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**Elliptic functions**

You probably know about trig functions like sin(x). These are the most basic functions that are periodic:

sin(x + 2π) = sin(x)

**Elliptic functions**are functions of two variables, x and y, that are periodic in two directions:

f(x + 2π,y) = f(x,y)

and

f(x,y + 2π) = f(x,y)

This movie is a way of illustrating an elliptic function.

What makes elliptic functions so special is that you can think of them as functions of a single complex variable:

z = x + iy

and then they have a derivative in the special sense you learn about in a course on complex functions!

It's a lot harder for a complex function to have a derivative than an ordinary real function! A function like

f(x,y) = sin(x) sin(y)

is periodic in two directions, but it doesn't have a derivative df/dz. Mysterious as this may sound, this is the reason elliptic functions are so special.

In the late 1800s, all the best mathematicians thought about elliptic functions, so there are 'Jacobi elliptic functions' and 'Weierstrass elliptic functions' and many more. Now they're less popular, but they're still incredibly important. You need to think about them if you want to deeply understand how long the perimeter of an ellipse is. They're also important in physics, and fundamental to the proof of Fermat's Last Theorem.

+Gerard Westendorp has been making mathematical illustrations for a long time, so if you like such things, circle him!

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cool

**Superionic ice**

There are over 15 kinds of ice. Different kinds are stable at different pressures and temperatures. Some of the weirdest kinds may exist inside giant icy planets like Uranus and Neptune. Most of what we know about them comes from computer simulations, because they only exist at very high pressures.

They're called

**superionic ices**, because while the oxygen atoms get locked in a crystal structure, the hydrogen atoms become ionized, breaking apart into protons and electrons. The protons can then

*move around like a liquid*between the oxygen atoms!

The first phase of superionic ice was predicted in 1999 by a group of Italian scientists. They predicted that this ice exists at pressures 500,000 times the atmospheric pressure here on Earth, and temperatures of a few thousand Kelvin. In this kind of ice, the oxygen atoms form a crystal called a body centered cubic.

In 2013, Hugh F. Wilson, Michael L. Wong, and Burkhard Militzer predicted the new phase shown here. This may show up above 1,000,000 times atmospheric pressure. The oxygen atoms, shown as blue spheres, lie in pattern called a face centered cubic. The protons are likely to be found in the orange regions.

Hugh Wilson said:

**Superionic water is a fairly exotic sort of substance. The phases of water we're familiar with all consist of water molecules in various arrangements, but superionic water is a non-molecular form of ice, where hydrogen atoms are shared between oxygens. It's somewhere between a solid and a liquid—the hydrogen atoms move around freely like in a liquid, while the oxygens stay rigidly fixed in place. It would probably flow more like a liquid, though, since the planes of oxygen atoms can slide quite freely against one another, lubricated by the hydrogens.**

These simulations are hard, and different papers are reporting different results. So, I wouldn't be surprised if people correct them a bit before we settle down to the truth.

You can also try to make superionic ice in the lab, but it's hard! In 2005 Laurence Fried tried to make it at the Lawrence Livermore National Laboratory in California. They smashed water molecules between diamond anvils while simultaneously zapping it with lasers. They seemed to find evidence for superionic ice.

You can read more here:

http://phys.org/news/2013-04-phase-dominate-interiors-uranus-neptune.html

and for some even newer results, try this:

https://chemistry.princeton.edu/news/scientists-predict-cool-new-phase-superionic-ice

Here's the paper on the first kind of superionic ice:

• C. Cavazzoni, G. L. Chiarotti, S. Scandolo, E. Tosatti, M. Bernasconi, and M. Parrinello, Superionic and metallic states of water and ammonia at giant planet conditions,

*Science*

**283**(1999), 44-46. Available free with registration at http://www.ncbi.nlm.nih.gov/pubmed/9872734.

and here's the second kind:

• Hugh F. Wilson, Michael L. Wong, Burkhard and Militzer, Superionic to superionic phase change in water: consequences for the interiors of Uranus and Neptune. Available free at http://arxiv.org/abs/1211.6482.

#spnetwork arXiv:1211.6482 #ice

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**Mt. Fuji Today 今日の富士山**

*SIGMA dp0 Quattro*

雲に隠れて見えない日が続きましたが、久しぶりに富士山が姿を表してくれました。

We hadn't seen Fujisan for a week but finally showed up today. It was a nice sunset =)

#CoolJapan

Hi all, hopefully getting back into G+ will net me some book suggestions somewhere. I'm pretty much burnt through everything I can find. I think.

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ha

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Suddenly I feel the need to reconnect with google plus. I'll think of something to post later.

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