Alan Turing is famous for cracking the Enigma code during the second World War, but he was a polymath, and worked on many other problems. In 1952, Turing published a paper, The Chemical Basis of Morphogenesis , presenting a mechanism of pattern formation. He developed a theory of how the chemistry in the cell influences factors such as hair colour.
Turing’s model included two chemical processes: reaction, in which chemicals interact to produce different substances; and diffusion, in which local concentrations spread out over time.
Minimal Posters - Five Great Mathematicians And Their Contributions.
See his site for more.
The technique used by Brand to create these pieces is not one of conventional holography. He meticulously controls the unique shape of thousands of tiny optical pieces placed on a surface creating a 3D effect when the light source or viewer moves. This is essentially a mathematical problem in differential geometry and combinatorial optimization. Brand was the first person to correctly describe this technique in 2008 even though it dates back as early as the 1930s (check out his paper for details).
Last time, What’s the deal with 360?, we asked the question “Why are there 360 degrees in a full circle?”. The answer we got was that there isn’t really one - 360 is a bit arbitrary, kind of plucked out of nowhere. So is there a better way to measure angles?
This is where radians come into the story. First things first radians are simply another way to measure angles; we have meters or feet to measure distance, Celsius or Fahrenheit to measure temperature etc, and now degrees or radians to measure angles.
So how do we define a radian? Well here goes: one radian is defined to be the angle inside a sector whose arc length is precisely equal to the radius of the circle. Phew. Confused? What this means is that we take a circle and measure the radius with a piece of string. We then wrap this string part way around the outside of the circle - the angle we create is what is defined to be one radian! Watch the animation below to see this visually, the red line is our piece of string…
This animation also shows us something else; there are 2pi radians in a full circle. This is because the circumference of a circle is given by 2pi times the radius! This gives us the conversions rates: 2pi radians = 360 degrees, 180 = pi, 90 = pi/2 and even the other way round; 1 radian = 180/pi degrees, or approximately 57.296 degrees.
So that’s defined a radian but now what’s the big deal about it? Why is it any better than using degrees?
The largest prime number yet has been discovered — and it’s 17,425,170 digits long. The new prime number crushes the last one discovered in 2008, which was a paltry 12,978,189 digits long.
The number — 2 raised to the 57,885,161 power minus 1 — was discovered by University of Central Missouri mathematician Curtis Cooper as part of a giant network of volunteer computers devoted to finding primes, similar to projects like SETI@Home, which downloads and analyzes radio telescope data in the Search for Extraterrestrial Intelligence (SETI). The network, called the Great Internet Mersenne Prime Search (GIMPS) harnesses about 360,000 processors operating at 150 trillion calculations per second. This is the third prime number discovered by Cooper.
“It’s analogous to climbing Mt. Everest,” said George Woltman, the retired, Orlando, Fla.-based computer scientist who created GIMPS. “People enjoy it for the challenge of the discovery of finding something that’s never been known before.”
In addition, the number is the 48th example of a rare class of primes called Mersenne Primes. Mersenne primes take the form of 2 raised to the power of a prime number minus 1. Since they were first described by French monk Marin Mersenne 350 years ago, only 48 of these elusive numbers have been found, including the most recent discovery.
After the prime was discovered, it was double-checked by several other researchers using other computers.
While the intuitive way to search for primes would be to divide every potential candidate by ever single number smaller than itself, that would be extremely time-consuming, Woltman told LiveScience.
“If you were to do it that way it would take longer than the age of the universe,” he said.
Instead, mathematicians have devised a much cleverer strategy, that dramatically reduces the time to find primes. That method uses a formula to check much fewer numbers.
The new discovery makes Cooper elligible for a $3,000 GIMPS research discovery award.
Pages from Albert Einstein’s Notebook.
Although I’m not graphologist, I find individual handwriting very interesting.
Synonymous to an artist’s sketchbook, a scientist expresses their thoughts into their notebook which overflows with creativity and innovation.
The smooth motion of rotating circles can be used to build up any repeating curve even one as angular as a digital square wave. Each circle spins at a multiple of a fundamental frequency, and a method called Fourier analysis shows how to pick the radiuses of the circles to make the picture work. Decomposing signals like this lies at the heart of a lot of signal processing. [more] [code]
Physics of a Broken Swing Image
Surely by now, someone online has clearly shown this image to be fake. Instead, let’s use this as an example looking at the way people think about forces and motion.
- Forces and Circular Motion
Suppose something is moving around in a circle. Maybe it is a amusement park swing. Here is a force diagram for one of the riders:
There are only two forces on the swinger. There is of course the gravitational force pulling down. The only other force is the tension in the chain pulling on the swinger in the direction of the chain. Then why does the swinger move in a circular path? A component of the tension force pulls up to counter act the gravitational force. The other part of the tension from the chain pulls towards the center of the circle. It is this part of the tension force that makes the swinger move in a circle.
If you like, you can break all forces into two types. If a force is in the same (or opposite) direction as the motion (velocity) of an object, that force will cause the speed to either increase or decrease. If the force is perpendicular to the direction of the velocity, this force will cause the object to change directions. Of course you can have a force that both speeds up an object and causes it to turn.
Really, that is it. That is the only physics that you need to get this swinger to move around in a circle. Sure, there is a relationship between the angle the swing is at and the speed that the swinger moves, but for now we can leave that alone.
- What Would Really Happen?
If the chain suddenly breaks, what happens next? Well, the force diagram becomes a little bit simpler. It would just look like this:
This gravitational force would cause the velocity to change in the downward direction. So, clearly, it would fall down since before the chain broke it wasn’t moving in the vertical direction at all. But what else would it do? Here is a diagram of the swinger from the top view.
In this view (after the chain broke), you can’t see the only force pulling on the swinger – the gravitational force is pulling down. Since there aren’t any forces pushing the swinger to the left or right, from the top the swinger would just go in a straight line.
In the faked image, the falling swinger is clearly moving away from the swing in a path perpendicular to the way he was originally moving.
- Why Would a Faker Get This Wrong?
Many people seem to think that if an object is moving in a circle there is a force pushing outward from the center of the circle. This is the fabulous and mythical centrifugal force. Even though the centrifugal force is fake, it can still be useful in some ways. However, the point is that if you are in a stationary frame then there IS NO force pushing outward.
If you account for the mass of the three quarks inside of the proton — then you’ve only accounted for 10% of the proton’s mass.
Though the rest is “empty space” and has no “particles” in it — it’s actually a boiling, bubbling brew of “virtual particles”, that pop in and out of existence in time-scales so short they can’t be measured.
This image is from an animation based on a mathematical model of what that other 90% looks like. These fields of “potential existence” [in constant flux] generate the lion’s share of the mass we observe in the proton.
Meaning: Your potential is roughly 90% — compared to your actual, which is only 10%.
In case it ever comes up.
When you worry about math, your brain feels pain
Using fMRI, researchers have found that the brain areas that are activated by the anticipation of math problems in highly math-anxious people overalp with the same brain regions that register threat and pain. The fMRI scans showed that the anticipation of math caused activity in the brain that was similar to physical pain.
Surprisingly, researchers found that the associated brain activation doesn’t occur during the actual math problem, but rather during the anticipation of the problem.
This latest study shows that math anxiety might need to be addressed just like any other phobia, as there is an indication of a real, negative psychological reaction to the prospect of doing math.
Although this is compelling - careful consideration of confounding factors is always important, and further careful research needs to be carried out to address these.
The vast violet region in the image is W44, the remnant of a supernova. This gas and dust were once the outer layers of a massive star that reached the end of its life in a spectacular explosion about 20,000 years ago. The star’s core survives as an ultra-dense, rotating neutron star - a pulsar - visible as bright blue point in the top left of the purple cloud.
Minimal Posters - Six Women Who Changed Science. And The World.
Music and Mathematics
Music theorists sometimes use mathematics to understand music. Mathematics is “the basis of sound” and sound itself “in its musical aspects… exhibits a remarkable array of number properties”, simply because nature itself “is amazingly mathematical”. Though ancient Chinese, Egyptians and Mesopotamians are known to have studied the mathematical principles of sound, the Pythagoreans of ancient Greece are the first researchers known to have investigated the expression of musical scales in terms of numerical ratios, particularly the ratios of small integers. Their central doctrine was that “all nature consists of harmony arising out of numbers”.
From the time of Plato, harmony was considered a fundamental branch of physics, now known as musical acoustics. Early Indian and Chinese theorists show similar approaches: all sought to show that the mathematical laws of harmonics and rhythms were fundamental not only to our understanding of the world but to human well-being. Confucius, like Pythagoras, regarded the small numbers 1,2,3,4 as the source of all perfection.
The attempt to structure and communicate new ways of composing and hearing music has led to musical applications of set theory, abstract algebra and number theory. Some composers have incorporated the golden ratio and Fibonacci numbers into their work.
“Want to get a visual high today? The infographic masterpiece Random Walk [random-walk.com] ask the question “What does Randomness Look Like?” It attempts to give the answer(s) by showing the mysterious interactions of chaos and the order in randomness by simulating randomness in visualizations which are easy to understand.
The portfolio webpage contains a collection of zoomable illustrations, with detailed explanations plus summary captions in the yellow speech balloons on the right. Experimental visualized datasets include the constant number pi, the so-called Poisson distribution, the empirical results behind the normal distribution, the distribution of prime numbers, the first-digit law also called Benford’s Law, the surface area calculation Monte Carlo Method, the Law of Large Numbers, an atom’s or molecule’s Brownian Motion, an atom’s Half Life, the chaotic motion of a double pendulum, pseudo random number generation, and many more.
The project’s author, Daniel A. Becker, adds this project to an already impressive portfolio, including the previously posted Barcode Plantage and the for-the-infosthetics-addict still unknown Visual DNA(discover!).”