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demonstration of quantum trapping

Photograph of the May 1919 solar eclipse captured by Arthur Eddington, which proved Einstein’s theory of general relativity. 

Credit: SSPL/Getty Images

9 years in the making, all our best geometry in one place. ❤, NakGeo

Four Planet System in Orbit, Directly Imaged | Planetary Landscapes Credit: Many Worlds

This evocative movie of four planets more massive than Jupiter orbiting the young star HR 8799 is a composite of sorts, including images taken over seven years at the W.M. Keck observatory in Hawaii.

Read more here.

A fascinating core / Source / by Hubble Space Telescope / ESA

Robot uses Human to create Art - Dragan Ilic

Substances don’t have to be a liquid or a gas to behave like a fluid. Swarms of fire ants display viscoelastic properties, meaning they can act like both a liquid and a solid. Like a spring, a ball of fire ants is elastic, bouncing back after being squished (top image). But the group can also act like a viscous liquid. A ball of ants can flow and diffuse outward (middle image). The ants are excellent at linking with one another, which allows them to survive floods by forming rafts and to escape containers by building towers. 

Researchers found the key characteristic is that ants will only maintain links with nearby ants as long as they themselves experience no more than 3 times their own weight in load. In practice, the ants can easily withstand 100 times that load without injury, but that lower threshold describes the transition point between ants as a solid and ants as a fluid. If an ant in a structure is loaded with more force, he’ll let go of his neighbors and start moving around.

When they’re linked, the fire ants are close enough together to be water-repellent. Even if an ant raft gets submerged (bottom image), the space between ants is small enough that water can’t get in and the air around them can’t get out. This coats the submerged ants in their own little bubble, which the ants use to breathe while they float out a flood. For more, check out the video below and the full (fun and readable!) research paper linked in the credits. (Video and image credits: Vox/Georgia Tech; research credit: S. Phonekeo et al., pdf; submitted by Joyce S., Rebecca S., and possibly others)

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!