
But why is a sphere's surface area four times its shadow? [video] - espeed
https://www.youtube.com/watch?v=GNcFjFmqEc8
======
quickthrower2
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.

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

~~~
empath75
The cube surface area is 4 times the average size of the shadow across all
rotations, according to the video.

~~~
rocqua
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.

~~~
pervycreeper
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](https://arxiv.org/pdf/1109.0595.pdf)

------
war1025
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.

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

~~~
Herodotus38
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.

~~~
thaumasiotes
> 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.)

~~~
amelius
> 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.

~~~
thaumasiotes
...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](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.

------
jjcm
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.

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

~~~
espeed
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](https://en.wikipedia.org/wiki/Brooks's_law)

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

~~~
espeed
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.

------
Xortl
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

~~~
zrobotics
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.

------
hanoz
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?

~~~
appden
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.

~~~
hanoz
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.

~~~
istjohn
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.

~~~
Twisol
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.

~~~
Twisol
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.

------
mxfh
Another 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/)

------
mncharity
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.

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

~~~
dmichulke
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"

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

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

------
espeed
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](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_roll...](https://golem.ph.utexas.edu/category/2013/06/g2_and_the_rolling_ball.html)

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)

