Summer is here and all over the world, people take the roads, airways, exploring the world around them. But our travels aren't just limited to terrestrial destinations this year. New Horizons is sending us postcards from Pluto and we've just received news of the discovery of an "Earth twin", a mere 1400 light years away, sighted by the deep space Kepler telescope, which is scanning the far reaches of the Universe like eagle-eyed RVers, phones at the ready, crisscrossing the United States. I love the audacity of the goals and the grandeur of the vision -- we're exploring the Universe! But what I also love about these achievements is that they celebrate the power of mathematics.
First off, let's just start with the name: The Kepler Telescope. While aesthetic (and even religious) influences caused earlier astronomers to claim that planetary orbits were circular. But a raft of data and some careful trigonometric reasoning enabled Johannes Kepler to show that in fact planetary orbits are elliptical, an observation which served as partial motivation for Isaac Newton's discovery of the law of gravity and his laws of motion. These in turn pushed Newton to invent the calculus. The snowball continues to roll, for with calculus in hand, the existence of celestial objects could be inferred, even when invisible through a telescope, by observing the deviations of visible planets (and other objects) from their expected orbits -- deviations that could be credited to some invisible heavenly body exerting its own gravitational pull. Hello, Pluto -- Thank you, calculus.
The discovery of our earth-twin (who goes by the futuristic name "452b") requires us to strengthen our mathematical eyesight. Surveying the Universe -- or even a slice of it -- how do we find other solar systems? How do we find these planets that live over 1000 light years away? The Kepler telescope scans deep space recording measurements of light intensity wherever it looks, just as you do when you look up at the night sky, noticing brighter and fainter points of light. Kepler has better vision than you do and is able to look farther and more carefully. Close investigation of those light points reveal that they are made up of multiple light sources moving in a synchronized fashion, akin to our own solar system. The disentangling of these mini kaleidoscopes of light ("light curves") is accomplished by the same math that enables us to disentangle the music of a symphony into basic sound waves - and then turn it into a digital recording - or extract patterns in the financial markets (which some of us then try to turn into money): Fourier analysis. Invented by the Frenchman, Joseph Fourier, chief scientific advisor to Napoleon, in the pursuit of explaining and modeling the cycles of cooling and warming that we call the seasons.
Now, Kepler had to actually get out there. Generations of research in air flow and turbulence and accompanying mathematical modeling form the foundations of our understanding of how to launch objects into the air and guide them into and through space. Kepler, however is not going, going, and going to be gone, but rather is meant to stay, roughly nearby, scanning the skies like a Klieg light at a Hollywood opening. For this, Kepler was put into a so-called "earth-traling orbit," engineered according to specifications determined by calculus-based tools. By slowly but surely falling behind the Earth orbit, Kepler gets a better view of the Universe.
Ok, we're there, looking at the sky, taking pictures. How do we get the information back to Earth? More generally, how do we stay in touch with a machine and its computers, whether it's three billion miles away like New Horizons or "only" 100 million miles away like Kepler? If you think you have problems with dropped calls here on Earth, they pale in comparison to the difficulties of having wireless conversations across the solar system. Once again, mathematics comes to the rescue. The messages that are sent between New Horizons and Earth (as well as pictures) arrive intact by virtue of their being packaged up using "error correcting codes". These are mathematical recipes that take the streams of 0s and 1s whose combinations form the original message and transform them into different messages able to withstand the deleterious effects of deep space travel. These codes are the mathematical analogy of the hull of the spacecraft itself: the equipment within the craft needs to be protected from the elements in order to fulfill its mission. Similarly, the digital transmission, packets of energy streaming through space, need to be "packaged" in order to withstand the vagaries of space travel. Rather than enclose them in a box, we enclose the digital messages in a mathematical "wrapper", which even if degraded and damaged by the radiation and other space phenomena, can still be opened upon arrival, with the embedded message decipherable. Sort of like the deciphering that you naturally do of a poorly executed closed captioning of a television show. In this case, the pressures of quickly turning out the words is the like the radiation of deep space and your brain is the computer that still manages to decode the original message. You perform "error correction" all the time!
Error correcting codes of the kind that our satellites use have as their foundation a kind of algebra that for centuries was studied for purely intrinsic mathematical reasons. It was only when digital communication became a reality, that connections were to means of creating message encodings that would be robust to degradation, were discovered and then, lo and behold, there was a whole world of results and understandings already in place that could form the foundation of a new and important discipline at the nexus of engineering and mathematics. Fast forward several decades, and here we are, exploring the solar system and the Milky Way on our laptops from the comforts of our own homes.
In fact, mathematics has always been an enabler of discovery and exploration. Our charting of the heavens via mathematics is but a modern updating of the use of the use of mathematics in seafaring or terrestrial navigation of explorers and cartographers from centuries in the past. And mathematics doesn't just help us explore the physical world. Mathematical engines power our exploration of the Internet via search engines, it maps the human body via medical imaging, and uncovers the structure of societies via network analysis. This is a story of the exploration of the Universe as a story of mathematics, but there's even a bigger story here (even bigger than the solar system!), one about the ways in which the exploration of ideas, be they mathematical or otherwise, takes us to all kinds of surprising and wondrous places. Exploration, be it of space or thought is one of the great natural human impulses. It can lead us anywhere, often with the help of mathematics.