Strange new twist: Researchers discover Mobius symmetry in metamaterials

Dec 20, 2010 by Lynn Yarris
Berkeley Lab researcher have discovered Möbius symmetry in metamolecular trimers made from metals and dielectrics. Credit: Image by Chih-Wei Chang

(PhysOrg.com) -- Möbius symmetry, the topological phenomenon that yields a half-twisted strip with two surfaces but only one side, has been a source of fascination since its discovery in 1858 by German mathematician August Möbius. As artist M.C. Escher so vividly demonstrated in his "parade of ants," it is possible to traverse the "inside" and "outside" surfaces of a Möbius strip without crossing over an edge. For years, scientists have been searching for an example of Möbius symmetry in natural materials without any success. Now a team of scientists has discovered Möbius symmetry in metamaterials – materials engineered from artificial "atoms" and "molecules" with electromagnetic properties that arise from their structure rather than their chemical composition.

Xiang Zhang, a scientist with the U.S. Department of Energy's Lawrence Berkeley National Laboratory (Berkeley Lab) and a professor at the University of California (UC) Berkeley, led a study in which electromagnetic Möbius symmetry was successfully introduced into composite metamolecular systems made from metals and dielectrics. This discovery opens the door to finding and exploiting novel phenomena in metamaterials.

"We have experimentally observed a new topological symmetry in electromagnetic metamaterial systems that is
equivalent to the structural symmetry of a Möbius strip, with the number of twists controlled by sign changes in the electromagnetic coupling between the meta-atoms," Zhang says. "We have further demonstrated that metamaterials with different coupling signs exhibit resonance frequencies that depend on the number but not the locations of the twists. This confirms the topological nature of the symmetry."

Working with metallic resonant meta-atoms configured as coupled split-ring resonators, Zhang and members of his research group assembled three of these meta-atoms into trimers. Through careful design of the electromagnetic couplings between the constituent meta-atoms, these trimers displayed Möbius C3 symmetry – meaning Möbius cyclic symmetry through three rotations of 120 degrees. The Möbius twists result from a change in the signs of the electromagnetic coupling constants between the constituent meta-atoms.

"The topological Möbius symmetry we found in our meta-molecule trimers is a new symmetry not found in naturally occurring materials or molecules." Zhang says. "Since the coupling constants of metamolecules can be arbitrarily varied from positive to negative without any constraints, the number of Möbius twists we can introduce are unlimited. This means that topological structures that have thus far been limited to mathematical imagination can now be realized using metamolecules of different designs."

Details on this discovery have been published in the journal , in a paper titled "Optical Möbius Symmetry in Metamaterials." Co-authoring the paper with Zhang were Chih-Wei Chang, Ming Liu, Sunghyun Nam, Shuang Zhang, Yongmin Liu and Guy Bartal.

Xiang Zhang is a principal investigator with Berkeley Lab's Materials Sciences Division and the Ernest S. Kuh Endowed Chaired Professor at UC Berkeley, where he directs the Center for Scalable and Integrated NanoManufacturing (SINAM), a National Science Foundation Nano-scale Science and Engineering Center.

In his “parade of ants,” Artist M.C. Escher demonstrated that it is possible to traverse the “inside” and “outside” surfaces of a Möbius band without crossing over an edge.

In science, symmetry is defined as a system feature or property that is preserved when the system undergoes a change. This is one of the most fundamental and crucial concepts in science, underpinning such physical phenomena as the conservation laws and selection rules that govern the transition of a system from one state to another. Symmetry also dictates chemical reactions and drives a number of important scientific tools, including crystallography and spectroscopy.

While some symmetries, such as spatial geometries, are easily observed, others, such as optical symmetries, may be hidden. A powerful investigative tool for uncovering hidden symmetries is a general phenomenon known as "degeneracy." For example, the energy level degeneracy of an atom in a crystal is correlated with the crystal symmetry. A three-body system, like a trimer, can be especially effective for studying the correlation between degeneracy and symmetry because, although it is a relatively simple system, it reveals a rich spectrum of phenomena.

"The unique properties of a three-body system make experimental investigations of hidden symmetries possible," says Chih-Wei Chang, a former post-doc in Zhang's group and the lead author of the paper in Physical Review Letters, says. "Intrigued by the extraordinary engineering flexibilities of metamaterials, we decided to investigate some non-trivial symmetries hidden beneath these metamolecules by studying their degeneracy properties"

The authors tested their metamaterials for hidden symmetry by shining a light and monitoring the optical resonances. The resulting resonant frequencies revealed that degeneracy is kept even when the coupling constants between meta-atoms flip signs.

"Because degeneracy and symmetry are always correlated, there must be some symmetry hidden beneath the observed degeneracy" says Chang.

The researchers showed that whereas trimer systems with uniform negative (or positive) coupling signs could be symbolized as an equilateral triangle, trimer systems with mixed signs of couplings could only be symbolized as a Möbius strip with topological C3 symmetry. Furthermore, in other metamolecular systems made of six meta-atoms, the authors demonstrated up to three Möbius twists.

Says Chang, now a faculty member at National Taiwan University in Taipei, "When going from natural systems to artificial meta-atoms and metamolecules, we can expect to encounter phenomena far beyond our conventional conceptions. The new symmetries we find in could be extended to other kinds of artificial systems, such as Josephson junctions, that will open new avenues for novel phenomena in quantum electronics and quantum optics."

Explore further: Better thermal-imaging lens from waste sulfur

Related Stories

Exotic symmetry seen in ultracold electrons

Jan 18, 2010

(PhysOrg.com) -- An exotic type of symmetry - suggested by string theory and theories of high-energy particle physics, and also conjectured for electrons in solids under certain conditions - has been observed ...

Pure mathematics behind the mechanics

Feb 07, 2008

Dutch researcher Peter Hochs has discovered that the same effects can be observed in quantum and classical mechanics, if quantisation is used.

Testing relativity in the lab

Jul 20, 2009

Even Albert Einstein might have been impressed. His theory of general relativity, which describes how the gravity of a massive object, such as a star, can curve space and time, has been successfully used to ...

Recommended for you

Could 'Jedi Putter' be the force golfers need?

2 hours ago

Putting is arguably the most important skill in golf; in fact, it's been described as a game within a game. Now a team of Rice engineering students has devised a training putter that offers golfers audio, ...

Better thermal-imaging lens from waste sulfur

17 hours ago

Sulfur left over from refining fossil fuels can be transformed into cheap, lightweight, plastic lenses for infrared devices, including night-vision goggles, a University of Arizona-led international team ...

User comments : 20

Adjust slider to filter visible comments by rank

Display comments: newest first

El_Nose
5 / 5 (1) Dec 20, 2010
stay tuned in for next week we investigate --- the klein bottle
michaelick
4.5 / 5 (8) Dec 20, 2010
"Möbius symmetry, the topological phenomenon that yields a half-twisted strip with two surfaces but only one side"

and again the editors of physorg have no idea what they are talking about. the möbius strip has also ONLY ONE surface, not two. that's what makes it special, together with the one edge.

and the reason they used the image of a penrose triangle instead of a möbius strip remains a mystery as well.
x646d63
3 / 5 (4) Dec 20, 2010
How is a mobius strip practically any different than a torus? A torus has only one "side" also, and all parts of it can be traversed without crossing "edges." A Mobius strip is just a "slice" of a torus. It's just not sliced on a plane.
beelize54
1 / 5 (7) Dec 20, 2010
This is actually feature of all dispersive materials, not just metamaterials. If you put the light bulb into a vessel covered with foam, whole volume of foam will glow, not just interior of vessel. Dispersion of light in vacuum is modeled with metamaterials, too.
BadMan
not rated yet Dec 20, 2010
A Mobius strip does have two surfaces, the wide side and the narrow edge.
lexington
not rated yet Dec 21, 2010
So... meta-atoms? What does that mean?
Husky
not rated yet Dec 21, 2010
i can see it now, a static bar magnet with a mobius field that resonate and have whirling/twisting magnetic field around it, with a moving north and south pole and glowing with hawking radiation or stuff like that
Quantum_Conundrum
1 / 5 (2) Dec 21, 2010
A Mobius strip does have two surfaces, the wide side and the narrow edge.

NOpe. That would be a peice of paper simulating a mobius strip.

A true mathematical mobius strip has thickness = zero.

that will open new avenues for novel phenomena in quantum electronics and quantum optics


I'm trying to envision a motherboard stretched like a rubber band and given a half-twist back upon itself.

No clue why you'd do that.

Although, if you made a motherboard as just a hollow, cyclindrical tube, the distance between some devices along the surface would be cut in half. If you then folded the tube into a torus, the distance between some of the other devices would be cut in half. Everything would be as little as half as far away from anything that is already over half the board's width in distance...

On the other hand, if the motherboard was a mobius strip, it would double the distance between devices...
El_Nose
5 / 5 (1) Dec 22, 2010
all the time i have spent on here I expected you guys to light into the guy that stated
How is a mobius strip practically any different than a torus? A torus has only one "side" also, and all parts of it can be traversed without crossing "edges." A Mobius strip is just a "slice" of a torus. It's just not sliced on a plane.
but nobody corrected him -- has this forum leanred tolerance or are people on here more inclined to physics than math/topology?
Quantum_Conundrum
1 / 5 (2) Dec 22, 2010
nobody corrected him -- has this forum leanred tolerance or are people on here more inclined to physics than math/topology?


Ok, a Torus is a 3-dimensional object which ends up having 1 surface and no edges. A torus is produced by revolving a circle into the third dimension around an axis outside itself, but parrallel to the plane of the original circle.

Although you could modify the location of the axis into the third dimension, but this would produce an eliptical toroid, or a toroidal elipse, depending on the orientation of the axis of revolution...

A mobius strip is a 2-dimensional object which has been half-twisted into the third dimension back upon itself, therefore making it a closed object with only one surface and one edge.
frajo
5 / 5 (4) Dec 22, 2010
a Torus is a 3-dimensional object which ends up having 1 surface and no edges.
No. A torus has two surfaces. Outside and inside. There is no continuous path wich connects both.
A Moebius strip, however, has only one surface. Any point P1 can be connected to any point P2 by a continuous path.
Quantum_Conundrum
2.3 / 5 (3) Dec 22, 2010
No. A torus has two surfaces. Outside and inside. There is no continuous path wich connects both.


See, that's only true of the torus is hollow, which would be an inner tube.

But in normal geometry, they are normally referring to a solid object when they use the term "torus". Nobody ever counts the "back" of a surface in geometry.

think about the formula for area of a square or circle.

X^2.
pi(r^2)

Surface area of a cube, which has six faces.
6x^2

Surface area of sphere, which has 1 surface
4pi(r^2)

See, nobody ever says anything like what you are saying. We treat the sphere as a solid, just as we treat a torus as a solid.

In technical terms, an inner tube is not a torus. It is a complex object formed by removing one torus from the center of another torus with a smaller little "r" component in the volume of a torus formula.

An inner tube has an internal surface, but a torus has only one surface, just like a sphere.
x646d63
not rated yet Dec 22, 2010
@El_nose: I was suggesting that a mobius strip could be cut from a torus, not that a torus could be formed into a mobius strip.

Take a line that measures the diameter of the solid ring of the torus. Move it through the solid part of the torus, rotating it 90 degrees until it returns to its starting point. You have traced a mobius strip.
Skeptic_Heretic
3 / 5 (2) Dec 23, 2010
all the time i have spent on here I expected you guys to light into the guy that stated
How is a mobius strip practically any different than a torus? A torus has only one "side" also, and all parts of it can be traversed without crossing "edges." A Mobius strip is just a "slice" of a torus. It's just not sliced on a plane.
but nobody corrected him -- has this forum leanred tolerance or are people on here more inclined to physics than math/topology?

I can only argue with an idiot for so long until they drag me down to their level of stupidity and beat me with experience in said stupidity.
ubavontuba
3 / 5 (2) Dec 24, 2010
all the time i have spent on here I expected you guys to light into the guy that stated
How is a mobius strip practically any different than a torus? A torus has only one "side" also, and all parts of it can be traversed without crossing "edges." A Mobius strip is just a "slice" of a torus. It's just not sliced on a plane.
but nobody corrected him -- has this forum leanred tolerance or are people on here more inclined to physics than math/topology?

A donut, or a twist. Both are good. Personally, I prefer the maple glazed, old-fashioned variety.
ubavontuba
1 / 5 (2) Dec 24, 2010
@El_nose: I was suggesting that a mobius strip could be cut from a torus, not that a torus could be formed into a mobius strip.

Take a line that measures the diameter of the solid ring of the torus. Move it through the solid part of the torus, rotating it 90 degrees until it returns to its starting point. You have traced a mobius strip.

You can't rotate a line on an axis 90 degrees and return it to its starting point.

Also, you can't cut a torus into a mobius strip. It will always have two sides (the inside, and the outside). This is because the two sides in a torus never meet, and you can't make them meet by cutting funny shapes out of the torus.
jsa09
2 / 5 (1) Dec 28, 2010

You can't rotate a line on an axis 90 degrees and return it to its starting point.

Also, you can't cut a torus into a mobius strip. It will always have two sides (the inside, and the outside). This is because the two sides in a torus never meet, and you can't make them meet by cutting funny shapes out of the torus.


Rubbish - the reason you cannot cut a torus into a mobius strip is because one is three dimensional object the other is 1 dimensional object.

You could however carve a shape that looked like a mobius strip out of a torus if you wanted to. You could also carve the same shape out of a cube so that is no test of anything.
ubavontuba
1 / 5 (1) Dec 29, 2010
Rubbish - the reason you cannot cut a torus into a mobius strip is because one is three dimensional object the other is 1 dimensional object.
Wrong. A mobius strip is a two-dimensional manifold.
You could however carve a shape that looked like a mobius strip out of a torus if you wanted to.
Wrong. You're thinking of a toroid (the solid form of a torus).
You could also carve the same shape out of a cube so that is no test of anything.
Sure, cubes are defined as being solid.
OmRa
2 / 5 (1) Dec 29, 2010
This is reality. This mobius twist.
From SpaceTime to TimeSpace. From within to without,
in one continous loop.
El_Nose
5 / 5 (2) Jan 04, 2011
Wow you guys got interesting there

ubavontuba - is the winner with a perfectly correct answer to all questions stated. We could get into the math - but who would want to read it?

More news stories

Could 'Jedi Putter' be the force golfers need?

Putting is arguably the most important skill in golf; in fact, it's been described as a game within a game. Now a team of Rice engineering students has devised a training putter that offers golfers audio, ...

Continents may be a key feature of Super-Earths

Huge Earth-like planets that have both continents and oceans may be better at harboring extraterrestrial life than those that are water-only worlds. A new study gives hope for the possibility that many super-Earth ...

Researchers successfully clone adult human stem cells

(Phys.org) —An international team of researchers, led by Robert Lanza, of Advanced Cell Technology, has announced that they have performed the first successful cloning of adult human skin cells into stem ...