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(Math) Dimensions: Shadows of the Fourth Dimension

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PostPosted: Sat Apr 08, 2006 12:50 pm    Post subject: (Math) Dimensions: Shadows of the Fourth Dimension Reply with quote

Science News Online
Week of April 8, 2006; Vol. 169, No. 14

Shadows of the Fourth Dimension
Ivars Peterson

Tony Robbin had started out as a painter. He created complex works filled with interwoven patterns and ambiguous figures to give the illusion of seeing more than one object in the same place at the same time.

"I was interested in ways of experiencing and depicting space—complex spaces, multiple spaces, paradoxical spaces," Robbin recalls. His fascination with finding ways to look beyond our three-dimensional universe inevitably led him into the domain of four-dimensional geometry.

In a fashion that was typical for him, Robbin set out to learn everything he could about the fourth dimension, even hiring a tutor to introduce him to space-time and Einstein's general theory of relativity. But it was a visit in 1979 to Brown University that persuaded him that mathematics could serve as his gateway to higher-dimensional spaces. During this visit, he saw Tom Banchoff's pioneering, computer-generated images of hypercubes—four-dimensional analogs of cubes.

"After years of hoping to visualize the four-dimensional cube, there it was," Robbin wrote in his 1992 book Fourfield. "By moving a joystick, I could turn the hypercube in my hand. The computer recalculated the position of the object thirty times a second, so the joystick was exquisitely sensitive to the touch, and the illusion of actually handling this beloved and mysterious object was very strong indeed."

"For three nights, I woke frequently from dreams of the images that I had seen on Banchoff's computer: the green screen, the quivering geometric figures," Robbin continued. "It seemed as if these images were imprinted on my mind. I had seen the fourth dimension directly."

"I realized that real mathematics was more liberating and richer, more complicated and more exhilarating, than my ignorant artist's fantasies about it," he noted. "I was dissatisfied with using artists' tricks to depict spaces. I decided to take the plunge."

Robbin's immersion in mathematics led to the creation of a series of works in which welded steel frames protrude from painted canvas to represent sections of hypercubes. Because the painted lines remain fixed and the relative positions of the rods change as the viewer walks past, such a work recreates in a novel fashion the experience of seeing the multiple faces of a three-dimensional shadow cast by a rotating hypercube.

Robbin also developed a strong appreciation of what a computer can do to help visualize complicated forms. "The lens allowed us to make recorded images of the things we see," he said. "The computer allows us to see things that we know are there but we can't see for ourselves."

Robbin went back to school to learn enough mathematics and computer programming to write his own software for creating images of four-dimensional geometric structures.

In later artworks, Robbin explored the dimension-bursting and space-twisting interactions of sculpture, painting, and light more fully. In some pieces, he took advantage of the shadows cast by steel-rod frameworks attached to canvas to complete four-dimensional geometric figures.

In other cases, Robbin shone red and blue light from different angles through wire frameworks, projecting colored shadows on a blank, white wall. He provided viewers with special glasses that fused the colored shadows to make figures that complete the hypercubic forms depicted in his artwork.

In his new book, Shadows of Reality, Robbin writes, "Mathematics can define and conquer the extra space and make four-dimensional geometry into a sensible world, perhaps even as sensible as the three-dimensional world."

Robbin's book is a provocative, illuminating adventure in a realm of many surprises, offering novel insights into both math and art history. He champions the powerful role that projective geometry, in particular, has played in the development of modern art, science, and math.


Check out Ivars Peterson's MathTrek blog at


Henderson, L. 1983. The Fourth Dimension and Non-Euclidean Geometry in Modern Art. Princeton, N.J.: Princeton University Press.

Peterson, I. 2001. Fragments of Infinity: A Kaleidoscope of Math and Art. New York: Wiley.

Robbin, T. 2006. Shadows of Reality: The Fourth Dimension in Relativity, Cubism, and Modern Thought. New Haven, Conn.: Yale University Press. See

______. 1992. Fourfield: Computers, Art, and the Fourth Dimension. Boston: Little, Brown and Company.

Tony Robbin has a Web site at Examples of his artworks are on display in Washington, D.C., at the gallery of the American Association for the Advancement of Science from April 4 to June 16, 2006. See

A collection of Ivars Peterson's early MathTrek articles, updated and illustrated, is now available as the Mathematical Association of America (MAA) book Mathematical Treks: From Surreal Numbers to Magic Circles. See
From Science News, Vol. 169, No. 14, April 8, 2006
Copyright (c) 2006 Science Service. All rights reserved.


Questions to explore further this topic:

What are dimensions?

Motion in one-dimension

Motion in two- and three-dimension

What are space-time diagrams?

What is special relativity?

What is theoretical physics?


Particles and relativity


What are the fourth and higher dimensions?

How can one view the fourth dimension in three dimension?

Geometry in higher dimensions

Visualizing multidimensional geometry

What is string theory?

The mathematics behind string theory

The story of string theory

History of string theory


How many string theories are there?


Are these string theories related to each other?


Is there a more fundamental theory?


The making of the elegant universe

Interviews on string theory

Experiments related to string theory



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PostPosted: Thu May 25, 2006 3:41 pm    Post subject: Scientists predict how to detect a fourth dimension of space Reply with quote

Duke University
25 May 2006

Scientists predict how to detect a fourth dimension of space

DURHAM, N.C. -- Scientists at Duke and Rutgers universities have developed a mathematical framework they say will enable astronomers to test a new five-dimensional theory of gravity that competes with Einstein's General Theory of Relativity.
Charles R. Keeton of Rutgers and Arlie O. Petters of Duke base their work on a recent theory called the type II Randall-Sundrum braneworld gravity model. The theory holds that the visible universe is a membrane (hence "braneworld") embedded within a larger universe, much like a strand of filmy seaweed floating in the ocean. The "braneworld universe" has five dimensions -- four spatial dimensions plus time -- compared with the four dimensions -- three spatial, plus time -- laid out in the General Theory of Relativity.

The framework Keeton and Petters developed predicts certain cosmological effects that, if observed, should help scientists validate the braneworld theory. The observations, they said, should be possible with satellites scheduled to launch in the next few years.

If the braneworld theory proves to be true, "this would upset the applecart," Petters said. "It would confirm that there is a fourth dimension to space, which would create a philosophical shift in our understanding of the natural world."

The scientists' findings appeared May 24, 2006, in the online edition of the journal Physical Review D. Keeton is an astronomy and physics professor at Rutgers, and Petters is a mathematics and physics professor at Duke. Their research is funded by the National Science Foundation.

The Randall-Sundrum braneworld model -- named for its originators, physicists Lisa Randall of Harvard University and Raman Sundrum of Johns Hopkins University -- provides a mathematical description of how gravity shapes the universe that differs from the description offered by the General Theory of Relativity.

Keeton and Petters focused on one particular gravitational consequence of the braneworld theory that distinguishes it from Einstein's theory.

The braneworld theory predicts that relatively small "black holes" created in the early universe have survived to the present. The black holes, with mass similar to a tiny asteroid, would be part of the "dark matter" in the universe. As the name suggests, dark matter does not emit or reflect light, but does exert a gravitational force.

The General Theory of Relativity, on the other hand, predicts that such primordial black holes no longer exist, as they would have evaporated by now.

"When we estimated how far braneworld black holes might be from Earth, we were surprised to find that the nearest ones would lie well inside Pluto's orbit," Keeton said.

Petters added, "If braneworld black holes form even 1 percent of the dark matter in our part of the galaxy -- a cautious assumption -- there should be several thousand braneworld black holes in our solar system."

But do braneworld black holes really exist -- and therefore stand as evidence for the 5-D braneworld theory?

The scientists showed that it should be possible to answer this question by observing the effects that braneworld black holes would exert on electromagnetic radiation traveling to Earth from other galaxies. Any such radiation passing near a black hole will be acted upon by the object's tremendous gravitational forces -- an effect called "gravitational lensing."

"A good place to look for gravitational lensing by braneworld black holes is in bursts of gamma rays coming to Earth," Keeton said. These gamma-ray bursts are thought to be produced by enormous explosions throughout the universe. Such bursts from outer space were discovered inadvertently by the U.S. Air Force in the 1960s.

Keeton and Petters calculated that braneworld black holes would impede the gamma rays in the same way a rock in a pond obstructs passing ripples. The rock produces an "interference pattern" in its wake in which some ripple peaks are higher, some troughs are deeper, and some peaks and troughs cancel each other out. The interference pattern bears the signature of the characteristics of both the rock and the water.

Similarly, a braneworld black hole would produce an interference pattern in a passing burst of gamma rays as they travel to Earth, said Keeton and Petters. The scientists predicted the resulting bright and dark "fringes" in the interference pattern, which they said provides a means of inferring characteristics of braneworld black holes and, in turn, of space and time.

"We discovered that the signature of a fourth dimension of space appears in the interference patterns," Petters said. "This extra spatial dimension creates a contraction between the fringes compared to what you'd get in General Relativity."

Petters and Keeton said it should be possible to measure the predicted gamma-ray fringe patterns using the Gamma-ray Large Area Space Telescope, which is scheduled to be launched on a spacecraft in August 2007. The telescope is a joint effort between NASA, the U.S. Department of Energy, and institutions in France, Germany, Japan, Italy and Sweden.

The scientists said their prediction would apply to all braneworld black holes, whether in our solar system or beyond.

"If the braneworld theory is correct," they said, "there should be many, many more braneworld black holes throughout the universe, each carrying the signature of a fourth dimension of space."
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PostPosted: Sat Oct 14, 2006 7:16 am    Post subject: Messiness Rules: In high dimensions, disorder packs tightest Reply with quote

Week of Oct. 14, 2006; Vol. 170, No. 16 , p. 244

Messiness Rules: In high dimensions, disorder packs tightest
Julie Rehmeyer

Should you find yourself with a 60-dimensional suitcase, the best way to pack it may be the easiest: Throw in everything in a jumble. That's the way to fit the most high-dimensional spheres into a fixed space, new research suggests.

The finding may be useful even to people without extra-dimensional luggage. It may improve the design of mathematical procedures called error-correcting codes used in computers to interpret noisy data.

Some 400 years ago, Johannes Kepler speculated that the best scheme for packing three-dimensional spheres is the way that grocers have always done it. Their orderly, pyramidal packing scheme piles the most oranges into the least space. Yet it took mathematicians until 1998 to prove Kepler right (SN: 8/15/98, p. 103:

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PostPosted: Sat Feb 03, 2007 8:56 am    Post subject: Physicists find way to 'see' extra dimensions Reply with quote

University of Wisconsin-Madison
2 February 2007

Physicists find way to 'see' extra dimensions

MADISON - Peering backward in time to an instant after the big bang, physicists at the University of Wisconsin-Madison have devised an approach that may help unlock the hidden shapes of alternate dimensions of the universe.

A new study demonstrates that the shapes of extra dimensions can be "seen" by deciphering their influence on cosmic energy released by the violent birth of the universe 13 billion years ago. The method, published today (Feb. 2) in Physical Review Letters, provides evidence that physicists can use experimental data to discern the nature of these elusive dimensions - the existence of which is a critical but as yet unproven element of string theory, the leading contender for a unified "theory of everything."

Scientists developed string theory, which proposes that everything in the universe is made of tiny, vibrating strings of energy, to encompass the physical principles of all objects from immense galaxies to subatomic particles. Though currently the front-runner to explain the framework of the cosmos, the theory remains, to date, untested.

The mathematics of string theory suggests that the world we know is not complete. In addition to our four familiar dimensions - three-dimensional space and time - string theory predicts the existence of six extra spatial dimensions, "hidden" dimensions curled in tiny geometric shapes at every single point in our universe.

Don't worry if you can't picture a 10-dimensional world. Our minds are accustomed to only three spatial dimensions and lack a frame of reference for the other six, says UW-Madison physicist Gary Shiu, who led the new study. Though scientists use computers to visualize what these six-dimensional geometries could look like (see image), no one really knows for sure what shape they take.

The new Wisconsin work may provide a long-sought foundation for measuring this previously immeasurable aspect of string theory.

According to string theory mathematics, the extra dimensions could adopt any of tens of thousands of possible shapes, each shape theoretically corresponding to its own universe with its own set of physical laws.

For our universe, "Nature picked one - and we want to know what that one looks like," explains Henry Tye, a physicist at Cornell University who was not involved in the new research.

Shiu says the many-dimensional shapes are far too small to see or measure through any usual means of observation, which makes testing this crucial aspect of string theory very difficult. "You can theorize anything, but you have to be able to show it with experiments," he says. "Now the problem is, how do we test it?"

He and graduate student Bret Underwood turned to the sky for inspiration.

Their approach is based on the idea that the six tiny dimensions had their strongest influence on the universe when it itself was a tiny speck of highly compressed matter and energy - that is, in the instant just after the big bang.

"Our idea was to go back in time and see what happened back then," says Shiu. "Of course, we couldn't really go back in time."

Lacking the requisite time machine, they used the next-best thing: a map of cosmic energy released from the big bang. The energy, captured by satellites such as NASA's Wilkinson Microwave Anisotropy Probe (WMAP), has persisted virtually unchanged for the last 13 billion years, making the energy map basically "a snapshot of the baby universe," Shiu says. The WMAP experiment is the successor to NASA's Cosmic Background Explorer (COBE) project, which garnered the 2006 Nobel Prize in physics.

Just as a shadow can give an idea of the shape of an object, the pattern of cosmic energy in the sky can give an indication of the shape of the other six dimensions present, Shiu explains.

To learn how to read telltale signs of the six-dimensional geometry from the cosmic map, they worked backward. Starting with two different types of mathematically simple geometries, called warped throats, they calculated the predicted energy map that would be seen in the universe described by each shape. When they compared the two maps, they found small but significant differences between them.

Their results show that specific patterns of cosmic energy can hold clues to the geometry of the six-dimensional shape - the first type of observable data to demonstrate such promise, says Tye.

Though the current data are not precise enough to compare their findings to our universe, upcoming experiments such as the European Space Agency's Planck satellite should have the sensitivity to detect subtle variations between different geometries, Shiu says.

"Our results with simple, well-understood shapes give proof of concept that the geometry of hidden dimensions can be deciphered from the pattern of cosmic energy," he says. "This provides a rare opportunity in which string theory can be tested."

Technological improvements to capture more detailed cosmic maps should help narrow down the possibilities and may allow scientists to crack the code of the cosmic energy map - and inch closer to identifying the single geometry that fits our universe.

The implications of such a possibility are profound, says Tye. "If this shape can be measured, it would also tell us that string theory is correct."

The new work was funded by grants from the National Science Foundation, the U.S. Department of Energy, and the Research Corp.
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PostPosted: Thu May 03, 2007 10:50 am    Post subject: Princeton physicists connect string theory with established Reply with quote

Princeton physicists connect string theory with established physics
1 May 2007
Princeton University

String theory, simultaneously one of the most promising and controversial ideas in modern physics, may be more capable of helping probe the inner workings of subatomic particles than was previously thought, according to a team of Princeton University scientists.

The theory has been highly praised by some physicists for its potential to forge the long-sought link between gravity and the forces that dominate within the atomic nucleus. But the theory -- which posits that all subatomic particles are actually tiny "strings" that vibrate in different ways -- has also drawn criticism for being untestable in the laboratory, and perhaps impossible to connect with real-world phenomena.

However, the Princeton researchers have found new mathematical evidence that some of string theory's predictions mesh closely with those of a well-respected body of physics called "gauge theory," which has been demonstrated to underlie the interactions among quarks and gluons, the vanishingly small objects that combine to form protons, neutrons and other, more exotic subatomic particles. The discovery, say the physicists, could open up a host of uses for string theory in attacking practical physics problems.

"These problems include describing the interactions among the quarks within everyday atomic nuclei," said Igor Klebanov, the Thomas D. Jones Professor of Mathematical Physics at Princeton and an author of a recent paper on the subject. "We have previously been able to study these interactions in detail only at the high-energy conditions within particle accelerators, but with these findings we may be able to describe what's happening inside the atoms that make up rocks and trees. We cannot do so yet, but it appears that the math of string theory could be what we need to bridge this gap."

The team's paper appears in the March 30 issue of the scientific journal Physical Review Letters. Klebanov's co-authors include graduate student Marcus Benna and postdoctoral fellows Sergio Benvenuti and Antonello Scardicchio.

For Klebanov, the findings represent a major success in the decades-old search for connections between strings and gauge theory, the latter of which -- to a particle physicist -- lays out the established laws that describe all ordinary matter.

The many facets of gauge theory add up to a well-established and coherent picture of the behaviors of quarks and gluons, which largely compose most familiar forms of matter. Decades of observations at particle accelerators have shown that gauge theory accounts for quark and gluon behavior quite well, at least at the very high energy levels that exist when two isolated particles are smashed together at nearly the speed of light.

At these high energies, the interaction force holding the quarks together grows weak, and scientists can break particles apart to observe their constituents. Unfortunately, these observations -- and even gauge theory itself when applied at these high energies -- do not reveal as much as physicists would like about everyday matter traveling at everyday speeds.

"The sad truth is that when these quarks and gluons start binding together into protons and neutrons, this interaction force grows very strong, and it is hard to use gauge theory to understand it," Klebanov said. "Basically, to understand how we are actually made of all this stuff, we need to understand quark and gluon behavior when the interaction force gets strong."

In the 1970s, physicists posited that when gauge theory loses its power to describe particle behavior as quarks bond together, string theory might be able to step in and handle the job. What string theorists needed was some indication that both theories were headed in the same direction.

The lucky break came in 1997 and early 1998 when a precise relation between the two was conjectured in the work of Princeton physicists Alexander Polyakov, Steven Gubser and Klebanov, as well as the Institute for Advanced Study's Juan Maldacena and Edward Witten. However, more work to explain this connection was needed.

"It was as though our understanding was a road that started at the point where the interaction between quarks was weak," Klebanov said. "We could follow it for a few miles through greater and greater interaction strengths, but then it stopped before reaching the great strengths that exist in the atoms of rocks and trees -- the section of road that string theory describes."

Between the two road sections lay a seemingly unbridgeable mathematical gulf, and Klebanov had little more than a hunch that there was any smooth transition between gauge and string theory.

"In simple terms, what we really wanted was some indication that such a smooth transition exists, which would hint that the two sections of road were parts of the same route," Klebanov said. "But we were having trouble finding any sort of connection at all."

String theory, for all its mathematical beauty, once again seemed too difficult to test -- until Niklas Beisert, an assistant professor of physics at Princeton, published a paper in late October of last year containing an equation that turned out to be a crucial piece of the puzzle.

"Beisert and his collaborators made an inspired guess based on sophisticated notions of gauge theory behavior," said Curtis Callan, the James S. McDonnell Distinguished University Professor of Physics at Princeton. "Their equation allowed Igor and his colleagues to work out the 'transition' between the two regimes. They demonstrated that it exactly matched string theory's predictions at the strong interaction limit. That was the hard part."

Beisert said that his team's work provided a useful abstract proof of the transition between weak and strong interaction strength, but that numerical evidence had been lacking.

"The result of Klebanov's group gives beautiful numerical evidence for the validity of our proposal," said Beisert, who is also junior research group leader at the Albert Einstein Institute in Potsdam, Germany. "All these studies now make us sure that string theory and well-established gauge theory are indeed two sides of the same coin."

Lance Dixon, a physicist at the Stanford Linear Accelerator, said the new paper by Klebanov's team provided a vital cross-check of the Beisert team's equation, which rested on a few insightful but unproven assumptions.

"The work by Igor Klebanov and his group really succeeded in removing all lingering doubts about the equation's validity," said Dixon, whose work provided other evidence in favor of the Beisert team's proposal. "The search is on now for a broad web of connections linking the high-energy behavior of quarks and gluons to that of strings, a web in which the first strand was laid down by (both teams') work."

Klebanov, while noting many other scientists' contributions, credits Beisert for "providing the techniques and writing down this fantastic equation," and said that his own team's findings were possible largely due to the Beisert team's paper.

"That a particular kind of gauge theory is in some sense 'exactly solvable' realizes a longstanding dream," Klebanov said. "This is almost like coming to a gap you expected to find between two sections of road and finding that someone else has built such a smooth connection between them that you don't notice it when you drive over it."

This is not to say that string theory is likely to become accepted as an overall explanation of subatomic physics anytime soon. Klebanov's team has found a bridge between established physics and a mathematical theory, which is only one step toward solid experimental proof that the world is actually constructed of tiny vibrating strings. And even this bridge applies to only one facet of gauge theory. Bridging this gap for other facets will be necessary to enable physicists to understand fundamentally the interiors of the protons and neutrons that make up the earth beneath our feet.

"I think there is hope that other facets of gauge theory are amenable to similar treatment," Klebanov said. "We don't know for sure if we can use this discovery to address other problems, but at least we now have new methods for bridging the gap between the weakly and strongly interacting regimes of the gauge theory.”

This research was sponsored in part by the National Science Foundation and the Princeton Center for Theoretical Physics.


A Test of the AdS/CFT Correspondence Using High-Spin Operators
M. K. Benna, S. Benvenuti, I. R. Klebanov, A. Scardicchio

In two remarkable recent papers the planar perturbative expansion was proposed for the universal function of the coupling appearing in the dimensions of high-spin operators of the [script N]=4 super Yang-Mills theory. We study numerically the integral equation derived by Beisert, Eden, and Staudacher, which resums the perturbative series. In a confirmation of the anti–de Sitter-space/conformal-field-theory (AdS/CFT) correspondence, we find a smooth function whose two leading terms at strong coupling match the results obtained for the semiclassical folded string spinning in AdS5. We also make a numerical prediction for the third term in the strong coupling series.
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PostPosted: Sat Nov 17, 2007 6:58 am    Post subject: Shadow World Reply with quote

Week of Nov. 17, 2007; Vol. 172, No. 20 , p. 315

Shadow World
How many dimensions space has could all be a matter of perspective

Davide Castelvecchi

In a school of thought that teaches the existence of extra dimensions, Juan Maldacena may at first sound a little out of place. String theory is physicists' still-tentative strategy for reconciling Einstein's theory of gravitation with quantum physics. Its premise is that the subatomic particles that roam our three-dimensional world are really infinitesimally thin strings vibrating in nine dimensions. According to Maldacena, however, the key to understanding string theory is not to add more dimensions but to cut their number down.

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