The NSA’s encryption-busting quantum computer

The NSA is working on the development of a quantum computer that could foil all public encryption systems.  The description of this technology, after the jump, combines weird physics, weird mathematics, and weird surveillance. [Read more...]

The geometric form of the year

Artist’s rendering of the amplituhedron, a newly discovered mathematical object resembling a multifaceted jewel in higher dimensions. Encoded in its volume are the most basic features of reality that can be calculated — the probabilities of outcomes of particle interactions.

[Read more...]

Weird math fact

A comic strip  (which you can see here after the jump) raised a mathematical conundrum that I’ve been trying to get my mind around.  Maybe you can help.

(1) 9 x 1/9 = 1.   Right?  That’s what the fraction 1/9 means.

(2)  1/9 = .1111111. . .  If you turn 1/9 into its decimal form, by dividing 9 into 1, the result is .1111111. . . [meaning a repetition into infinity]

(3)  9 x (.1111111. . .) = 1    Substitute the decimal form for  1/9 in equation (1).

(4)  .9999999. . . = 1      Do the calculation in equation (3).   9 x .111111. . .  is .999999. . ., another repetition into infinity.  The number by itself is short of 1, infinitely short, but you could go into infinity and it would never be 1.  AND YET, in the equation, .99999 in an infinite regress EQUALS 1.

How can that be? [Read more...]

Degrees of separation

Forget Google Earth, Google Street View, and those proposed Google goggles.  Here is Google’s coolest feature:  Google will now calculate how many degrees of separation a person is from Kevin Bacon.

It is the party game beloved of cinephiles everywhere, one which rewards detailed knowledge of the career of one of the finest actors never to receive an Oscar nomination. And now it is even easier to play: Google has built Six Degrees of Kevin Bacon into its search system.

Devised in January 1994 by a trio of students at Pennsylvania’s Albright college, the original game was based on the idea that it is always possible to connect every movie actor in the world back to the Footloose star in no more than six associations. A website, board game and book later emerged and an initially reluctant Bacon eventually embraced the phenomenon by launching his own site, SixDegrees.org, to foster charitable donations.

To use Google’s system, the user simply types in the words “Bacon number” followed by the name of the actor. By way of example, typing “Bacon number Simon Pegg” reveals that Bacon and the British actor are linked by Tom Cruise, because the latter appeared in 1992′s A Few Good Men with Bacon and in 2006′s Mission: Impossible III with Pegg. Pegg therefore has a Bacon number of two, indicating two degrees of separation.

Lead engineer Yossi Matias said the project was about showcasing the power of Google’s search engine by flagging up the deep-rooted connections between people in the film industry. “If you think about search in the traditional sense, for years it has been to try and match, find pages and sources where you would find the text,” he told the Hollywood Reporter. “It’s interesting that this small-world phenomena when applied to the world of actors actually shows that in most cases, most actors aren’t that far apart from each other. And most of them have a relatively small Bacon number.”

By way of example, type French Oscar-winner Marion Cotillard’s name into the system and it is revealed that she has a Bacon number of two, while Humphrey Bogart, who died in 1957, nevertheless has a Bacon number of just three.

via Google builds Six Degrees of Kevin Bacon into its search system | Film | guardian.co.uk.

The Google algorithm (someone please explain how it works)  just works for Hollywood figures, which is the original game, though one variation tries to connect anyone on earth to Mr. Bacon.  I, for example, have a Bacon number of four:  (1) The wife of a former colleague is the daughter of the man who did the make-up on Citizen Kane.  (2)  He worked for Orson Welles, who produced, directed, and acted in Citizen Kane.  (3)  Google tells me that Orson Welles appeared with Jack Nicholson in A Safe Place.  (4)  And that Jack Nicholson and Kevin Bacon appeared in A Few Good Men.

Another variation proposes that there are no more than six degrees of separation (or maybe a few more) between any two people in the world.  For example, what chain of people who have met or have had a direct personal contact with each other might connect me to, say, a hypothetical Chinese peasant named Chen who lives in the Jiangxi province?  (1) When I was in high school, I shook the hand of Senator Eugene McCarthy.  (2)  He shook the hand of Richard Nixon.  (3)  Nixon shook the hand of Mao Zedong.  (4)  Chairman Mao knew the members of his Communist Politburo.  (5)  The representative of the Jiangxi province worked with the Secret Police.  (6) One of whose members doubtless spied on Chen and his parents.

I don’t know if that always works, of course.  I suppose it’s based on the exponential calculations that perhaps someone could explain for us.  (Is it that if each person has met a thousand people, six degrees would mean 1000 to the 6th power, or 1,000,000,000,000,000,000, a number that would require a great deal of overlap since the world’s population is only about 7,000,000,000.)

I think it shouldn’t count if your only contact with a person is reading a book or article that was written by that person.  But if you comment on the person’s blog, that does count, like talking to someone over the phone.  So you can add onto my degrees of separation with Kevin Bacon.

Do any of you have any other interesting degrees of separation that the rest of us could then appropriate?

There’s always room at the Hilbert Hotel

I stumbled upon this series on mind-blowing math facts from a couple of years ago.  It’s by Cornell mathematician Steven Strogatz and treats things like the weirdness of pi, the quirks of probability, Zeno’s paradox, and some of the fun things you can do with calculus.

(Homeschoolers and other educators, take note:  Recovering mathematics and its different applications is urgently needed today and is the missing piece of a truly classical education.  We are doing things with the language part, the trivium, but we now must bring back the mathematics part, the quadrivium, which is far more than just Saxon math.  What Strogatz does here is show that math is far more than memorizing tables and working out problems, showing that it is wonderful, mysterious, philosophical, and imaginative, something that students need to realize.)

Anyway, here he treats the mathematics of infinity, along with the paradox that some sets are more infinite than others:

Some of its [infinity's] strangest aspects first came to light in the late 1800s, with Georg Cantor’s groundbreaking work on “set theory.” Cantor was particularly interested in infinite sets of numbers and points, like the set {1, 2, 3, 4,…} of “natural numbers” and the set of points on a line. He defined a rigorous way to compare different infinite sets and discovered, shockingly, that some infinities are bigger than others.

At the time, Cantor’s theory provoked not just resistance, but outrage. Henri Poincaré, one of the leading mathematicians of the day, called it a “disease.” But another giant of the era, David Hilbert, saw it as a lasting contribution and later proclaimed, “No one shall expel us from the Paradise that Cantor has created.”

My goal here is to give you a glimpse of this paradise. But rather than working directly with sets of numbers or points, let me follow an approach introduced by Hilbert himself. He vividly conveyed the strangeness and wonder of Cantor’s theory by telling a parable about a grand hotel, now known as the Hilbert Hotel.

It’s always booked solid, yet there’s always a vacancy.

For the Hilbert Hotel doesn’t merely have hundreds of rooms — it has an infinite number of them. Whenever a new guest arrives, the manager shifts the occupant of room 1 to room 2, room 2 to room 3, and so on. That frees up room 1 for the newcomer, and accommodates everyone else as well (though inconveniencing them by the move).

Now suppose infinitely many new guests arrive, sweaty and short-tempered. No problem. The unflappable manager moves the occupant of room 1 to room 2, room 2 to room 4, room 3 to room 6, and so on. This doubling trick opens up all the odd-numbered rooms — infinitely many of them — for the new guests.

Later that night, an endless convoy of buses rumbles up to reception. There are infinitely many buses, and worse still, each one is loaded with an infinity of crabby people demanding that the hotel live up to its motto, “There’s always room at the Hilbert Hotel.”

The manager has faced this challenge before and takes it in stride.

First he does the doubling trick. That reassigns the current guests to the even-numbered rooms and clears out all the odd-numbered ones — a good start, because he now has an infinite number of rooms available.

But is that enough? Are there really enough odd-numbered rooms to accommodate the teeming horde of new guests? It seems unlikely, since there are something like “infinity squared” people clamoring for these rooms. (Why infinity squared? Because there were an infinite number of people on each of an infinite number of buses, and that amounts to infinity times infinity, whatever that means.)

This is where the logic of infinity gets very weird.

via The Hilbert Hotel – NYTimes.com.

Does it ever.   Including a set of guests that there is no room for after all.  Read the whole post and the whole series (which is reportedly coming out as a book).

The “God particle”

Michael Gerson gives the most lucid explanation I have found for what the Higgs boson–a.k.a. the “God particle”–is.  He also explores the implications of the strange fact that mathematics, which is a function of the human mind, can actually predict what things exist in the external world:

Modern physics can explain just about everything, except why anything has mass. The Standard Model of physics, which emerged four decades ago, employs an elegant mathematical formula to account for most of the elemental forces in the universe. It correctly predicted the discovery of various leptons and quarks in the laboratory.

But the equation doesn’t explain gravity. So the Standard Model requires the existence of some other force that seized the massless particles produced by the Big Bang and sucked them into physicality. The detection of Higgs bosons would confirm this theory — which is why scientists are smashing protons into one another in a 17-mile round particle accelerator and picking through the subatomic wreckage.

It will take a few more years for definitive results. But most scientists don’t seem to appreciate the glorious improbability — and philosophic implications — of the entire enterprise.

In 1928, theoretical physicist Paul Dirac combined the mathematical formulas for relativity and quantum mechanics into a single equation and predicted the existence of antimatter. Antimatter was duly discovered in 1932. But why should a mathematical equation — the product of brain chemistry — describe physical reality? It is not self-evident that there should be any correspondence between mathematical formulas and the laws of the universe. Modern physics does not consist of measured phenomena summarized in elegant equations; it consists of elegant equations that predict measured phenomena. This has been called “the unreasonable effectiveness of mathematics.” However unreasonable, it led to the construction of the Large Hadron Collider along the border of France and Switzerland, the largest machine ever built by human beings.

Dr. Ard Louis, a young physicist teaching at the University of Oxford, recalls his first encounter with Dirac’s equation. “How can mathematics demand something so fantastical from nature? I was sure it couldn’t be true and spent many hours trying to find a way out. When I finally gave up and saw that there was no way around Dirac’s result, it gave me goose bumps. I remember thinking that even if I never used my years of physics training again, it would have been worth it just to see something so spectacularly beautiful.”

Louis describes a cumulative case for wonder. Not only does the universe unexpectedly correspond to mathematical theories, it is self-organizing — from biology to astrophysics — in unlikely ways. The physical constants of the universe seem finely tuned for the emergence of complexity and life. Slightly modify the strength of gravity, or the chemistry of carbon, or the ratio of the mass of protons and electrons, and biological systems become impossible. The universe-ending Big Crunch comes too soon, or carbon isn’t produced, or suns explode.

The wild improbability of a universe that allows us to be aware of it seems to demand some explanation. This does not require theism. Some physicists favor the theory of the multiverse, in which every possible universe exists simultaneously. If everything happens, it is not surprising that anything happens. But this is not a theory that can be scientifically tested. Other universes, by definition, are not accessible. The multiverse is metaphysics — just as subject to the scientific method as the existence of heaven.

One reasonable alternative — the one advocated by Louis — is theism. It explains a universe finely tuned for life and accessible to human reason. It accounts for the cosmic coincidences. And a theistic universe, unlike the alternatives, also makes sense of free will and moral responsibility.

via The search for the God particle goes beyond mere physics – The Washington Post.

I love that:  “sucked into physicality.”  Also the “unreasonable effectiveness of mathematics.”  Also “Modern physics does not consist of measured phenomena summarized in elegant equations; it consists of elegant equations that predict measured phenomena.”

Intelligent design is not just predicated on one thing or another showing evidence of having been designed by a primal mind.  It seems to me to go much deeper than that.  Mathematics is mind, and that mathematics applies to nature is evidence of a mind behind nature.   Isn’t it?


CLOSE | X

HIDE | X