# Fractal Antennas

*This article was initially published in the August 2021 issue of IEEE-USA InSight*

Mobile phones perform multiple tasks by transmitting and receiving on many different frequencies. For example, when the user first dials, there is a carrier frequency that the phone and the base station communicate with each other to set up the call, establish which cell tower the phone is in range of, and choose which frequency or frequencies to use for the call. Some mobile phone systems use frequency-shift keying, which means the zeros of the digital signal are sent on one frequency and the ones are sent on another. BlueTooth, WiFi, and other applications use yet more frequencies.

Prior to the late 1980s, this would have required mobile phones to use an antenna for each frequency. Mobile phones would have needed many different antennas sticking out, each sized according to wavelength. Instead, today’s mobile phones owe their sleek design in part to antennas whose shapes are determined by fractals.

In 1982, mathematician Benoit Mandelbrot published his influential book, The Fractal Geometry of Nature. Some of the earliest applications of fractals were in computer graphics. Mandelbrot gave a paper on the landscapes of an imaginary planet at the 99th Colloquium of the International Astronomical Union, held in Balaton, Hungary in June of 1987. Also giving a paper at the conference was IEEE member and radio astronomer Prof. Nathan Cohen of Boston University. Cohen was a ham radio operator, and Mandelbrot’s talk intrigued him. He wondered how an antenna shaped according to fractal geometry would work.

Cohen found that it worked very well. However, his then landlord had a strict policy about not attaching antennas to the building. Cohen was working with 2-meter FM, meaning that a conventional antenna would need to be about one meter, and thus quite visible. Having found that his fractal antennas could be made much smaller, Cohen made a fractal microstrip antenna out of aluminum foil, glue, and construction paper using the pagoda motif, and attached it to the railing of his apartment. It was about six inches square.

Despite its diminutive size and looking more like a doily than an antenna, (as well as being up on the 27th floor), the landlord learned of its existence and cut it down while Cohen was at work, teaching a math class. Cohen found its sad remains in the snow on the ground below a few days later. Given that the antenna’s descendants now function in hundreds of millions of devices around the world, the severing of the prototype must surely rank as one of the most infelicitous acts of technological obstruction since the Visigoths dismantled the Roman aqueducts. In addition to their uses in mobile phones and computers, fractal antennas have important applications in RFIDs and in vehicular radar and collision-avoidance systems. They reduce scattering of the signal and enhance the radar reflection of highway tags. As one measure of fractal antennas’ importance, IEEE has published more than 2,100 papers on them.

“Fractal antennas not only shrink antenna sizes, but also control multiband performance, enable wideband use, and actually increase realized gain in small sizes. The gain attribute arises from being able to produce multiple current maxima, such as on a fractal perimeter, in a highly compact area. Constructive interference can happen in regimes far smaller than a ¼ wave,” Cohen explains.

Fractal Antenna Systems, the company Cohen founded, filed US patent 6,452,553 in August of 1995 for the fractal antenna. The earliest adopters of the technology were government customers. “Today,” notes Cohen, “you would be challenged to get an x-ray, use radios or phones in a public building, or fly on a commercial jet without fractal antennas. Fractals are a big part of keeping the world connected.”

In addition to being a very widely-used technology, fractal antennas are visually intricate and beautiful. Because they are small enough to be hidden inside the technologies that they make work, most of us are not aware of them. The author hopes that this article has raised their visibility.