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Read more comics like this smbc-comics.com
Like this strip? Get a print!
Adams Carvalho illustration.
1. Take a circle and draw some points on the boundary. For every point you draw, you must also draw its antipode (point on the opposite side of the circle).
2. Draw some points in the interior wherever you want. 3. Label the points either +1, -1, +2, or -2 as you wish. The only stipulation is that antipodes must have opposite sign.
4. Draw triangles however you want without crossing lines.
Tucker’s Lemma says that you will ALWAYS end up with at least one line that has endpoints of either +1 and -1 or +2 and -2. Try it! More info and proof here.
Can you flatten a sphere?
The answer is NO, you can not. This is why all map projections are innacurate and distorted, requiring some form of compromise between how accurate the angles, distances and areas in a globe are represented.
This is all due to Gauss’s Theorema Egregium, which dictates that you can only bend surfaces without distortion/stretching if you don’t change their Gaussian curvature.
The Gaussian curvature is an intrinsic and important property of a surface. Planes, cylinders and cones all have zero Gaussian curvature, and this is why you can make a tube or a party hat out of a flat piece of paper. A sphere has a positive Gaussian curvature, and a saddle shape has a negative one, so you cannot make those starting out with something flat.
If you like pizza then you are probably intimately familiar with this theorem. That universal trick of bending a pizza slice so it stiffens up is a direct result of the theorem, as the bend forces the other direction to stay flat as to maintain zero Gaussian curvature on the slice. Here’s a Numberphile video explaining it in more detail.
However, there are several ways to approximate a sphere as a collection of shapes you can flatten. For instance, you can project the surface of the sphere onto an icosahedron, a solid with 20 equal triangular faces, giving you what it is called the Dymaxion projection.
The Dymaxion map projection.
The problem with this technique is that you still have a sphere approximated by flat shapes, and not curved ones.
One of the earliest proofs of the surface area of the sphere (4πr2) came from the great Greek mathematician Archimedes. He realized that he could approximate the surface of the sphere arbitrarily close by stacks of truncated cones. The animation below shows this construction.
The great thing about cones is that not only they are curved surfaces, they also have zero curvature! This means we can flatten each of those conical strips onto a flat sheet of paper, which will then be a good approximation of a sphere.
So what does this flattened sphere approximated by conical strips look like? Check the image below.
But this is not the only way to distribute the strips. We could also align them by a corner, like this:
All of this is not exactly new, of course, but I never saw anyone assembling one of these. I wanted to try it out with paper, and that photo above is the result.
It’s really hard to put together and it doesn’t hold itself up too well, but it’s a nice little reminder that math works after all!
Here’s the PDF to print it out, if you want to try it yourself. Send me a picture if you do!
Word of the Day: Mephistopheles
n. A fiendish person, esp. one who traps another into adopting a disastrous or destructive course of action; a tempter.
Image credit: Mephistopheles flying over Wittenberg, in a lithograph by Eugène Delacroix. Public Domain via Wikimedia Commons.
A 75 × 112 rectangle can be cut up into 13 squares, which can be rearranged into the initial rectangle in two different ways. One way was found by Brooks, who was so pleased with this dissection that he made a jigsaw puzzle of it, each piece being a square. His mother then tackled the puzzle and eventually succeeded in putting the pieces together, but not in the way Brooks has found!
Allgäu | Fabian Krueger
Photographers second set of images from their most recent trip to the Allgäu, part of Bavaria, Germany. Showcasing the beautiful scenery and shades of light.
Got bored
Harmonograph, H. Irwine Whitty, 1893
“The facts that musical notes are due to regular air-pulses, and that the pitch of the note depends on the frequency with which these pulses succeed each other, are too well known to require any extended notice. But although these phenomena and their laws have been known for a very long time, Chladni, late in the last century, was the first who discovered that there was a connection between sound and form.”
source here
1979 Mercedes-Benz S Class
Roads I’d like to ride: photo taken from an account of riding from Åndalsnes to Oslo in Norway over on the Rapha website.
2016 Bugatti Chiron
2016 Audi A3 S-line
Roads I’d like to ride: Jered Gruber captures a cyclist on the Gavia Pass.
1973 BMW 2002 Cabrio von Baur
Laurine van Riessen of the Netherlands rides on the barrier wall with Virginie Cueff of France after avoiding a crash during the second heat of the keirin race first round, at the Olympic velodrome in Rio de Janeiro. The Dave Hunt photograph was published in The Guardian’s Sport picture of the day.