University Park, PA, USA. The new discovery of a sixth type of structural symmetry is a very large advance toward understanding the structure of proteins, polymers, minerals, and engineered materials.

Discovery of rotation reversal symmetry by two Penn State scientists has broad relevance for many development efforts involving the biological, chemical, engineering, and physical disciplines — including implications for new approaches to DNA, biomaterials and process interventions.


Previously, scientists and engineers had five different types of symmetries to use as tools for understanding the structures of materials whose building blocks are arranged in fairly regular patterns. Four types of symmetries had been known for thousands of years — called rotation, inversion, rotation inversion, and translation. A fifth type — called time reversal — had been discovered about 60 years ago.



Repeating Molecular Structures

The lattice shown here is composed of columns of squares representing repeating molecular structures.

One of the structures is rotated clockwise (colored blue) and the other is counterclockwise (colored orange) with respect to each other.

Click Pic for Details

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In mathematics, symmetry is a self-similarity in a regular pattern. That is, the whole of an object has the same shape as one or more of its parts. Even more precisely, the object is exactly or approximately similar to a part of itself.

The statistical self-similarity of lengthy coastlines and the commonly known fractals are examples of symmetrical objects. Any one part is statistically representative of the whole.

Scientists have found applications for this mathematically precise and well-defined concept throughout the natural world.

In biology, most multicellular organisms exhibit symmetry in their body plans (commonly known as bilateral, radial, or spherical). This highly observable symmetry can be more obviously approximate in the daily world due to variations introduced by their manufacture, the environment, and the effects of aging.

There is a small minority of organisms for which no symmetry is known to exist (ie., they are asymmetric).Venkatraman Gopalan and Daniel B. Litvin have added a sixth type of symmetry: rotation reversal. As a result, the number of known ways in which the components of such crystalline materials can be combined in symmetrical ways has multiplied from no more than 1,651 before to more than 17,800 now. The scientists report on their findings in the journal Nature Materials.

"It is important to look at symmetries in materials because symmetry dictates all natural laws in our physical universe."

— Daniel B. LitvinThe rules of a formal system can be applied to prove and/or demonstrate symmetry, usually through geometry or physics. Daniel B. Litvin says "We mathematically combined the new rotation-reversal symmetry with the previous five symmetries and now we know that symmetrical groups can form in crystalline materials in a much larger number of ways." Litvin, himself a distinguished professor of physics, coauthored the study with Venkatraman Gopalan, professor of materials science and engineering.

The newly found rotation-reversal symmetry enriches the mathematical language that researchers use to describe a crystalline material's structure and to predict its properties. "Rotation reversal is an absolutely new approach that is different in that it acts on a static component of the material's structure, not on the whole structure all at once," Litvin said.

The simplest type of symmetry — rotation symmetry — is obvious, for example, when a square shape is rotated around its center point: the square shows its symmetrical character by looking exactly the same at four points during the rotation: at 90 degrees, 180 degrees, 270 degrees, and 360 degrees.

Gopalan and Litvin say their new rotation-reversal symmetry is obvious, as well, if you know where to look. The pivotal moment of the discovery occurred when Gopalan recognized that the simple concept of reversing the direction of a spiral-shaped structure from clockwise to counterclockwise opens the door to a distinctly new type of symmetry.

Just as a square shape has the quality of rotation symmetry even when it is not being rotated, Gopalan realized that a spiral shape has the quality of rotation-reversal symmetry even when it is not being physically forced to rotate in the reverse direction.

Their further work with this rotation-reversal concept revealed many more structural symmetries than previously had been recognized in materials containing various types of directionally oriented structures. Many important biological molecules, for example, are said to be either right handed or left handed, including DNA, sugars, and proteins.

"We found that rotation-reversal symmetry also exists in paired structures where the partner components lean toward each other, then away from each other in paired patterns symmetrically throughout a material," Gopalan said.

These "tilting octahedral" structures are common in a wide variety of crystalline materials, where all the component structures are tightly interconnected by networks of shared atoms. The researchers say it is possible that components of materials with rotation-reversal symmetry could be engineered to function as on/off switches for a variety of novel applications.

Given the handedness of DNA, the new discovery open up new and different research avenues for exploring the potential of engineered bioswitches that control the on/off states of gene expression.

The now-much-larger number of possible symmetry groups also is expected to be useful in identifying materials with unusual combinations of properties. The researchers point to expanded possibilities for discovering or designing materials with desired properties, including the search for advanced ferroelectric ferromagnet materials for next-generation ultrasound devices and computers.

"For example, the goal in developing a ferroelectric ferromagnet is to have a material in which the electrical dipoles and the magnetic moments coexist and are coupled in the same material — that is, a material that allows electrical control of magnetism — which would be very useful to have in computers," Gopalan said. The addition of rotation-reversal symmetry to the materials-science toolbox may help researchers to identify and search for structures in materials that could have strong ferroelectric and ferromagnetic properties.

Gopalan and Litvin said a goal of their continuing research is to describe each of the more than 17,800 different combinations of the six symmetry types to give materials scientists a practical new tool for significantly increasing the efficiency and effectiveness in finding novel materials.

The team also plans to conduct laboratory experiments that make use of their theoretical work on rotation-reversal symmetry. "We have done some predictions; we will test those predictions experimentally," Litvin said.

"We are in the very early stages of implementing the results we have described in our new theory paper." Gopalan said, for example, that he has predicted new forms for optical properties in the commonplace quartz crystals that are used widely in watches and electronic equipment, and that his group now is testing these predictions experimentally.

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