Jakub Sliacan
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55 followers
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Hmm, interesting...
The densest way to pack octagons

This image made by shows the densest way to pack equal-sized regular octagons in the plane.  The cool part: the density is slightly less than the best you can do for circles!

You can pack equal-sized circles with a density of at most

90.68996%

or so.  For equal-sized regular octagons, the best you can do is

90.61636%

That's just about 0.07% worse, but it's enough to prove that a circle isn't the pessimal plane packer - that is, the shape whose densest packing is the lowest of all.

And one reason that is interesting is that Stanislaw Ulam conjectured that in 3 dimensions, the sphere is the pessimal packer!  That conjecture is still open.

For more about this story, visit my blog Visual Insight:

http://blogs.ams.org/visualinsight/2014/10/15/packing-regular-octagons/

#geometry
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Hyperreal numbers: infinities and infinitesimals

When Leibniz invented calculus, he also invented infinitesimals - numbers that are bigger than zero but smaller than 1/2, 1/3, 1/4, ... and so on.  Many people were uncomfortable with these, so they figured out how to do calculus without infinitesimals.   That's how it's usually taught now.

But it turns out you can do calculus with infinitesimals in a perfectly rigorous way... and in some ways, it's easier!   Here's a free online textbook that teaches calculus this way:

• H. Jerome Keisler, Elementary Calculushttp://www.vias.org/calculus/ or for a PDF version, https://www.math.wisc.edu/~keisler/calc.html.

The picture here is from this book.  There's a tiny little infinitesimal number ε, pronounced epsilon.  And 1/ε is infinitely big!  These aren't 'real numbers' in the usual sense.  Sometimes they're called hyperreal numbers.

You can calculate the derivative, or rate of change, of a function f by doing

(f(x+ε) - f(x)) / ε

and then at the end throwing out terms involving ε.  For example, suppose

f(x) = x²

Then to compute its derivative we do

((x+ε)² - x²) / ε

Working this out, we get

(x² + 2εx + ε² - x²) / ε = (2εx + ε²) / ε  = 2x + ε

At the end, we throw out the term involving ε.  So, we get 2x.  This is the rate of change of the function x².

The book will teach you calculus this way, from scratch.  If you had trouble understanding 'limits' in calculus, you might prefer this way.  Or, you might just enjoy seeing another approach.

The details of this subject are infinitely interesting, but I'll just say an infinitesimal amount.  In 1961 the logician Abraham Robinson showed that hyperreal numbers are just as consistent as ordinary real numbers, and that the two systems are compatible in a certain precise sense.  In 1976, Jerome Keisler, a student of the famous logician Tarski, published this elementary textbook that teaches calculus using hyperreal numbers.

Now it's free, with a Creative Commons copyright!
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Warwick's Martin Hairer was awarded a Fields Medal. Another recipient is a lady - for the first time, Maryam Mirzakhani. Great news!
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Lemma 1. Under various conditions,
(1) ...
(2) ...
(3) ...
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Holy hell, this could be an amazing thing. Even their reasoning of what boils down to "we literally can't build/sell cars fast enough to effect the level of change we need in emissions, so let's allow other companies to use our tech to speed things along" is an amazing thing.

Would have loved to sit on on the meeting where that was originally brought up.

I just really hope no one sets them up the bomb.
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Turing test passed