Breakfast Far Above The Clouds, Pokut, Turkey

From @dai0.0: “【 #もずくさんのお風呂シリーズ 】 もずくさん🐱の. あかすりサービスはじめました. 湯加減チエックに続いて. あかすりをしてくれるように. なりました(*´▽｀*) 「手をこっちによこすニャ🐱❤️ あかすりするニャ🐱」” #catsofinstagram #twitterweek [source: http://ift.tt/2FpgmM0 ]

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!

Want her waist

The Bortle Scale and Light Pollution

The Bortle Scale is used by astronomers to rate the darkness of our skies. It ranges from 1 (darkest) to 9 (brightest). For most of us, our daily lives are spent beneath a radiance level of between 5 and 8 and rarely venture into areas ranked 3 or darker- and what a shame that is.

Light pollution, while a testament to our technological advances, has blanketed our view of the universe and decoupled our relationship with the cosmos. For the millions of people living in areas where less than 20 stars can be seen in the night sky, it is practically impossible to imagine a natural sky blanketed with upwards of 2,500 stars backed by great ribbons of billions of stars which can be found in our Galaxy: The Milky Way.

What are the effects of light pollution?

Continuar lendo

Helioceras Heteromorph Ammonite

“Love Is” In Small Things

Heteromorph Ammonites with spikes (Emericiceras alpinum, Ancyloceras vandenbecki, Emericiceras barremense) Cretaceous, Lower Barremian -  Agadir, Morocco