Particles in Love: Quantum Mechanics Explored in New Study

Here’s a love story at the smallest scales imaginable: particles of light. It is possible to have particles that are so intimately linked that a change to one affects the other, even when they are separated at a distance.

This idea, called “entanglement,” is part of the branch of physics called quantum mechanics, a description of the way the world works at the level of atoms and particles that are even smaller. Quantum mechanics says that at these very tiny scales, some properties of particles are based entirely on probability. In other words, nothing is certain until it happens.

Testing Bell’s Theorem

Albert Einstein did not entirely believe that the laws of quantum mechanics described reality. He and others postulated that there must be some hidden variables at work, which would allow quantum systems to be predictable. In 1964, however, John Bell published the idea that any model of physical reality with such hidden variables also must allow for the instantaneous influence of one particle on another. While Einstein proved that information cannot travel faster than the speed of light, particles can still affect each other when they are far apart according to Bell.

Scientists consider Bell’s theorem an important foundation for modern physics. While many experiments have taken place to try to prove his theorem, no one was able to run a full, proper test of the experiment Bell would have needed until recently. In 2015, three separate studies were published on this topic, all consistent with the predictions of quantum mechanics and entanglement.

“What’s exciting is that in some sense, we’re doing experimental philosophy,” said Krister Shalm, physicist with the National Institute of Standards and Technology (NIST), Boulder, Colorado. Shalm is lead author on one of the 2015 studies testing Bell’s theorem. “Humans have always had certain expectations of how the world works, and when quantum mechanics came along, it seemed to behave differently.”

Continue to the next page for an awesome cartoon that explains quantum entanglement and a full explanation of how it has been experimentally verified…

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Four New Elements Officially Added to Periodic Table

The International Union of Pure and Applied Chemistry (IUPAC), which verifies claims of newly discovered elements, has announced that four new super-heavy elements have cleared the verification process and are now officially recognized.

The recognition of these four new elements completes the 7th row of the Periodic Table of the Elements, a table which most people studied in their high school chemistry class. Researchers consider the discovery and verification of new elements to be an honor greater than winning an Olympic gold medal, given that it is such a rare occurrence.  In fact, researchers are already hard at work attempting to create elements in the lab that would begin an 8th row of the Periodic Table.

Teams from around the globe had created all four of these elements using particle accelerators to smash atoms together. None of them are stable elements, decaying into lighter particles almost instantly after their creation.

A press release from IUPAC provides the details of the announcement:

The fourth IUPAC/IUPAP Joint Working Party (JWP) on the priority of claims to the discovery of new elements has reviewed the relevant literature for elements 113, 115, 117, and 118 and has determined that the claims for discovery of these elements have been fulfilled, in accordance with the criteria for the discovery of elements of the IUPAP/IUPAC Transfermium Working Group (TWG) 1991 discovery criteria. These elements complete the 7th row of the periodic table of the elements, and the discoverers from Japan, Russia and the USA will now be invited to suggest permanent names and symbols. The new elements and assigned priorities of discovery are as follows:

Element 113 (temporary working name and symbol: ununtrium, Uut)
The RIKEN collaboration team in Japan have fulfilled the criteria for elementZ=113 and will be invited to propose a permanent name and symbol.

Elements 115, 117, and 118 (temporary working names and symbols: ununpentium, Uup; ununseptium, Uus; and ununoctium, Uuo)
The collaboration between the Joint Institute for Nuclear Research in Dubna, Russia; Lawrence Livermore National Laboratory, California, USA; and Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA have fulfilled the criteria for element Z=115, 117 and will be invited to propose permanent names and symbols.

The collaboration between the Joint Institute for Nuclear Research in Dubna, Russia and Lawrence Livermore National Laboratory, California, USA have fulfilled the criteria for element Z=118 and will be invited to propose a permanent name and symbol.

The priorities for four new chemical elements are being introduced simultaneously, after the careful verification of the discoveries and priorities. The decisions are detailed in two reports by the Joint Working Party (JWP), which includes experts drawn from IUPAC and IUPAP (the International Union of Pure and Applied Physics). These reports will be published in an early 2016 issue of the IUPAC journal Pure and Applied Chemistry (PAC).The JWP has reviewed the relevant literature pertaining to several claims of these new elements. The JWP has determined that the RIKEN collaboration have fulfilled the criteria for the discovery of element with atomic numbers Z=113. Several studies published from 2004 to 2012 have been construed as sufficient to ratify the discovery and priority.

In the same PAC report, the JWP also concluded that the collaborative work between scientists from the Joint Institute for Nuclear Research in Dubna, Russia; from Lawrence Livermore National Laboratory, California, USA; and from Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA (the Dubna-Livermore-Oak Ridge collaborations), starting in 2010, and subsequently confirmed in 2012 and 2013, have met the criteria for discovery of the elements with atomic numbersZ=115 and Z=117.

Finally, in a separate PAC article the Dubna–Livermore collaboration started in 2006 is reported as having satisfied the criteria for discovery of element Z=118.

“A particular difficulty in establishing these new elements is that they decay into hitherto unknown isotopes of slightly lighter elements that also need to be unequivocally identified” commented JWP chair Professor Paul J. Karol, “but in the future we hope to improve methods that can directly measure the atomic number, Z“.

“The chemistry community is eager to see its most cherished table finally being completed down to the seventh row. IUPAC has now initiated the process of formalizing names and symbols for these elements temporarily named as ununtrium, (Uut or element 113), ununpentium (Uup, element 115), ununseptium (Uus, element 117), and ununoctium  (Uuo, element 118)” said Professor Jan Reedijk, President of the Inorganic Chemistry Division of IUPAC.

The discoverers now have the opportunity to propose permanent names that conform to IUPAC’s guidelines, so sometime later in 2016 the new official Periodic Table will be published with those newly named elements. See the press release on the IUPAC website for additional details.

Source: IUPAC.org – “Discovery and Assignment of Elements with Atomic Numbers 113, 115, 117 and 118

Physicist Shows How He Creates Identical Twin Snowflakes [Video]

Conventional wisdom has always said that no two snowflakes are alike—until now.

Kenneth Libbrecht, a Caltech physics professor, has created nearly identical “designer snowflakes” out of ice crystals in his laboratory, where he studies the molecular dynamics of crystal growth, especially ice crystal growth.

Libbrecht also takes amazing photos of natural snowflakes and has produced videos of designer snowflakes growing in the lab.

In this video, he explains how he does it:

 

Republished as a derivative work from Futurity.org under the Attribution 4.0 International license. Original article published on Futurity by .

Featured Image Credit:  Kenneth Libbrecht/SnowCrystals.com

Large Hadron Collider sees tantalizing hints of a new particle that could revolutionise physics

Harry Cliff, University of Cambridge

At the start of December a rumour swirled around the internet and physics lab coffee rooms that researchers at the Large Hadron Collider had spotted a new particle. After a three-year drought that followed the discovery of the Higgs boson, could this be the first sign of new physics that particle physicists have all been desperately hoping for?

Researchers working on the LHC experiments remained tight-lipped until December 14 when physicists packed out CERN’s main auditorium to hear presentations from the scientists working on CMS and ATLAS experiments, the two gargantuan particle detectors that discovered the Higgs boson in 2012. Even watching the online webcast, the excitement was palpable.

Everybody was wondering if we would witness the beginning of a new age of discovery. The answer is … maybe.

Baffling bump

The CMS results were revealed first. At first the story was familiar, an impressive range of measurements that again and again showed no signs of new particles. But in the last few minutes of the presentation a subtle but intriguing bump on a graph was revealed that hinted at a new heavy particle decaying into two photons (particles of light). The bump appeared at a mass of around 760GeV (the unit of mass and energy used in particle physics – the Higgs boson has a mass of about 125 GeV) but was far too weak a signal to be conclusive on its own. The question was, would ATLAS see a similar bump in the same place?

The ATLAS presentation mirrored the one from CMS, another list of non-discoveries. But, saving the best for last, a bump was unveiled towards the end, close to where CMS saw theirs at 750GeV – but bigger. It was still too weak to reach the statistical threshold to be considered solid evidence, but the fact that both experiments saw evidence in the same place is exciting.

The discovery of the Higgs back in 2012 completed the Standard Model, our current best theory of particle physics, but left many unsolved mysteries. These include the nature of “dark matter”, an invisible substance that makes up around 85% of the matter in the universe, the weakness of gravity and the way that the laws of physics appear fine-tuned to allow life to exist, to name but a few.


Could supersymmetry one day crack the mystery of all the dark matter lurking in galaxy clusters?
NASA/wikimedia

A number of theories have been proposed to solve these problems. The most popular is an idea called supersymmetry, which proposes that there is a heavier super-partner for every particle in the Standard Model. This theory provides an explanation for the fine-tuning of the laws of physics and one of the super-partners could also account for dark matter.

Supersymmetry predicts the existence of new particles that should be in reach of the LHC. But despite high hopes the first run of the machine from 2009-2013 revealed a barren subatomic wilderness, populated only by a solitary Higgs boson. Many of the theoretical physicists working on supersymmetry have found the recent results from the LHC rather depressing. Some had begun to worry that answers to the outstanding questions in physics might lie forever beyond our reach.

This summer the 27km LHC restarted operation after a two-year upgrade that almost doubled its collision energy. Physicists are eagerly waiting to see what these collisions reveal, as higher energy makes it possible to create heavy particles that were out of reach during the first run. So this hint of a new particle is very welcome indeed.

A cousin of Higgs?

Andy Parker, head of Cambridge’s Cavendish Laboratory and senior member of the ATLAS experiment, told me: “If the bump is real, and it decays into two photons as seen, then it must be a boson, most likely another Higgs boson. Extra Higgs are predicted by many models, including supersymmetry”.

Perhaps even more exciting, it could be a type of graviton, a hypothesised particle associated with the force of gravity. Crucially, gravitons exist in theories with additional dimensions of space to the three (height, width and depth) we experience.

For now, physicists will remain sceptical – more data is needed to rule this intriguing hint in or out. Parker described the results as “preliminary and inconclusive” but added, “should it turn out to be the first sign of physics beyond the standard model, with hindsight, this will be seen as historic science.”

Whether this new particle turns out to be real or not, one thing that everyone agrees on is that 2016 is going to be an exciting year for particle physics.

The Conversation

Harry Cliff, Particle physicist and Science Museum fellow, University of Cambridge

This article was originally published on The Conversation. Read the original article.

Even Your Face Mites Are Too Big To Race These NanoCars [Video]

Face mites are microscopic and everyone has them, but they are still not small enough to drive the nano-cars that will be racing in the first-ever international NanoCar race. No one will be able to directly watch this race, although the tiny cars from five teams will be visible through sophisticated microscopes developed for the event.

Time trials will determine which nanocar is the fastest, though there may be head-to-head races with up to four cars on the track at once, according to organizers of the miniature grand prix. The event will take place next November in Toulouse, France.

A nanocar is a single-molecule vehicle of 100 or so atoms that incorporates a chassis, axles, and freely rotating wheels. Each of the entries will be propelled across a custom-built gold surface by an electric current supplied by the tip of a scanning electron microscope. The track will be cold at 5 kelvins (minus 450 degrees Fahrenheit) and in a vacuum.

nanocar_1170-770x440
This illustration shows a nanocar design. The first nanocars consisted of a chassis, two axles, and four wheels, all part of a single molecule. (Credit: Tour Group/Rice)

The Rice University entry will be a new model and the latest in a line that began when James Tour, professor of computer science and of materials science and nanoengineering, and his team built the world’s first nanocar more than 10 years ago.

“It’s challenging because, first of all, we have to design a car that can be manipulated on that specific surface,” Tour says. “Then we have to figure out the driving techniques that are appropriate for that car. But we’ll be ready.”

http://dai.ly/x3fnpxi

Graduate student Victor Garcia is building what Tour calls his group’s Model 1, which members of Professor Leonhard Grill’s group at the University of Graz in Austria will drive. The labs are collaborating to optimize the design.

The Center for Materials Elaboration and Structural Studies (CEMES) of the French National Center for Scientific Research (CNRS) is organizing the event.

Christian Joachim, a senior researcher at CNRS, and Gwénaël Rapenne, a professor at Paul Sabatier University, first proposed the race in a 2013 paper in the journal ACS Nano.

Joining Rice are teams from Ohio University; Dresden University of Technology; the National Institute for Materials Science, Tsukuba, Japan; and Paul Sabatier University.

 

Republished from Futurity.org as a derivative work  under the Creative Commons Attribution 4.0 International license. Original article posted to Futurity by .

Featured Photo Credit: Leap Kye/flickr, CC BY ND 2.0

LUX Detector Hasn’t Found Dark Matter Yet, Even Though it’s the Most Sensitive Instrument Ever Built

A recent press release from the Stanford Underground Research Facility (SURF) says that the Large Underground Xenon (LUX) detector is now 20 times more sensitive, which should help in detecting any dark matter that happens to pass through it. Scientists have increased the calibration of the detector by injecting various other substances that mimic the expected reaction that would happen if dark matter collided with the liquid Xenon in the detector.

According to a press release from SURF:

The Large Underground Xenon (LUX) dark matter experiment, which operates nearly a mile underground at the Sanford Underground Research Facility (SURF) in the Black Hills of South Dakota, has already proven itself to be the most sensitive detector in the hunt for dark matter, the unseen stuff believed to account for most of the matter in the universe. Now, a new set of calibration techniques employed by LUX scientists has again dramatically improved the detector’s sensitivity.

Researchers with LUX are looking for WIMPs, or weakly interacting massive particles, which are among the leading candidates for dark matter. “We have improved the sensitivity of LUX by more than a factor of 20 for low-mass dark matter particles, significantly enhancing our ability to look for WIMPs,” said Rick Gaitskell, professor of physics at Brown University and co-spokesperson for the LUX experiment. “It is vital that we continue to push the capabilities of our detector in the search for the elusive dark matter particles,” Gaitskell said.

LUX improvements, coupled to advanced computer simulations at the U.S. Department of Energy’s Lawrence Berkeley National Laboratory’s (Berkeley Lab) National Energy Research Scientific Computing Center (NERSC) and Brown University’s Center for Computation and Visualization (CCV), have allowed scientists to test additional particle models of dark matter that now can be excluded from the search. NERSC also stores large volumes of LUX data–measured in trillions of bytes, or terabytes–and Berkeley Lab has a growing role in the LUX collaboration.

Scientists are confident that dark matter exists because the effects of its gravity can be seen in the rotation of galaxies and in the way light bends as it travels through the universe. Because WIMPs are thought to interact with other matter only on very rare occasions, they have yet to be detected directly.

“We have looked for dark matter particles during the experiment’s first three-month run, but are exploiting new calibration techniques better pinning down how they would appear to our detector,” said Alastair Currie of Imperial College London, a LUX researcher. “These calibrations have deepened our understanding of the response of xenon to dark matter, and to backgrounds. This allows us to search, with improved confidence, for particles that we hadn’t previously known would be visible to LUX.”

The new research is described in a paper submitted to Physical Review Letters. The work reexamines data collected during LUX’s first three-month run in 2013 and helps to rule out the possibility of dark matter detections at low-mass ranges where other experiments had previously reported potential detections.

LUX consists of one-third ton of liquid xenon surrounded with sensitive light detectors. It is designed to identify the very rare occasions when a dark matter particle collides with a xenon atom inside the detector. When a collision happens, a xenon atom will recoil and emit a tiny flash of light, which is detected by LUX’s light sensors. The detector’s location at Sanford Lab beneath a mile of rock helps to shield it from cosmic rays and other radiation that would interfere with a dark matter signal.

So far LUX hasn’t detected a dark matter signal, but its exquisite sensitivity has allowed scientists to all but rule out vast mass ranges where dark matter particles might exist. These new calibrations increase that sensitivity even further.

One calibration technique used neutrons as stand-ins for dark matter particles. Bouncing neutrons off the xenon atoms allows scientists to quantify how the LUX detector responds to the recoiling process.

“It is like a giant game of pool with a neutron as the cue ball and the xenon atoms as the stripes and solids,” Gaitskell said. “We can track the neutron to deduce the details of the xenon recoil, and calibrate the response of LUX better than anything previously possible.”

The nature of the interaction between neutrons and xenon atoms is thought to be very similar to the interaction between dark matter and xenon. “It’s just that dark matter particles interact very much more weakly–about a million-million-million-million times more weakly,” Gaitskell said.

The neutron experiments help to calibrate the detector for interactions with the xenon nucleus. But LUX scientists have also calibrated the detector’s response to the deposition of small amounts of energy by struck atomic electrons. That’s done by injecting tritiated methane–a radioactive gas–into the detector.

“In a typical science run, most of what LUX sees are background electron recoil events,” said Carter Hall a University of Maryland professor. “Tritiated methane is a convenient source of similar events, and we’ve now studied hundreds of thousands of its decays in LUX. This gives us confidence that we won’t mistake these garden-variety events for dark matter.”

Another radioactive gas, krypton, was injected to help scientists distinguish between signals produced by ambient radioactivity and a potential dark matter signal.

“The krypton mixes uniformly in the liquid xenon and emits radiation with a known, specific energy, but then quickly decays away to a stable, non-radioactive form,” said Dan McKinsey, a UC Berkeley physics professor and co-spokesperson for LUX who is also an affiliate with Berkeley Lab. By precisely measuring the light and charge produced by this interaction, researchers can effectively filter out background events from their search.

“And so the search continues,” McKinsey said. “LUX is once again in dark matter detection mode at Sanford Lab. The latest run began in late 2014 and is expected to continue until June 2016. This run will represent an increase in exposure of more than four times compared to our previous 2013 run. We will be very excited to see if any dark matter particles have shown themselves in the new data.”

McKinsey, formerly at Yale University, joined UC Berkeley and Berkeley Lab in July, accompanied by members of his research team.

The Sanford Lab is a South Dakota-owned facility. Homestake Mining Co. donated its gold mine in Lead to the South Dakota Science and Technology Authority (SDSTA), which reopened the facility in 2007 with $40 million in funding from the South Dakota State Legislature and a $70 million donation from philanthropist T. Denny Sanford. The U.S. Department of Energy (DOE) supports Sanford Lab’s operations.

Kevin Lesko, who oversees SURF operations and leads the Dark Matter Research Group at Berkeley Lab, said, “It’s good to see that the experiments installed in SURF continue to produce world-leading results.”

The LUX scientific collaboration, which is supported by the DOE and National Science Foundation (NSF), includes 19 research universities and national laboratories in the United States, the United Kingdom and Portugal.

“The global search for dark matter aims to answer one of the biggest questions about the makeup of our universe. We’re proud to support the LUX collaboration and congratulate them on achieving an even greater level of sensitivity,” said Mike Headley, Executive Director of the SDSTA.

In late 2016, LUX will be decommissioned to make way for an even more sensitive detector that uses 10 tons of liquid Xeon. One might wonder what it will mean for the theory of dark matter if neither experiment is able to detect a dark matter interaction.

 

Source: Eurekalert.com – “New results from world’s most sensitive dark matter detector” 

Featured Photo Credit: Photo by Matthew Kapust/Sanford Underground Research Facility

This Incredible Device is Testing Space-Time to See if it Might be Pixelated

The theory is that if these researchers discover that the resolution of space-time is not continuous, but is actually pixelated, then we might be living in a two-dimensional hologram. As strange as that sounds, scientists at Fermilab have actually built a device that can measure the resolution of space-time and tell us if we are living in The Matrix.

According to a fantastic article on GeekWire, so far, it hasn’t yet found evidence that we aren’t living in 3D:

The experiments are being conducted at Fermilab in Illinois, using a gnarly-looking device known as the Holometer. The apparatus is designed to measure the smoothness of spacetime at lengths down to a billionth of a billionth of a meter. Put another way, that’s a thousand times smaller than the size of a proton.

The standard view is that the fabric of reality is continuous – but some theories propose that spacetime is pixelated, like a digital image. If that’s the case, there’s a built-in limit to the “resolution” of reality.

The Holometer uses a pair of high-power laser interferometers to look for tiny discontinuities in movements that last only a millionth of a second. Such discontinuities would provide evidence of holographic noise, or quantum jitters, in spacetime.

This week, a research team reported finding no discontinuities. But Craig Hogan, a professor at the University of Chicago who heads Fermilab’s Center for Particle Astrophysics, said that doesn’t yet rule out the holographic hypothesis.

“This is just the beginning of the story,” he said in a Fermilab report on the experiment. “We’ve developed a new way of studying space and time that we didn’t have before. We weren’t even sure we could attain the sensitivity we did.”

In fact, the level of sensitivity of this instrument is really quite amazing, in that it can filter out random vibrations like large trucks driving by and other jittery “space-time noise.” See the amazing article on the GeekWire website for more incredible details.

 

Source: GeekWire.com – “Holometer finds no evidence we’re living in a Matrix-like hologram … so far

Featured Photo Credit: Reidar Hahn / Fermilab

 

As World Leaders Debate Climate Policy, This Company is Close to a Fusion Solution [Video]

The energy source that many believe will be the “Holy Grail” of energy sources is fusion. In theory, if we we can create a controlled fusion reaction here on Earth, then we will have an energy source that is as perpetual and abundant as the sun.

And although fusion research has had its difficulties over time, it turns out that a UK company might not be so far away from achieving that Holy Grail reaction, according to a very interesting video published by Reuters:

A ‘Cool’ New Application for a Laser

Normally, when people think of the intense light of lasers, what comes to mind is heat and destruction. Now, for the first time, scientists have figured out how to use an infrared laser to cool water by about 36 degrees Fahrenheit.

“Typically, when you go to the movies and see Star Wars laser blasters, they heat things up. This is the first example of a laser beam that will refrigerate liquids like water under everyday conditions,” says Peter Pauzauskie, an assistant professor of materials science and engineering at the University of Washington. “It was really an open question as to whether this could be done because normally water warms when illuminated.”

As they are cooled by the laser, nanocrystals emit a reddish-green “glow” that can be seen by the naked eye. (Credit: Dennis Wise/ University of Washington)
As they are cooled by the laser, nanocrystals emit a reddish-green “glow” that can be seen by the naked eye. (Credit: Dennis Wise/ University of Washington)

To achieve the breakthrough, the team used a material commonly found in commercial lasers but essentially ran the laser phenomenon in reverse. They illuminated a single microscopic crystal suspended in water with infrared laser light to excite a unique kind of glow that has slightly more energy than that amount of light absorbed.

This higher-energy glow carries heat away from both the crystal and the water surrounding it.

The laser refrigeration process was first demonstrated in vacuum conditions at Los Alamos National Laboratory in 1995, but it has taken nearly 20 years to demonstrate this process in liquids.

Continue on to see exactly how they did it…

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Physics Professor Dreams Up a Homework Problem and then Discovers This

Carl Hagen, a particle physicist at the University of Rochester, decided to have his students apply a computational technique for approximating the energy states of quantum systems (such as molecules) to the hydrogen atom.  As he worked out the solution to the problem set himself, he noticed an interesting trend, and it turns out that he accidentally discovered a link between quantum physics and pure mathematics for pi.

John Wallis, an English mathematician, derived a pure mathematical formula for pi in his book “Arithmetica Infinitorum” published in 1655 (see featured image for this article, above). Until now, this derivation of pi has lived only in the world of mathematics, but, as a very interesting press release from EurekAlert! reports, almost 400 years later, the same formula he derived was discovered by Hagan in a happy accident:

“We weren’t looking for the Wallis formula for pi. It just fell into our laps,” said Carl Hagen, a particle physicist at the University of Rochester. Having noticed an intriguing trend in the solutions to a problem set he had developed for students in a class on quantum mechanics, Hagen recruited mathematician Tamar Friedmann and they realized this trend was in fact a manifestation of the Wallis formula for pi.

“It was a complete surprise – I jumped up and down when we got the Wallis formula out of equations for the hydrogen atom,” said Friedmann. “The special thing is that it brings out a beautiful connection between physics and math. I find it fascinating that a purely mathematical formula from the 17th century characterizes a physical system that was discovered 300 years later.”

“At the lower energy orbits, the path of the electron is fuzzy and spread out,” Hagen explained. “At more excited states, the orbits become more sharply defined and the uncertainty in the radius decreases.”

From the formula for the limit of the variational solution as the energy increased, Hagen and Friedmann were able to pull out the Wallis formula for pi.

Interestingly, the Wallis formula had been around for hundreds of years and the theory of quantum mechanics dates from the early 20th century, but the pi connection had remained hidden for all these years.  For additional details, check out the informative press release on Eureka Alert, and for the really adventurous, you can access the technical paper that was published in the Journal of Mathematical Physics on the AIP Scitation website.

 

Source: EurekaAlert.org – “New derivation of pi links quantum physics and pure math” 

Featured Image Credit: Digitized by Google

The featured image is two pages from the book “Arithmetica Infinitorum,” by John Wallis. In the table on the left page, the square that appears repeatedly denotes 4/pi, or the ratio of the area of a square to the area of the circumscribed circle. Wallis used the table to obtain the inequalities shown at the top of the page on the right that led to his formula.