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<item rdf:about="https://complex-analysis.com/">
    <title>Complex Analysis</title>
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    <link>https://complex-analysis.com/</link>
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<item rdf:about="https://bollu.github.io/a-hackers-guide-to-numerical-analysis.html">
    <title>A Hacker’s Guide to Numerical Analysis</title>
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<item rdf:about="https://jamesmccaffrey.wordpress.com/2017/11/30/the-epsilon-greedy-algorithm/">
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<item rdf:about="http://jamiehannaford.com/mathematics/the-beauty-of-recurring-decimals/">
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<item rdf:about="http://www.lisazhang.ca/2011/12/galois-theory-in-1500-words.html">
    <title>A Notebook: Galois Theory in 1500 Words</title>
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<item rdf:about="http://www.geogebra.org">
    <title>Geogebra</title>
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    <link>http://www.geogebra.org</link>
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<item rdf:about="http://scienceblogs.com/evolutionblog/2011/08/hilberts_hotel.php">
    <title>Hilbert's Hotel</title>
    <dc:date>2011-08-23T04:19:01+00:00</dc:date>
    <link>http://scienceblogs.com/evolutionblog/2011/08/hilberts_hotel.php</link>
    <dc:creator>jbcrail</dc:creator><description><![CDATA[One upside to my recent convalescence has been that I have had plenty of time for reading.  Currently I'm working my way through Graham Oppy's book Philosophical Perspectives on Infinity, published by Cambridge University Press in 2006.  Oppy is best known as a philosopher of religion, writing from a generally atheistic perspective.  His book Arguing About Gods is really excellent, thought it definitely does not make for light reading.


As for the present book, I'm only through the first two chapters so far.  I think I'm going to like the rest, though, since the preface contains remarks like this:



Part of my interest in philosophy of religion stems from the conviction that it must be possible to convince reasonable religious believers that traditional monotheistic arguments for the existence of God are worthless.  Hence, not surprisingly, one of the subsidiary goals for the projected larger work is to make some contribution to the case for supposing that reasonable religious believers ought to recognise that the arguments for the existence of God provide no reason at all for reasonable nonbelievers to change their minds.


My kind of guy!


I've written about infinity a couple of times before (here and here.)  But there is plenty more to say.  So how about we take a quick look at one of the more intriguing thought experiments in mathematics.  I refer, of course, to Hilbert's Hotel.
 Read the rest of this post... | Read the comments on this post...]]></description>
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<item rdf:about="http://www.wired.com/wiredscience/2011/08/crumpled-paper-physics/">
    <title>The Cutting-Edge Physics of a Crumpled Paper Ball</title>
    <dc:date>2011-08-22T19:00:22+00:00</dc:date>
    <link>http://www.wired.com/wiredscience/2011/08/crumpled-paper-physics/</link>
    <dc:creator>jbcrail</dc:creator><description><![CDATA[

Take a piece of paper. Crumple it. Before you sink a three-pointer in the corner wastebasket, consider that you’ve just created an object of extraordinary mathematical and structural complexity, filled with mysteries that physicists are just starting to unfold. 

“Crush a piece of typing paper into the size of a golf ball, and suddenly it becomes a very stiff object. The thing to realize is that it’s 90 percent air, and it’s not that you designed architectural motifs to make it stiff. It did it itself,” said physicist Narayan Menon of the University of Massachusetts Amherst. “It became a rigid object. This is what we are trying to figure out: What is the architecture inside that creates this stiffness?”

Menon’s expedition into the shadowy heart of a crumpled sheet — of aluminum foil, to be precise — was undertaken with fellow Amherst physicist Anne Dominique Cambou and published in an August 23 Proceedings of the National Academy of Sciences article. The pair think they’ve mapped the mathematical underpinnings of its rigidity. 



The geometry of a conically distorted sheet of paper, painted and viewed through cross-polarized lenses that reveal subtle variations in wavelengths of reflected light. Cerda et al./Nature.
Of course, it may seem surprising that a balled-up sheet of paper or foil should contort itself beyond knowledge. But Menon noted that when physicists finally described the precise dynamics of conical crumpling, which you can achieve by laying a sheet of paper over a coffee cup and poking down with one finger,  it was regarded as a mathematical tour-de-force.

A crumpled cone is a far simpler example of the tendencies that produce a crumpled ball: when a flat plane is subjected to distortional stress but only permitted to bend, not stretch, it transforms suddenly and unpredictably into a landscape of folds and facets, each representing an entirely new surface. It’s what researchers call a “far from equilibrium” process, guided by strange rules and non-linear effects. The mechanics of an individual crease are understood, but when physicists try to predict where that crease will appear or how it will influence the next, understanding goes dim. 

Trying to peer inside a crumpled ball by simulating the process in three dimensions is “mathematically nasty,” a problem that quickly pushes lab-grade computers to their limits, said Menon. And trying to reverse-engineer structure from patterns revealed upon unfolding just isn’t possible. What happens in a crumpled ball stays in a crumpled ball. 

‘I love it that these simple-looking problems are so nasty sometimes.’
“If you’re not talking about simulation, but mathematical understanding of these things, that’s one step harder,” said Menon. “We understand the underlying equations of the mechanics of a thin sheet very well. Those have been around for a century. But solving those equations, to produce a physical understanding, is difficult even in simple cases. If you’re talking about a structure that owes its properties to 1,000 or more of these structures, interacting in complicated ways, that’s asking more than we can do now.” 

To look into crumpled balls, Menon and Cambou used X-ray microtomography, an imaging technique that, like a medical CT scan, assembles three-dimensional images from thousands of two-dimensional, cross-section snapshots. They imaged dozens of balls of different sizes, searching for statistical patterns in their internal geometries.

Internal snapshot of a simulated crumpled plastic sheet. Tallinen et al./Nature
They found that a crumpled ball is most dense in its outer regions, and least dense in its core. Once inside its folds, there’s no way of knowing from their shape which direction is out and which is in (as, for example, one can determine from an onion, which has layers of skin arranged in curves parallel to its outer surface.) “If I was a creature that lived inside this ball, could I make my way out by looking at the way things are arranged? The answer is no,” said Menon. 

When he and Cambou studied arrangements of creases and folds, they found a distinctive pattern. Planes often lie flat against other planes. “It’s a fairly uniform object, though you’ve created it by a random, not-so-uniform process,” said Menon. “That’s the most surprising thing. There is no real geometrical reason why things should stack and layer in that way.” But if the researchers don’t know why this happens, they can speculate as to its effect: strength. 

Multiple layers of a thin sheet soon become walls. Per the lack-of-orientation observation, these walls are aligned in thousands of random directions. Press down and, from any angle, you’re pressing against down columns. “It can resist being crushed in all different directions,” said Menon. 

To explore why this happens, he and Cambou are now using transparent plastic sheets to make three-dimensional movies of crumpling. The implications extend far beyond Menon’s lab. “You’ve heard of crumple zones,” he said. “I’m just as interested in understanding leaves, or thin membranes of animal tissue, or the conformation of the Earth’s crust when it’s folded into mountains. I love it that these simple-looking problems are so nasty sometimes.” 



Images: 1) Reconstructed cross-section image of a foil ball approximately 4 inches in diameter. (Menon & Cambou/PNAS) 2) Turinboy/Flickr

See Also:


To Understand the Blueprint of Life, Crumple It
Physics of Pruney Fingers Revealed
Video: Secrets of Swimming in Sand Revealed
Video: The Different Shapes of Iciclology
Snowflakes Under an Electron Microscope
Video: Mysterious Patterns Reveal Self-Organizing Muscle Fibers

Citation: “Three-dimensional structure of a sheet crumpled into a ball.” By Anne Dominique Cambou and Narayanan Menon. Proceedings of the National Academy of Science, Vol. 108 No. 33, August 23, 2011.
]]></description>
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    <title>A First Course in Linear Algebra</title>
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    <title>The Status of the P Versus NP Problem</title>
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<item rdf:about="http://zimmer.csufresno.edu/~larryc/proofs/proofs.html">
    <title>How To Write Proofs</title>
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<item rdf:about="http://projecteuler.net/">
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    <title>SAGE</title>
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<item rdf:about="http://www.de.ufpe.br/~toom/">
    <title>Andre Toom</title>
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    <title>What Every Computer Scientist Should Know About Floating-Point Arithmetic</title>
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<item rdf:about="http://mathforum.org/dr.math/">
    <title>The Math Forum - Ask Dr. Math</title>
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<item rdf:about="http://www.polyomino.f2s.com/david/haskell/main.html">
    <title>Haskell for Maths</title>
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<item rdf:about="http://www-groups.mcs.st-andrews.ac.uk/~john/analysis/index.html">
    <title>Real Analysis Course</title>
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    <title>Converting a repeating decimal to a fraction</title>
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<item rdf:about="http://joshua.smcvt.edu/linearalgebra/">
    <title>Linear Algebra</title>
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    <link>http://joshua.smcvt.edu/linearalgebra/</link>
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<item rdf:about="http://math.arizona.edu/~kerl/doc/square-root.html">
    <title>How to manually find a square root</title>
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<item rdf:about="http://steve-yegge.blogspot.com/2006/03/math-for-programmers.html">
    <title>Stevey's Blog Rants: Math For Programmers</title>
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<item rdf:about="http://epsilondelta.net/">
    <title>Epsilon-Delta</title>
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    <title>This Week's Finds in Mathematical Physics</title>
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<item rdf:about="http://mathworld.wolfram.com/">
    <title>MathWorld</title>
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<item rdf:about="http://obsessivemathsfreak.org/wordpress/">
    <title>Obsessive Maths Freak</title>
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<item rdf:about="http://order.ph.utexas.edu/chaos/">
    <title>What is Chaos?</title>
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<item rdf:about="http://www.math.vanderbilt.edu/~schectex/commerrs/">
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