Physicists reported this week the discovery of a jewel-like geometric object that dramatically simplifies calculations of particle interactions and challenges the notion that space and time are fundamental components of reality.
“This is completely new and very much simpler than anything that has been done before,” said Andrew Hodges, a mathematical physicist at Oxford University who has been following the work.
The revelation that particle interactions, the most basic events in nature, may be consequences of geometry significantly advances a decades-long effort to reformulate quantum field theory, the body of laws describing elementary particles and their interactions. Interactions that were previously calculated with mathematical formulas thousands of terms long can now be described by computing the volume of the corresponding jewel-like “amplituhedron,” which yields an equivalent one-term expression.
“The degree of efficiency is mind-boggling,” said Jacob Bourjaily, a theoretical physicist at Harvard University and an author of the first of two papers detailing the new idea. “You can easily do, on paper, computations that were infeasible even with a computer before.”
The new geometric version of quantum field theory could also facilitate the search for a theory of quantum gravity that would seamlessly connect the large- and small-scale pictures of the universe. Attempts thus far to incorporate gravity into the laws of physics at the quantum scale have run up against nonsensical infinities and deep paradoxes. The amplituhedron, or a similar geometric object, could help by removing two deeply rooted principles of physics: locality and unitarity.
“Both are hard-wired in the usual way we think about things,” said Nima Arkani-Hamed, a professor of physics at the Institute for Advanced Study in Princeton, N.J., and the lead author of the two new papers, which were posted on the physics preprint site arXiv.org, one last December, and one last week. “Both are suspect.”
"Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without the gorgeous trappings of painting or music." —Betrand Russell
Hidden can track your missing device's location, take photos of the person using it and screenshots of what they're up to. All without them knowing you're watching.
This fantastic video by Vi Hart shows you what the math of music looks like in a visual representation — or, should that be
"what visual frieze patterns sound like when turned into music"?
Frieze patterns are symmetrical repeating patterns that show up in architecture, art, and even our model of DNA. According to Hart, this video is:
A visual and musical expression of mathematical symmetry groups. The transformations done to the video are equivalent to the transformations done to the notes.
In mathematics, two quantities are in the silver ratio if the ratio between the sum of the smaller plus twice the larger of those quantities and the larger one is the same as the ratio between the larger one and the smaller (see below). This defines the silver ratio as an irrationalmathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is denoted by δS.
Out of all of the infinite numbers in the world, there are precious few that are given their own letter from the all-too-finite Greek alphabet. The golden ratio, also known by the letter φ, or phi (usually pronounced “fie” in English), is one of those few. An irrational number that begins 1.618…, it describes an important kind of geometrical proportion—specifically, an elegant way to divide a line segment. Imagine we divide a segment (a) into a longer part (b) and a shorter part (c). If the ratio of a to b is the same as b to c, then that single ratio is golden. A rectangle whose sides are lengths a and b is called a golden rectangle, and it’s found in the geometry of a regular pentagon and the Platonic solids, five fundamental 3-D shapes, including the cube. The golden ratio is also tightly connected with the mathematically important Fibonacci sequence: The ratios of successive numbers in the Fibonacci sequence converge to the golden ratio.
So, like fractals, the golden ratio unites different areas of mathematics together. Interestingly, it is also found in many places in nature. For example, evidence of the golden ratio has been detected at the quantum level, where magnetic atoms linked together seem to vibrate at frequencies described by φ. On the macroscopic scale, the Fibonacci sequence and golden ratio describe the natural arrangements of seeds and leaves on many plants. If you examine the packing of seeds on the head of a sunflower, there are a series of clockwise and counterclockwise spirals, which generally appear in successive Fibonacci numbers. The plants are precise enough with their math that one especially robust sunflower reportedly had exactly 233 spirals of seeds one way and 144 the other. In addition, the golden ratio can be converted into an angular version (approximately 137.5 degrees), which specifies the rotational gap between successive sunflower seeds, and also between leaves growing from the stems of many other plants. In both these cases, it is thought that the golden angle allows for the most efficient packing scheme possible, as described by Mario Livio in his book The Golden Ratio. Phi is sometimes called “the most irrational number,” meaning that it is the hardest to approximate with a ratio of rational numbers. This means that by offsetting each leaf by the golden angle from the one below it, the leaves will not line up in stacks, with one on top of another and gaps in between—they instead come in the tightly packed spirals we see in nature.
Phi has also had an interesting role in aesthetics. For centuries, some people have argued that rectangles that have golden proportions are the most pleasing to the human eye. This has inspired artists like Salvador Dali and the architect Le Corbusier to use the golden ratio in their art. Dali’s Sacrament of the Last Supper, for instance, is painted on a large golden-rectangle canvas, and Jesus and his apostles are framed by a somewhat incongruous dodecahedron, one of the Platonic solids. Le Corbusier developed an entire system of measurement called Modulor based on the golden ratio and the human body, which he used in many of his architectural drawings.
For all of the appearances of the golden ratio, there many be even more erroneous sightings of it. The spiral of the nautilus’ shell is often said to fit precisely within a golden rectangle regardless of its size. But that is untrue. Each nautilus shell does maintain the same proportions throughout the animal’s life (that is, it’s a logarithmic spiral), but that proportion is generally not the golden ratio. Many have also claimed that the golden ratio is found in the proportions of various parts of the human body, the shape of the Gutenberg Bible, the Mona Lisa, and the Parthenon. None of these assertions have stood up to skeptical scrutiny, yet these myths stick with us. The mathematician Keith Devlin once gave a talk about the golden ratio, discussing numerous misunderstandings and debunking them, but when a radio station re-broadcast a portion of his lecture, it crucially omitted the fact that the examples were all false. Why does this myth persist? What makes it so resilient and appealing?
Since there are 360 degrees in a circle and Pi is intimately connected with the circle, some mathematicians were delighted to discover that the number 360 is at the 359th digit position of Pi.
In the Greek alphabet, π is the sixteenth letter. In the English alphabet p os also sixteenth letter.
Instructor, Roderic Watson lead a team of 20 high school students, from YES Prep., in the creation of Sacred Geometry designs. The designs will be incorporated into ceramic tile mosaic sidewalk inlays that will be a part of the new Greenspoint Bridge in Houston Texas. The project is sponsored by the Greenspoint Redevelopment Authority. This is another public art project by the Museum of Cultural Arts Houston. Visit MOCAH online at www.mocah.org or call 713.224.2787 for more information.
Vedic mathematics is a system of mathematics consisting of a list of 16 basic sutras, or aphorisms, that allegedly encompass all mathematics. They were presented by a Hindu scholar and mathematician, Jagadguru Swami Sri Bharati Krishna Tirthaji Maharaja, during the early part of the 20th century (Trivedi, 1965).
