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Jeff Doak
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Sorry to be a little off topic and jokey, but you guys are the best audience for this post ... I'm just going to leave these concept book covers here for your consideration #someTruthHere 
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A 3-dimensional golden star

Here Greg Egan has drawn a dodecahedron with 5 tetrahedra in it.  This picture is 'left-handed': if you look at where the 5 tetrahedra meet, you'll see they swirl counterclockwise as you go out!  If you view this thing in a mirror you'll get a right-handed version. 

Putting them together, you get a dodecahedron with 10 tetrahedra in it.   You can see it here:

The two kinds of tetrahedra are colored yellow and cyan.  Regions belonging to both are colored magenta.  It's pretty - but it's hard to see the tetrahedra, because they overlap a lot!

A cube has 8 corners.  If you take every other corner of the cube, you get the 4 corners of a tetrahedron.  But you can do this in 2 ways.  If you choose both, you get a cube with 2 tetrahedra in it:

This picture is from Frederick Goodman's book Algebra: Abstract and Concrete.

All this is just the start of a much more elaborate and beautiful story which also involves the golden ratio, the quaternions, and 4-dimensional shapes like the 4-simplex, which has 5 tetrahedral faces, and the 600-cell, which has 600 tetrahedral faces!   You can read it here:

I learned some of this story from Adrian Ocneanu at Penn State University.  Greg Egan and I figured out the rest... or most of the rest.  There's an unproven conjecture here, which needs to be true to make the whole story work.  Can you prove it?

Puzzle: If you take a regular 4-simplex whose vertices are unit quaternions, with the first equal to 1, can you prove the other 4 vertices generate a free group on 4 elements?

Hmm, I see that this puzzle has been solved by +Ian Agol and someone else on Mathoverflow:

I don't understand the solution yet, because I don't know what a 'Bass-Serre tree' is... but I'll try to learn about this.  Math is infinite, there's always more to learn.

#geometry #4d  
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Yes, there is ample supply of people supporting rote memorization and antiquated skills... at least as long as they are taught to children...

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The Sun at night

This picture of the Sun is hardly high-definition. But, in its own way, it is extraordinary. Why? Because it was taken at night. It was taken looking down through the Earth. And it was taken not with light but with neutrinos.
Neutrinos are ghostly subatomic particles which are created in abundance by the sunlight-generating nuclear reactions in the core of the Sun. To them solid matter is as transparent as a pane of glass.
Hold up your hand. You would never know it but about a 100 million million neutrinos are passing through every square centimeter of your flesh every second. That’s why it is possible to image the Sun on the other side of the Earth by looking down through almost 13,000 kilometers of rock.
This picture was obtained by the Japanese Super-Kamiokande neutrino detector, situated in the Kamioka metal mine in the Japanese Alps. While sunlight takes about 30,000 years to work its way out from the center to the surface of the Sun, neutrinos take just two seconds.
Once at the surface, it is only another eight-odd minutes of free-flight before they get to the Earth.

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Important history
High Stakes Lab

In my introductory physics courses, I occasionally implement what I call “high stakes labs.” Take a set of measurements for a devices such as a ball launcher, use those results and your knowledge of physics to predict an outcome (such as where the ball will land) and then test your prediction. The catch is that the grade for the lab depends upon the accuracy of your prediction. It is stressful and challenging, but it demonstrates a common aspect of science and engineering. You can’t simply look up the answer in the book. You have to test your ideas in the real world.

In terms of high stakes labs, the highest of stakes was likely the early manned space program. Not only does everything from the propulsion system to life support need to function correctly, the rocket needs to land in the correct location.  Calculating trajectories is not easy, as anyone who has played Kerbal Space knows.  It involves complex mathematics such as analytic geometry, and it absolutely needs to be correct.

When NASA prepared to launch Alan Shepard as the first American in space, they used a computer to calculate his capsule’s trajectory. Her name was Katherine Johnson. Johnson’s forte was analytic geometry, and she was very, very good at it.  She began her work as part of a pool of “computers” that worked through the complex mathematics necessary for orbital predictions. Her skill as a mathematician was so impressive that when NASA first began using electronic computers to calculate trajectories, Johnson was asked to verify the results.

Johnson worked for NASA’s Langley Research Center for more than three decades, co-authored 26 papers, and ensured that American astronauts reached their destination. She also happened to be a black woman in a field dominated by white men, which was a whole other kind of high stakes lab.

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The first man-made, biologically functional leaf has been created. It takes in carbon dioxide, water, and light and releases oxygen. That's good news for space travelers:

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The weather forecast did not mention mayhem

#Godzilla #Cloud
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