Hexagonal Hyperbolic Honeycombs

Azimuth

This post is just for fun.

Roice Nelson likes geometry, and he makes plastic models of interesting objects using a 3d printer. He recently created some great pictures of ‘hexagonal hyperbolic honeycombs’. With his permission, I wrote about them on my blog Visual Insight. Here I’ve combined those posts into a single more polished article.

But the pictures are the star of the show. They deserve to be bigger than the 450-pixel width of this blog, so please click on them and see the full-sized versions!

The {6,3,3} honeycomb

This is the {6,3,3} honeycomb.

How do you build this structure? Take 4 rods and glue them together so their free ends lie at the corners of a regular tetrahedron. Make lots of copies of this thing. Then stick them together so that as you go around from one intersection to the next, following the rods, the shortest possible…

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Kolmogorov’s power law for turbulence

What's new

Many fluid equations are expected to exhibit turbulence in their solutions, in which a significant portion of their energy ends up in high frequency modes. A typical example arises from the three-dimensional periodic Navier-Stokes equations

$latex displaystyle partial_t u + u cdot nabla u = nu Delta u + nabla p + f&fg=000000$

$latex displaystyle nabla cdot u = 0&fg=000000$

where $latex {u: {bf R} times {bf R}^3/{bf Z}^3 rightarrow {bf R}^3}&fg=000000$ is the velocity field, $latex {f: {bf R} times {bf R}^3/{bf Z}^3 rightarrow {bf R}^3}&fg=000000$ is a forcing term, $latex {p: {bf R} times {bf R}^3/{bf Z}^3 rightarrow {bf R}}&fg=000000$ is a pressure field, and $latex {nu > 0}&fg=000000$ is the viscosity. To study the dynamics of energy for this system, we first pass to the Fourier transform

$latex displaystyle hat u(t,k) := int_{{bf R}^3/{bf Z}^3} u(t,x) e^{-2pi i k cdot x}&fg=000000$

so that the system becomes

$latex…

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El Niño Project (Part 1)

Azimuth

A bunch of Azimuth Project members like to program, so they started the Azimuth Code Project… but now it’s getting more lively! We’re trying to understand and predict the climate phenomenon known as El Nio.

Why? Several reasons:

? It’s the biggest source of variability in the Earth’s climate on times scales between a year and a decade. It causes weather disturbances in many regions, especially near the Pacific Ocean. The last really big one happened in 1997-1998, and we’re waiting for the next.

? It’s hard to predict for more than 6 months in advance. It’s not periodic: it’s a quasi-periodic phenomenon that occurs across the tropical Pacific Ocean every 3 to 7 years.

? It matters for global warming. A lot of heat gets stored in the ocean, and a lot comes back into the atmosphere during an El Nio. So, the average surface air temperature of…

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Simple Mass Calculation

What's inside the Proton?

Hadrons, particles composed of quarks, can be sorted into two different categories: Baryons and Mesons. The proton and the neutron are baryons.

According to the Standard Model, baryons contain three quarks, and no other massive particles. Comay’s model assumes that baryons have a massive core and that the three observable quarks occupy outer energy shell.
With regard to mesons, both models agree that they are composed of quark-antiquark pair and contain no other massive particles. It is further agreed that according to Field Theory, there is a probability of finding additional quark-antiquark pairs inside the baryons and the mesons.

Useful Energy Notions
Electrons within the atom are bound to the nucleus, and the Hadrons are bound states of quarks. In Physics, three different notions relate to the energy of these states:

A bound state can only occur if “attraction forces” apply between the system’s components. The energy derived from theses…

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From Griffiths to Peskin: a lit review for beginners

An American Physics Student in England

a.k.a. “How to get started learning QFT as an undergraduate.”

5 Jan 07: Slight updates (added more details) to a few of the reviews, and an added caveat.

Quantum Field Theory (QFT) plays a key role in all branches of theoretical physics. For students interested in high energy theory, exposure to QFT at any early stage is slowly becoming the standard for top American graduate schools. This is already the case for the Mathematics Tripos at Cambridge.

However, the ‘standard’ American undergraduate physics curriculum doesn’t typically encourage Quantum Field Theory, nor do smaller liberal arts college regularly offer QFT courses. It is often expected that a student will take a second year of ‘graduate’ quantum mechanics before taking QFT.

An inspired student with adequate background should be able to take quantum mechanics in his/her second or third year and then progress directly to a ‘real’ QFT course with a bit…

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Eddy Who?

Azimuth

Or: A very short introduction to turbulence

guest post by Tim van Beek

Have a look at this picture:

Then look at this one:

Do they look similar?

They should! They are both examples of a Kelvin-Helmoltz instability.

The first graphic is a picture of billow clouds (the fancier name is altostratus undulatus clouds):

The picture is taken from:

? C. Donald Ahrens: Meteorology Today, 9th edition, Brooks/Cole, Kentucky, 2009.

The second graphic:

shows a lab experiment and is taken from:

? G.L. Brown and A. Roshko, online available Density effects and large structure in turbulent mixing layers, Journal of Fluid Mechanics64 (1974), 775-816.

Isn’t it strange that clouds in the sky would show the same pattern as some gases in a small laboratory experiment? The reason for this is not quite understood today. In this post, I would like to talk a little bit…

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