But why is a sphere's surface area four times its shadow? [video] | Hacker News Reader

But why is a sphere's surface area four times its shadow? [video]

youtube.com

141 pointsespeed61 comments

Haven't watched yet but an obvious lower bound is 2 times it's shadow (one for each side) - in fact a coin achieves this.

I don't see why this is getting downvoted. It's a perfectly valid argument that's very much in the spirit of 3b1b.

And 6x is a good upper bound (achieved by a cube). It's nice that the correct answer is halfway between these.

I guess that only works if you are considering convex shapes (which sphere and cube obviously are)

The cube surface area is 4 times the average size of the shadow across all rotations, according to the video.

In 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.

Not a surprise to see pi in there really, since we're averaging over "surfaces" of circles/spheres. In general (spoiler), it does generalize to arbitrary dimensions. We get a rational factor for odd dimensions, and 1/pi * rational for even.

See sections 6-9 here for demonstration: https://arxiv.org/pdf/1109.0595.pdf

and Graham's number is a correct but very weak upper bound -- as usual.

It 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.

Fun fact: by the same logic, the perimeter of a circle is the derivative of its area.

Does 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.

Well 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.

As the other commenter noted, the generalization to 1-dimensional hyperspheres is not super interesting. However, it generalizes perfectly to higher-dimensional hyperspheres.

Seems to me that a one-dimensional circle would be two points on a number line separated by distance 2*r.

Hopefully I can justify the reasoning behind this with the following:

A circle is defined as the set of all points in a 2-dimensional space equidistant to a given center point (the distance being the circle's radius).

A sphere is defined as the set of all points in a 3-dimensional space equidistant to a given center point.

A line in 1-d does not analogize to this definition but a point does.

> Does this relationship generalize?

Yes, the fundamental theorem of calculus guarantees this will be true. It's the same phenomenon as that if your "volume" is the definite integral of f(t) from fixed a to variable x, then the rate of change of the volume (= the surface area) is f(x), the length of the infinitesimal sliver you're about to add to the volume.

You can think of the solid sphere as an infinite stack of concentric infinitely thin hollow spheres. Each layer contributes volume to the solid sphere equal to its own area. (And this is what it means to calculate the volume of a sphere as the integral of the area of a spherical shell as the radius goes from 0 to r.)

It also generalizes to any object that can be seen as an infinite stack of concentric infinitely thin objects of the same shape (that set includes all convex polyhedra)

For example:

- for squares, area is 4r², circumference is 8r

- for cubes, volume is 8r³, area is 24r²

Correction: that rule isn’t as generic as I stated. It doesn’t apply to blocks of width 2r, breadth 2r, and height 4r, for example (volume is 16r³, area 40r²)

> You can think of the solid sphere as an infinite stack of concentric infinitely thin hollow spheres.

This is dangerous logic. For example a similar argument might lead you to think that a staircase in 2d between two points would have the same length as the Pythagorean distance between the two points, as you can make the steps arbitrarily small.

...no? The same logic will tell you that you can think of a triangle as being a linear stack of infinitely thin line segments, and there, as here, you'd be completely correct. The logic you're calling "dangerous" is just the principle behind https://en.wikipedia.org/wiki/Shell_integration .

But no similar argument will let you call the diagonal line an aggregation of little right-angled jags. A one-dimensional line is an aggregation of zero-dimensional points, not other one-dimensional lines.

Its not dangerous as long one knows what one is doing.

This is a poster boy of an example to illustrate the concept that if you have a sequence of functions f_i that becomes indistinguishable to the eye (being vague on purpose) from some function f, it does not mean that integral of that sequence will be close to the integral of f -- mathematical rigor does not matter except when it does

That’s literally what calculus is.

Not sure the exact nuances that go into “generalizable” but please consider:

To go from a radius to a circle, you can rotate a radius 360deg. To go from a circle to a sphere, you can rotate a circle on its axis 360deg.

This sort of “take a shape and move it” to produce an N+1 dimensional shape is definable using calculus.

https://en.m.wikipedia.org/wiki/Stokes%27_theorem

I'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.

I 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.

To 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.

I 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.

Yup, 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.

The 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.

It 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...

If I recall correctly 3blue1brown is actually a team of people, and it's founder created a python library which helps generate the animations.

This makes it no less impressive, IMO

That 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

I didn't realize that happened. Kind of a shame. But he did get that interactive page for visualizing quaternions built with a collaborator.

He'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.

As 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

History: see The Great War channel.

Geopolitics: 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.

Applied 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.

Search for Eugene Khutoryansky on YouTube.

For 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

[2] https://www.youtube.com/channel/UCtwKon9qMt5YLVgQt1tvJKg

Dan Carlin's hardcore history and common sense podcasts are worth checking out.

www.dancarlin.com

Very 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?

Imagine 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.

Thanks, 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.

When 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?

They may do if I measure each shadow separately only when the sun is level with the slat which casts it - I don't know.

You're intuition is correct that the rectangles would overlap if projected by a light like shadow puppets. Instead, imagine a series of lasers up and down the vertical axis positioned so that their beams are parallel to the plane that contains the sphere's equator. And imagine that the lasers spin 360 degrees around the vertical axis but their beams always stay parallel to the plane of the sphere's equator. That's the projection being described here.

Right. Effectively, we're projecting every horizontal slice of the sphere laterally to a fixed distance from the vertical axis. It's a stack of 1D projections, not a single 2D projection.

I realized after the fact that the more useful terms would be “orthogonal projection” and “perspective projection”. The novelty of the orthogonal projection at hand is that it’s projecting (flattening) in a cylindrical space.

With an orthogonal projection, you can usually think of it as taking two planes and squashing whatever object your want to project between them. In the scenario here, the ambient space has been wrapped up, so one of these squashing planes has been wrapped into a cylinder, and the other has been wrapped into a line (the degenerate case).

In either event, an orthogonal projection is indeed a collection of orthogonal projections of one dimension less. But that’s not really the whole picture.

Imagine if two non-overlapping patches on the sphere could overlap once they get projected. Now consider some single point in the overlapped region. It came from two distinct places on the original sphere, so this projection is not 1-to-1 but many-to-1.

In principle there is nothing wrong with a many-to-1 projection. But in this particular case, I hope it's pretty intuitive that the projection from the sphere to the containing cylinder is indeed one-to-one except at the poles (where it is actually 1-to-many).

Another considerably unexpected, not to say mindblowing property, in that context:

VCone + VHemisphere = VCylinder

https://en.wikipedia.org/wiki/Cavalieri%27s_principle

http://blog.zacharyabel.com/tag/cavalieris-principle/

In 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.

We all know? I think that's a fact that isn't well known.

Reminds 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"

Those are just stepping stones to "It can be shown" and "Left as an exercise for the reader."

Thank 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.

Careful with areas... Volumes are grasped “intuitevly” but areas are much much harder to “know”.

FYI, if you like 3b1b's videos, you can support him on Patreon: https://www.patreon.com/3blue1brown/posts

While 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

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

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

[2] Horopter https://en.wikipedia.org/wiki/Horopter , Hering–Hillebrand deviationn https://en.wikipedia.org/wiki/Hering-Hillebrand_deviation

[3] Brachistochrone curve https://en.wikipedia.org/wiki/Brachistochrone_curve

[4] 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

[6] Spinor https://en.wikipedia.org/wiki/Spinor

[7] Epicycloid https://en.wikipedia.org/wiki/Epicycloid

[8] G2 and the Rolling Ball 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

Eric Weinstein explains Gauge Symmetry [video] https://www.youtube.com/watch?v=2xiEEtoa-_4

Reply on news.ycombinator.com