Quantum Leaps (eBook)

CHAPTER 1

Where Are We Now?

The Maths of Location and Navigation

A mathematical model is, essentially, an approximation of the real world, which allows us to use mathematical tools to analyse real-world events. From antiquity, mathematicians have developed methods to try to generate better and better estimates and representations of the real world. Think of how the ancient Greeks found ways to estimate the area of a circle, the volume of a sphere, or the square root of 2, long before modern mathematics provided us with the concept of irrational numbers and calculus allowed us to deal with curved objects more accurately. Mathematical models today underpin everything from our attempts to predict the weather to space travel. The maths used to identify our location in the world is a good place to start exploring the concept of mathematical modelling.

Navigating our Environment

How do we know where we are in the world? Cognitive maps have been studied in a variety of ways over the decades, but in 2005 the neuroscientists May-Britt and Edvard Moser built on earlier work by John O’Keefe to show that there is something similar to a GPS system in the brain. In a study of rats, they identified ‘grid cells’ which would fire according to the specific location the rats found themselves in. This suggests that there is effectively a grid in the brain that matches up to the real world: obviously the cells aren’t actually laid out in a grid, but particular neurons do respond to particular places within an environment.

It has been known for millennia that there is an association between memory and location; one of the earliest mnemonic devices, the method of loci (or memory palace), allowed users to retain large amounts of information by associating them with rooms or spaces in a building well known to the mnemonist. But progress in the way mathematics dealt with location had more or less stopped after the creation of maps and measurement of distances, directions and elevations. It took the genius of René Descartes to find a clearer mathematical definition of our location in space.

The story is that he was idly watching a fly wander around on his ceiling and wondering how best to describe its location at a given time. He realized that he could use one corner of the ceiling as a reference point and use the distance along each of the two adjacent walls to create an exact reference for location.

This led to his creation of Cartesian coordinates, which are the underlying principle of graphs, with the corner of the room becoming the origin and the two adjacent walls the axes (which can also be continued indefinitely in either direction). The rats in the experiment described above were essentially using a rudimentary xy-coordinate system. The addition of a z axis, perpendicular to the other two, allows for an accurate description of location within a three-dimensional space. In addition, once we start mapping out equations on the graph, we can create paths and shapes that occupy that space. For instance, on a two-dimensional graph, x2 + y2 = r2 will draw a circle with its centre on the origin and a radius of r. Below is the graph of x2 + y2 = 52. Every point on the circle gives a pair of values x and y that are a solution to the equation. The only solution in which both x and y are positive whole numbers is the Pythagorean triple, x = 3, y = 4, r = 5. (A Pythagorean triple is a set of three whole numbers that can form the lengths of the sides of a right-angled triangle.)

The Satnav World

The concept of Cartesian coordinates is fundamental when it comes to defining the location of a person or vehicle using GPS. The history of automobile navigation devices takes us on a weird and wonderful journey through some bizarre devices. The earliest was patented in 1909. It was called the Jones Live-Map and involved a turntable with a pointer on it, which measured both direction and, through a cable connected to the wheels, distance. You could place a paper disc on the turntable with directions for a single route; the idea was that the disc would turn as you travelled so that an arrow lined up with the current step in the directions, but it was pretty ineffective, especially on bumpy roads, which were ubiquitous at the time. A variety of strange mechanical devices followed, all featuring modifications on the basic idea, but none of them worked well.

There was a bit of a leap forward in 1981 when Honda introduced the Electro Gyrocator as an extra feature you could choose in an Accord car. It was computerized, so not easily jogged out of position by uneven roads. However, as with the Jones Live-Map, it was a single route device which responded to distance and direction, and which would quickly start giving erroneous instructions if you altered your route in any way. Toyota followed, with a similar CD-ROM navigation system in 1987.

Meanwhile the ingredients for the modern satnav were starting to be assembled: the US military had been developing the GPS (Global Positioning System) since the 1960s, but it was made available for the public and corporations to use by President Reagan in the 1980s. The essence of GPS is that signals are exchanged with several satellites, which send out information on their position. By continually interpreting this data, and comparing different satellites’ information, it is possible to come to a highly accurate definition of your position on the planet.

A small British firm, Nextbase, set up by early home PC enthusiasts created the AutoRoute journey planner, a digitized road map of Britain, in 1988. This was the tech that was required to map actual routes along the roads, and to measure their distances (as opposed to the distance as the crow flies).

Finally, in 1990, the Mazda Eunos Cosmo became the first car with built-in GPS. And from there the satnavs and navigation systems have multiplied until today, when it isn’t uncommon in the city to walk into a teenager who, having grown up with a mobile phone, is only able to find their way by staring at a screen rather than looking around them and working out where they are.

The maths is of course rooted in Cartesian coordinates. You can define a place in the world using an xyz grid, but there are some complexities that lead to alternative geographic coordinate systems being used. One common choice of coordinates is latitude, longitude and elevation.

Specifying latitude and longitude requires a map projection. This is the means by which a curved surface is accurately rendered on a plane (a flat surface). The distortions this can cause lead most satnav systems to use a system called Universal Transverse Mercator (UTM) rather than relying purely on latitude and longitude (although some do continue to use the latter).

The UTM splits the planet into sixty latitudinal zones, which are generally six degrees across, and projects each of these onto a plane. Imagine an orange neatly sliced into sixty segments: effectively, if you tried to flatten out the peel from each segment, you would get something close to the UTM. The location is thus defined by the zone number plus x and y coordinates within that zone. Specifying a location means specifying the zone and the (x, y ) coordinates in that zone.

Because the lines of latitude converge (get closer together) towards the poles, where they all meet, every UTM zone is wider at the centre than at the top and bottom. The zones are about 666,000 metres across at the point where the central line of latitude crosses the equator. Then they narrow to approximately 115,000 metres across at 80° South and 70,000 metres at 84° North. (Alternative polar navigation systems are used beyond that point, but the GPS systems used in the average car are unlikely to be called upon to navigate their owners on a journey to the poles.)

How To Eat Pizza the Gaussian Way

The problem of projecting a map of a globe-shaped planet onto a flat plane is an old one with many imperfect solutions. However you attempt to do it, you end up with some areas being stretched while others are shrunk. Look at this map of the world.

You wouldn’t know from looking at this map that Greenland is only one-eighth the size of South America, or that Africa is over twice the size of Antarctica: the land in the polar regions has been increased the most while the land closer to the equator has been shrunk in order to come up with a map that makes sense in two dimensions.

We know there isn’t a perfect solution to this problem because the great mathematician Carl Friedrich Gauss proved it in his Theorem Egregium (Remarkable Theorem). Here’s a rough explanation of how it works. Imagine taking a rectangular piece of paper and rolling it into a cylinder. It looks curved, right? The challenge Gauss set himself was to define the curvature of a surface in such a way that bending the surface doesn’t actually change the curvature.

The line b is ‘flat’, c is more ‘curved’ than b, and a is the most ‘curved’.

To make sense of that slightly odd statement, imagine that the fly from Descartes’ ceiling lands on the exterior surface of the cylinder. There are various paths that it could take, which would involve varying levels of curvature. It can take a circular route horizontally around the...