So here I am, trained as a topologist and geometric group theorist, starting a job that involves mainly digital signal processing. Today I was perusing the magazines on the shelf at my new job, and what do I see? The cover of Microwave Journal is emblazoned with “SUPER MOBIUS RESONATOR” and features a Superhero whose crest is a Mobius Strip! Is this really the unorientable surface with boundary that I have grown to love? Why yes! The first few paragraphs of the multi-page article discuss the very familiar properties of a Mobius strip using familiar words like “developable” and “anholonomy”. While I am still a bit mystified as to why the Mobius strip makes such a good resonator and what exactly is meant by “planar” Mobius Strip, I am certainly intrigued. Apparently signals can travel unimpeded around the Mobius strip in a way that is not possible around an oriented loop. And to think, the first week on the job I entertained my office mate by cutting a mobius strip in half! (More about that later.)
So this sent me on a little Mobius hunt around the web. Is the idea of a Mobius strip being used to make a resonator related to he presence of the Mobius strip in meta-materials as impelemented in 2010? Researcher Xiang Zhang stated in this article “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.”
It’s so exciting to me to see such connections, and I’m looking forward to fleshing them out. As I searched for more information on recent applications involving the Mobius Strip, I found this wonderful video from the Royal Institution of Great Britain of a superconductor floating along a Mobius racetrack made of magnets.
And if this racetrack made you wish you had a bigger model of a Mobius strip, look no further than Dutch architect Janjaap Ruijssenaars who worked with a mathematically trained sculptor to create a model of a Mobius house. This house will be created using a 3D printer and will be 12,000 square feet.
Lastly, even if you have cut a Mobius band in half (or in thirds or fourths, etc) along its length, you may not have tried cutting up a Klein bottle. For a guided tour of that experience, check out the November 15th post by Science writer Matthew R. Francis at Galileo’s Pendulum. Francis suggests with his post that with all this cutting, we might convince even those who claim to dislike mathematics to be swayed to our side. My computer scientist office-mate was certainly entertained!
The Mobius strips, Mobius coupling and Mobius Symmetry is very interesting and I am following Dr. Ajay Poddar work on this topic (Author of Microwave journal 2011, Nov issue) and previous publications (1992). Dr. Poddar’s work and demonstration in IEEE publication is very interesting: Mobius transformation f can be expressed as a composition of magnifications, rotations, translations and inversions. This leads to
f maps the extended complex plane onto itself, f maps the class of circled and lines to itself, and f is conformal at every point except its pole.
As per Dr. Poddar, these unique properties are blessing in disguise for constructing multi-band VCOs using Mobius-metamaterials resonator, reported in IEEE IMS 2014 at Tampa and IEEE Wamicon 2014 CONFERENCE.
Möbius symmetry is a topological observable fact characterized by a half-twisted strip with two surfaces but only one side. Mobius strip has been a source of fascination since its discovery in 1858 by German mathematician August Möbius because it is possible to traverse the “inside” and “outside” surfaces of a Möbius strip without crossing over an edge.
Dr. Poddar’s reported [http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6697691] the resonator of this invention is characterized by having a twist or a cross-over in the surface structure of a resonator so that the surface structure provides a continuous electromagnetic path to a wave, multiple waves, or a fraction of a wave. This is a real breakthrough. I have attended Dr. Poddar’s workshop in 2010, the invention is unique for modern communication system