Which has the largest land area out of DR Congo, Greenland and Saudi Arabia? They are in order of size with Congo the largest, but all three are similar. Why don’t our maps seem to show that – read on.
Can we measure the earth’s curvature? Yes, in fact the tunnels bored for the LIGO gravity wave experiments had to take this into account. Over a metre in 4 km length!
A number line is one dimensional because you only need a single coordinate to define where you are. Two dimensional space is what we see on paper and computer screens. We now need 2 coordinates, one for the x direction and one for the y direction to define where we are. In this space we multiply in the 2 dimensions to get areas, which is a good way to see that we are in 2 dimensional space. It seems simple but we get complex very quickly again! A straight line can be displayed in 2 dimensional space as we saw with y = 5x+7, but this is still one dimensional. In maths this is called a 1 manifold in 2 space.
The surface of the earth is a 2 manifold in 3 space. The earth is 3 dimensional, but the surface is 2 dimensional. We can verify this by looking at how a point is defined on the surface of the earth. It just needs 2 coordinates – eg latitude and longitude. If you look at a globe you can run your finger around a line of longitude and find a particular latitude to uniquely define a point. The surface is 2 dimensional but not flat. The surface is a 2 dimensional sphere and the 3 dimensional shape is a spheroid which is a 3 manifold in 3 space.
We do most of our work in 2D space, so we need a mathematical way to transform or project from a 3 dimensional view to a 2 dimensional view (go from a 2 manifold in 3 space to a 2 manifold in 2 space). The most common method of this “conformable mapping” is to use a Transverse Mercator Projection. It is like peeling an orange in a number of lens shapes (or lunes) and pushing these flat.
In the standard UTM projection there are 60 of these lunes that are then flattened (mathematically) to create a 2 manifold view. In each of these zones we can now measure Eastings and Northings in metres with the zero points (0,0) at the equator and left edge of each lune / rectangle (called zones). Adelaide is in Zone 54 and, because it is in the Southern Hemisphere its northings are negative.
It is pretty obvious (but we always seem to forget) that this projection magnifies things toward the poles and therefore does not show real areas. There are many other projections that can be used each being more accurate in some ways and less in others. A soccer ball projection (made of pentagons and hexagons) will keep areas constant for example but looks strange stretched out.
Mathematically we can have as many dimensions as we want, although they are hard to visualise. In physics Einstein gave us the 4 dimensions of spacetime, which includes time as the fourth dimension. So we now have 3 dimensional space moving through time. String theory gives us 9 space dimensions and 1 time dimension while M theory gives us 10 space dimensions. These theories are attempts at better quantifying our universe but are unproven and hence only theories.
We have looked at waves regularly through these blogs as they are a fundamental way to move energy about. Light, sound, surfers and the quantum nature of matter itself all use waves to travel. So let’s look at the basics of waves.
The diagram shows a sine and a cosine wave that are 90 degrees out of phase (called quadrature phase). A simple wave has an amplitude (measured from the zero line), a phase, a wavelength (when measuring distance travelled), a period (time taken to travel one wavelength), a frequency (generally in Hertz which are cycles per second) and a velocity. A complex wave is made up of a number of simple waves.
We know that Velocity is Distance / Time (V = X/T), so we can easily calculate one property from another. It is possible to define waves in the Amplitude / Time domain by taking amplitude measurements at regular and close spaced time intervals. Alternatively we could measure the frequency at regular and close spaced distance intervals. To move between the two we use a Fourier Transform, in a similar way that we used a projection or transform to go from a sphere to a plane.
Taking amplitudes at close spaced time intervals is digital sampling and is the method we do most things these days. If we recorded the entire wave form as a continuous signal (eg by recording onto a graph) then that is an analogue signal. To record an accurate digital signal we need at least 2 samples per wavelength. Audio CDs sample at 41000 times per second (hz) to accurately play frequencies up to 20 kHz.
So we record sounds digitally by sampling at an appropriate interval and we can then manipulate the recordings digitally by working in the frequency domain instead of the time domain. This way we can erase certain frequencies that we consider as noise and boost frequencies we consider pleasing.
Complex waves can be split into their component simple waves by Fourier Analysis and transformed between domains by Fourier Transforms (or Laplace Transforms).
To work with Fourier Transforms in the frequency domain we work in what is called Complex Space (or the S Plane). Put simply it is a 2 dimensional space where one dimension is in units of the Imaginary number i and the other in real units. Complex numbers are a mix of the two eg (2i+3).
The imaginary dimension accounts for the phase differences of the individual simple sine waves.
The Fourier Transform from Frequency to Time is
This is a basic property of the mathematics and analysis of waves.
The complex plane gives us one of the fun bits of maths discovered in recent times. In the 1980s Benoit Mandelbrot came up with the idea of fractals, where patterns repeat on smaller and smaller scales and have been described as the most complex things in existence.
The Mandelbrot set is produced by iterating the complex equation
z = over and over until a limit is reached. In this equation both z and c are complex numbers (choose any 2 you like). After each iteration the old z is replaced by the new one. If you choose numbers that create a converging set and paint the various members of the set in different colours you get the beautiful patterns of the Mandelbrot set.
Check out Mandelbrot zoom in Google and you get some pretty amazing views.
These views show fractal geometries where the same sorts of shapes repeat over and over as you zoom to smaller and smaller scales. In fact it reminds me of a theory proposed by my father when I was young. He always thought that atoms were like planets and that you could zoom in and out to find the same things on different scales. He thought it pretty funny that we might be living on the equivalent of atoms with giants peering down microscopes and looking at us.
While modern physics has shown this not to be the case, there are theories that we live in some sort of fractal universe. The spiral arms in the zoomed Mandelbrot set are reminiscent of spiral galaxies in the Universe.
It’s crazy that we have to come up with imaginary numbers and mystical complex planes to describe the Universe and many of its principles. It is a complex, magic place which is full of truths coming from our imagination.
William Blake, the poet summed it all up pretty well in the first verse of Auguries of Innocence in about 1803
To see a world in a grain of sand
And a heaven in a wild flower,
Hold infinity in the palm of your hand,
And eternity in an hour.