huge chunk of people on this site aren't as educated and as intelligent as you'd like them to be.
- dmurray 8y agoAnd 6x is a good upper bound (achieved by a cube). It's nice that the correct answer is halfway between these.
- fred256 8y agoI guess that only works if you are considering convex shapes (which sphere and cube obviously are)
- empath75 8y agoThe cube surface area is 4 times the average size of the shadow across all rotations, according to the video.
- rocqua 8y agoIn fact, the video states this for all convex shapes. I've been trying something similar for 2D, but there it doesn't quite seem to hold. Consider a very thin rectangle of size 1 by epsilon. Then it has circumference 2 (ignoring the epsilon). The shadow it casts at angle phi has size |sin phi|. Now, if we average |sin phi| from 0 to 180 degrees, (or 0 to 360 or 0 to 90) we get (2 / pi). I haven't checked whether this average holds for things other than thin rectangles, but I'd imagine so. I then find it weird we get a trancendental number in 2D but an integer in 3D.
- hanoz 8y agoVery well done, as always. As someone who is probably below the intended audience level my one complaint is that it wasn't clear to me why the fact that the square on the sphere has the same area as its projection on the cylinder implies that the total of all these areas is the same, i.e. why do we know the projections don't overlap, when each is cast from a light at a different height on the pole, as it were?
- appden 8y agoImagine the sphere’s surface is divided into one million little rectangles. Their total area is meant to approximate the surface area of the sphere. If each one of those were projected onto the cylinder, and we showed that each of those projected rectangles has the same area and that they don’t overlap, then the total area of all million of the projected rectangles would be there same as those on the sphere. Therefore, the cylinder has the same area, since it is comprised of all those projected rectangles.
- hanoz 8y agoThanks, so it's the overlapping issue I'm left struggling with (sorry, I elaborated on this in an edit after you'd already started replying, by the looks of it). It's not obvious to me they don't overlap, if the 'light' is at a different point for each one.
- coherentpony 8y agoWhen the sun shines through your blinds, do the shadows of each of the individual chinks overlap? How about throughout the day when the sun moves? :)
- hanoz 8y agoThey may do if I measure each shadow separately only when the sun is level with the slat which casts it - I don't know.
- istjohn 8y ago
- war1025 8y agoIt maybe answers a different question, but I always thought it was neat that the surface area for a sphere is just the derivative of its volume. Beyond that, I guess I never thought about it much.
- azernik 8y agoFun fact: by the same logic, the perimeter of a circle is the derivative of its area.
- Herodotus38 8y agoDoes this relationship generalize? Hypothetically carrying further, if you take the derivative of the area (wrt radius) you get 2pi. A one dimensional circle is a line segment of length 2*r right? So how is 2 pi related to a line segment? Is it in some way analogous to the way perimeter and area are for 2d or surface area and volume? I don't know the answer.
- war1025 8y agoWell in the sphere and circle cases, it works because if you are expanding size, it expands at the perimeter. In the line case, if your line is 2r, the derivative is 2. So expanding infinitesimally expands the line by 2 times that amount. So I guess it holds, but maybe isn't all that interesting.
- azernik 8y agoAs the other commenter noted, the generalization to 1-dimensional hyperspheres is not super interesting. However, it generalizes perfectly to higher-dimensional hyperspheres.
- baddox 8y agoSeems to me that a one-dimensional circle would be two points on a number line separated by distance 2*r.
- 8y ago
- espeed 8y agoWhile we're on the topic of curves and spheres... Does anyone here know of any theorems that relate the curve of a cycloid [1] to the curve of the horopter [2] (in particular, the empirical horopter)? In my continued quest to connect curious properties of the cycloid, I noticed hints of potential correspondences between the two curves beyond just their shape, particularly in the ways the curves of the cycloid and horopter both relate to the path of light. A few years back, Grant did a 3Blue1Brown video with with Steven Strogatz on the cycloid Brachistochrone curve : https://www.youtube.com/watch?v=Cld0p3a43fU https://www.youtube.com/watch?v=Cld0p3a43fU And Vsauce did one with Adam Savage on the Brachistochrone [3] where they build a mechanical model of one that shows it's the fastest/optimal path among different curves, and their experiment also shows the cycloid Tautochrone [4] invariant property where objects begin up the curve at different distances apart and yet all arrive together simultaneously in constant time. https://www.youtube.com/watch?v=skvnj67YGmw https://www.youtube.com/watch?v=skvnj67YGmw Some other interesting properties and places the cycloid shows up... * The arclength of the cycloid curve is 8R, a rational value given a rational radius. * The shape of the closed universe [5]. While we don't yet know if the shape of the universe is open or closed, we do know that if the universe is closed the shape of its evolution is precisely the shape of a cycloid. * Spinors [6], octonions, and the epicycloid [7]. Electrons, protons, neutrinos, and quarks are spinors. The Rolling Spinor is like a ball, but "thanks to the 'double' in the double cover SU(2)→SO(3), a 360∘ rotation does not act like the identity. Instead, we need to rotate by 720∘ degrees to get back where we started" [8]. The two balls have a 3-to-1 ratio, and the path traced out around the larger ball is an epicycloid. [1] Cycloid https://en.wikipedia.org/wiki/Cycloid https://en.wikipedia.org/wiki/Cycloid [2] Horopter https://en.wikipedia.org/wiki/Horopter https://en.wikipedia.org/wiki/Horopter , Hering–Hillebrand deviationn https://en.wikipedia.org/wiki/Hering-Hillebrand_deviation https://en.wikipedia.org/wiki/Hering-Hillebrand_deviation [3] Brachistochrone curve https://en.wikipedia.org/wiki/Brachistochrone_curve https://en.wikipedia.org/wiki/Brachistochrone_curve [4] Tautochrone curve https://en.wikipedia.org/wiki/Tautochrone_curve https://en.wikipedia.org/wiki/Tautochrone_curve [5] MIT 8.286 The Early Universe: Introduction to Non-Euclidean Space [video] https://www.youtube.com/watch?v=YfbXB_MSkSY https://www.youtube.com/watch?v=YfbXB_MSkSY [6] Spinor https://en.wikipedia.org/wiki/Spinor https://en.wikipedia.org/wiki/Spinor [7] Epicycloid https://en.wikipedia.org/wiki/Epicycloid https://en.wikipedia.org/wiki/Epicycloid [8] G2 and the Rolling Ball https://golem.ph.utexas.edu/category/2013/06/g2_and_the_rolling_ball.html https://golem.ph.utexas.edu/category/2013/06/g2_and_the_roll... Split Octonions and the Rolling Ball, Dr. John Baez [video] https://www.youtube.com/watch?v=xvflQcHT5C4 https://www.youtube.com/watch?v=xvflQcHT5C4 Eric Weinstein explains Gauge Symmetry [video] https://www.youtube.com/watch?v=2xiEEtoa-_4 https://www.youtube.com/watch?v=2xiEEtoa-_4
- mncharity 8y agoIn a spirit of rough-quantitative reasoning... Because pi is 3, modulo a 5% error we'll so easily ignore, we all know the volume and area of a ball are 1/2 those of the box it came in. And the area and perimeter of a circle are 3/4 those of its bounding square. So a unit-wide ball in a box has an area of 3, and its shadow on the bottom of the box has an area of 3/4. QED. How often does education leave students struggling with half-remembered "was it 4/3 pi r^3... or was it 4 pi r hmm...", and bereft of any clue that the ball's simply half its box. It's like education never uses it to engage with the physical world, so there's no selection pressure for being able to apply it as other than a math tidbit.
- krapht 8y agoWe all know? I think that's a fact that isn't well known.
- dmichulke 8y agoReminds me of the math lectures during university: "Since we all know that ... it is obvious that ..." and I kept thinking "no we don't and no it isn't"
- LanceH 8y agoThose are just stepping stones to "It can be shown" and "Left as an exercise for the reader."
- mncharity 8y agoThank you for posting this comment. I intended "we all know" as a device, an "obvious" counterfactual to emphasize the point, that there's this bizarre gap in what's taught. I wasn't quite happy with it, and later worried it might burn someone ("I guess it's just me that doesn't know that"), but it was late, and I punted. So, my thanks for the save.
- pfortuny 8y agoCareful with areas... Volumes are grasped “intuitevly” but areas are much much harder to “know”.
- jrauser 8y agoFYI, if you like 3b1b's videos, you can support him on Patreon: https://www.patreon.com/3blue1brown/posts https://www.patreon.com/3blue1brown/posts
- Xortl 8y agoAs a huge fan and patron of 3blue1brown's videos, does anyone have recommendations for similar high-quality channels covering other topics? Personally I'm interested in at least: * History/biographies * Sciences (physics, chem, astronomy) * In-depth nonpartisan analysis of political situations, especially current ones. The breaking news cycle does a poor job parsing out useless or incorrect information. * How we've managed to create the insanely complex technology we have today, starting at the basics
- dandare 8y agoHistory: see The Great War channel.
- dandare 8y agoGeopolitics: the Caspian Report is really interesting and if I remember correctly some Wikileaks documents link in to Stratfor. Interesting combination of "amateur" presenter and private intelligence agency pretending not to push a certain worldview.
- zrobotics 8y agoApplied science. He doesn't cover things in documentary style, but recreates quite a lot of interesting science that one would assume is beyond the level of garage science, and does an excellent job explaining all the concepts.
- MarekKnapek 8y agoSearch for Eugene Khutoryansky on YouTube.
- sohkamyung 8y agoFor Chemistry, I enjoy Periodic Videos [1] For the history of interesting objects at the Royal Society (and other occasional places), see Objectivity [2] [1] https://www.youtube.com/user/periodicvideos https://www.youtube.com/user/periodicvideos [2] https://www.youtube.com/channel/UCtwKon9qMt5YLVgQt1tvJKg https://www.youtube.com/channel/UCtwKon9qMt5YLVgQt1tvJKg
- simagule 8y ago
- mxfh 8y agoAnother considerably unexpected, not to say mindblowing property, in that context: VCone + VHemisphere = VCylinder https://en.wikipedia.org/wiki/Cavalieri%27s_principle https://en.wikipedia.org/wiki/Cavalieri%27s_principle http://blog.zacharyabel.com/tag/cavalieris-principle/ http://blog.zacharyabel.com/tag/cavalieris-principle/