Dsungaripterus. Cretaceous. China

Work

smithsoniannmnh When calcium leached out from this scallop some 40 million years ago, it formed a halo that solidified the sediment around it. Unearthed in Washington state, the specimen now resides in our research collections. Happy #FossilFridayfrom our Department of Paleobiology and @SIxDIGI, who are well on their way to digitizing #1MillionFossils!

Ammonite Fossil (Goniatite, Devonian) - Atlas Mtns, Morocco

Another one…

fms.fossils

Here was a lucky little find from today’s fossil hunt on Charmouth beach. This is a pyrite ammonite of the species Echioceras raricostatum. They are preserved in very soft shale, as you can see, which means that they come out in one piece very easily. Usually, the sea pulls them out of the rocks and dumps them on the beach. This can easily damage or destroy them. This one has a thin calcite shell on top of the pyrite which would have been removed by the sea almost instantly if it had got to it before I did. This fossil is around 196 million years old.

Panda tree!

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!

Gray wolves by Chad Barry

Art By IG: @bryanthegirl Instagram: @artwoonz