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« : Sierpień 23, 2010, 19:44:43 »


Music Has Its Own Geometry, Researchers Find

ScienceDaily (Apr. 18, 2008) — The connection between music and mathematics has fascinated scholars for centuries. More than 2000 years ago Pythagoras reportedly discovered that pleasing musical intervals could be described using simple ratios.
And the so-called musica universalis or "music of the spheres" emerged in the Middle Ages as the philosophical idea that the proportions in the movements of the celestial bodies -- the sun, moon and planets -- could be viewed as a form of music, inaudible but perfectly harmonious.

Now, three music professors -- Clifton Callender at Florida State University, Ian Quinn at Yale University and Dmitri Tymoczko at Princeton University -- have devised a new way of analyzing and categorizing music that takes advantage of the deep, complex mathematics they see enmeshed in its very fabric.

The figure shows how geometrical music theory represents four-note chord-types -- the collections of notes form a tetrahedron, with the colors indicating the spacing between the individual notes in a sequence. In the blue spheres, the notes are clustered, in the warmer colors, they are farther apart. The red ball at the top of the pyramid is the diminished seventh chord, a popular 19th-century chord. Near it are all the most familiar chords of Western music. (Credit: Dmitri Tymoczko, Princeton University)

Writing in the April 18 issue of Science, the trio has outlined a method called "geometrical music theory" that translates the language of musical theory into that of contemporary geometry. They take sequences of notes, like chords, rhythms and scales, and categorize them so they can be grouped into "families." They have found a way to assign mathematical structure to these families, so they can then be represented by points in complex geometrical spaces, much the way "x" and "y" coordinates, in the simpler system of high school algebra, correspond to points on a two-dimensional plane.

Different types of categorization produce different geometrical spaces, and reflect the different ways in which musicians over the centuries have understood music. This achievement, they expect, will allow researchers to analyze and understand music in much deeper and more satisfying ways.

The work represents a significant departure from other attempts to quantify music, according to Rachel Wells Hall of the Department of Mathematics and Computer Science at St. Joseph's University in Philadelphia. In an accompanying essay, she writes that their effort, "stands out both for the breadth of its musical implications and the depth of its mathematical content."

The method, according to its authors, allows them to analyze and compare many kinds of Western (and perhaps some non-Western) music. (The method focuses on Western-style music because concepts like "chord" are not universal in all styles.) It also incorporates many past schemes by music theorists to render music into mathematical form.

"The music of the spheres isn't really a metaphor -- some musical spaces really are spheres," said Tymoczko, an assistant professor of music at Princeton. "The whole point of making these geometric spaces is that, at the end of the day, it helps you understand music better. Having a powerful set of tools for conceptualizing music allows you to do all sorts of things you hadn't done before."

Like what?

"You could create new kinds of musical instruments or new kinds of toys," he said. "You could create new kinds of visualization tools -- imagine going to a classical music concert where the music was being translated visually. We could change the way we educate musicians. There are lots of practical consequences that could follow from these ideas."

"But to me," Tymoczko added, "the most satisfying aspect of this research is that we can now see that there is a logical structure linking many, many different musical concepts. To some extent, we can represent the history of music as a long process of exploring different symmetries and different geometries."

Understanding music, the authors write, is a process of discarding information. For instance, suppose a musician plays middle "C" on a piano, followed by the note "E" above that and the note "G" above that. Musicians have many different terms to describe this sequence of events, such as "an ascending C major arpeggio," "a C major chord," or "a major chord." The authors provide a unified mathematical framework for relating these different descriptions of the same musical event.

The trio describes five different ways of categorizing collections of notes that are similar, but not identical. They refer to these musical resemblances as the "OPTIC symmetries," with each letter of the word "OPTIC" representing a different way of ignoring musical information -- for instance, what octave the notes are in, their order, or how many times each note is repeated. The authors show that five symmetries can be combined with each other to produce a cornucopia of different musical concepts, some of which are familiar and some of which are novel.

In this way, the musicians are able to reduce musical works to their mathematical essence.

Once notes are translated into numbers and then translated again into the language of geometry the result is a rich menagerie of geometrical spaces, each inhabited by a different species of geometrical object. After all the mathematics is done, three-note chords end up on a triangular donut while chord types perch on the surface of a cone.


The broad effort follows upon earlier work by Tymoczko in which he developed geometric models for selected musical objects.

The method could help answer whether there are new scales and chords that exist but have yet to be discovered.

"Have Western composers already discovered the essential and most important musical objects?" Tymoczko asked. "If so, then Western music is more than just an arbitrary set of conventions. It may be that the basic objects of Western music are fantastically special, in which case it would be quite difficult to find alternatives to broadly traditional methods of musical organization."

The tools for analysis also offer the exciting possibility of investigating the differences between musical styles.

"Our methods are not so great at distinguishing Aerosmith from the Rolling Stones," Tymoczko said. "But they might allow you to visualize some of the differences between John Lennon and Paul McCartney. And they certainly help you understand more deeply how classical music relates to rock or is different from atonal music."

http://www.sciencedaily.com/releases/2008/04/080417142454.htm

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« Odpowiedz #1 : Sierpień 31, 2010, 23:23:04 »




Star and flower-shaped moulds tell stem cells what to be
* 07:00 02 March 2010 by Andy Coghlan

Growing cells in star and flower-shaped moulds may sound frivolous. But this difference is enough to dictate whether human cells become fat or bone.

It's the first time geometry alone has been shown to shape a cell's fate, and could lead to new ways of coaxing stem cells into becoming specific tissues for transplant into patients, using physical, not chemical, cues.

Milan Mrksich at the University of Chicago in Illinois and colleagues created moulds, 50 micrometres wide, rather like miniature cookie cutters, in shapes including stars, flowers, squares, pentagons and circle. Into each, they placed a single, human bone marrow cell. A type of stem cell, these immature cells are precursors to blood, bone and fat. The moulds were bathed in a broth of chemicals that encourage cells to become fat and bone.

Despite experiencing identical chemical conditions, 70 per cent of cells sitting in angular moulds, such as stars or thin rectangles, matured into bone. However, cells grown in more curvy moulds, such as circles or flower shapes, were more likely to become fat.
From flower to fat

Mrksich's team conclude that the deciding factor is the availability of corners, which provide points for cells to push against and sprout internal "stress filaments". These springy nanocables span cells and form their inner skeletons.

In star-shaped moulds, the sharp corners trigger the growth of long strong filaments, leading to hard bone cells, says Mrksich. By contrast, the rounded edges of flower-shaped moulds encourage the cells to grow short soft filaments, leading to fat cells.

Geometry almost certainly influences the fate of cells in their natural setting in the body. "Cells change their geometries and mechanics throughout development and as they move through the body," says bioengineer Christopher Chen of the University of Pennsylvania, Philadelphia. Cells may have evolved to use these changes as triggers for switching on genes that guide their development.

Physical forces such as pressure from moving fluidMovie Camera or air are also known to alter cells' fate.

Źródło: http://www.newscientist.com/article/dn18593-star-and-flowershaped-moulds-tell-stem-cells-what-to-be.html?DCMP=OTC-rss&nsref=online-news

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« Odpowiedz #2 : Marzec 06, 2015, 12:23:13 »


Strange Stars Pulsate According to the "Golden Ratio"
February 9, 2015 |By Clara Moskowitz
http://www.scientificamerican.com/article/strange-stars-pulsate-according-to-the-golden-ratio/


Astronomers have discovered variable stars that periodically dim and brighten at frequencies close to the famed golden mean


Scholars have seen the golden ratio in nautilus shells, the Parthenon, da Vinci paintings and now in stars. A new study of variable stars observed by the Kepler space telescope found four stars that pulsate at frequencies whose ratio is near the irrational number 0.61803398875, known as the Greek letter phi, or the golden ratio (which is also sometimes referred to as the inverse of that number, 1.61803398875…).
 
The golden ratio had not turned up in the celestial sphere before astronomer John Linder of The College of Wooster in Ohio and his colleagues analyzed the Kepler data. The researchers looked at a class of stars called RR Lyrae that are known for their variability. Unlike the sun, which shines at a near constant brightness (a good thing for life on Earth!), these stars brighten and dim as their atmospheres expand and contract due to periodic pressure changes. Each star pulses with a primary frequency and also shows smaller brightness fluctuations occurring on a secondary frequency. The ratios between these two frequencies “are very important,” says astronomer Róbert Szabó of the Konkoly Observatory in Hungary, who was not involved in the study, “because they are characterized by the inner structure of stars—and if a star exhibits many modes, then observation of the frequencies gives very strict constraints to stellar models.” For four of the six RR Lyrae stars the researchers analyzed, the ratio of the primary to secondary frequencies was near the golden mean—within 2 percent of its value in the case of the star KIC 5520878, for example.



The golden ratio has been a source of fascination to mathematicians, scientists and artists since the days of Pythagoras and Euclid in ancient Greece, although whether or not it is actually present in all of the places people have claimed to find it is debatable. “The golden ratio has a long history in disciplines ranging from the physics of crystals to visual arts,” says astrophysicist Mario Livio of the Space Telescope Science Institute in Baltimore, who wrote the 2002 book The Golden Ratio: The Story of Phi, The World's Most Astonishing Number. Two numbers have the golden mean if the ratio between them is the same as the ratio between their sum and the larger of the two numbers—in rectangular terms, long is to short as the whole is to the long. “The golden ratio is special in that it is in some sense the most irrational of all irrational numbers,” Livio says. An irrational number is one that cannot be expressed as a ratio of whole numbers. But some irrational numbers are easy to approximate using rational numbers whereas others are hard. The golden ratio is the irrational number that is hardest to approximate with rational numbers.
 
The connection between the golden ratio and these variable stars could be meaningful or it could just be a fluke. “Many claims about natural phenomena and the golden ratio are exaggerated,” says mathematician and computer scientist George Markowsky of the University of Maine, Orono. “I refuse to accept anything off by 2 percent or more as evidence of the golden ratio. After all, around any real number there are infinitely many other real numbers. People don't seem to write papers about the mystic properties of .6 (which is very close to .618....).” Astronomer Szabó, who leads the working group studying Kepler data on RR Lyrae stars, says he is not yet convinced that the golden ratio in this case is more than a coincidence, but that characterizing the stars’ oscillation frequencies is important. “This paper is a significant contribution to the topic,” he says.
 
Although the sample of stars in this study was very small, the researchers noticed an intriguing pattern among the four stars with pulsation frequencies close to the golden ratio. These stars all exhibited fractal behavior—never-ending patterns that repeat on continuously smaller scales—whereas the two non–golden ratio stars did not. “That suggests there might be a pattern,” Linder says. “What we need is more data.” An example of a fractal is a jagged coastline, which reveals more and more wiggles in its outline as you zoom in from any vantage point. “It’s the same with the frequencies in these stars,” Linder says. “As we lower the threshold we see more and more frequencies.”
 
The golden stars are actually the first examples outside of a laboratory of what’s called “strange nonchaotic dynamics.” The “strange” here refers to a fractal pattern, and nonchaotic means the pattern is orderly, rather than random. Most fractal patterns in nature, such as weather, are chaotic, so this aspect of the variable stars came as a surprise. “If you look in the literature, you see lots of examples of strange chaotic behavior,” Linder says. “I think our paper is going to bring this overlooked type of dynamics to the foreground.” If the same pattern is seen in more stars with golden ratio frequencies, it might help astronomers better understand and predict the detailed physics of stellar pulsations. “From a dynamics perspective,” Livio says, “it is quite intriguing to understand why systems would be attracted to this ratio.”

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