Fractals! Script Revisited
What do snowflakes and cell phones have in common?
Snowflakes have intricate detail no matter how closely you look. In mathematics, such shapes are called fractals.
Fractals are never-ending patterns that on any scale, on any level of zoom, look roughly the same. Computer scientists can program these infinite patterns by repeating an often simple mathematical process over and over.
To model a snowflake, for example, start by drawing an equilateral triangle. Divide each side into three equal parts and build another equilateral triangle on top of each side.
Take out the middle, and repeat the process, this time with 1, 2, 3, 4 times 3, which is 12 sides. Eventually, the shape will look something like this:
This curve is called in mathematics a Koch Snowflake. It is a fractal because, if you zoom in, youâll get this same pattern (show pattern) again and again.
The Koch fractal is useful not only for modeling snowflakes.
In 1990âs, a radio astronomer named Nathan Cohen used the Koch snowflake to revolutionize wireless communications.
At the time, Cohen was having troubles with his landlord. The man wouldnât let him put a radio antenna on the roof! So, Cohen decided to make a more compact, fractal radio antenna instead (show wire bent as Koch snowflake)
The landlord didnât notice it, and it worked better than the ones before!
Working further, Cohen designed a new version, this time using a fractal called âthe Menger Spongeâ (build a tangible model of the Sponge, and show it). The fractalâs infinite âsponginessâ allowed the antennae to let through multiple different signals.
(soapy sponge used as prop; maybe host is in shower?)
The Menger Sponge is not really the sponge youâd be scrubbing your back with, but you can still think of it kind of like that. Imagine both water and soap getting through your spongeâs holes, except the water is Wi Fi and the soap is, say, Bluetooth.
The fractal shape allows the antenna to operate well at many different frequencies simultaneously. Before Cohenâs invention, antennas had to be "cut" for one necessary frequency. That was the only frequency they could operate at.
So, without Cohenâs âsponge,â your cell phone would have to look something like a hedgehog to receive different signals, including the radio signal that allows you to hear your friends when they call (illustrate: phone with several antennas glued on). As Cohen later proved, only fractal shapes could work with such a wide range of frequencies.
Today, millions of wireless communication devices, such as laptops and barcode scanners, also use Cohenâs fractal antenna.
Cohenâs genius invention, however, was not the first application of fractals in the world. Turns out, nature has been doing it the whole time!
Natural selection has allowed it to create the most efficient systems and organisms, most of which (if not all) evolved to a fractal shape.
The spiral fractal, for example, is present in seashells, broccoli, and hurricanes.
So, many natural systems previously thought off-limits to mathematicians were suddenly explained in terms of fractals. The fractal tree for example, is relatively easy to program, and allows mathematicians to study anything from river systems to blood vessels and lightning bolts.
Thus, fractals allow us to learn natureâs best practices, and then apply them to solve real world problems. Much like Cohenâs antenna revolutionized telecommunications, other fractal research is changing fields of medicine, weather prediction, and many more. Here at MIT (move camera from host and Charles river to MIT dome right across) and everywhere in the world.
Look around you. What beautiful patterns do you see?












