Series turn up everywhere in the natural world, with one of the most famous being the Fibonacci series 0,1,1,2,3,5,8,13,21,34….. You probably already know that each number is the sum of the previous 2 numbers. This can be written in functional notation
(which just says that the next number is equal to the previous 2 numbers added together)
Things that are additive in biology for example follow Fibonacci, including the patterns of tree branches and leaves. Artichokes and pineapples also follow this scheme. If we put squares together in a Fibonacci series we get a “golden” rectangle with quite special properties.
The squares form a golden spiral and in the limit, the ratio of long to short side is a “magic” number called phi ψ and is the irrational number 1.61803…. (but not transcendental). Dan Brown’s character, Robert Langdon is pretty keen on the number phi in the Da Vinci Code.
The ratio is considered “magical” and has been decribed as having “divine proportions”. Artists and architects have used it for millenia to create images that we find beautiful. Da Vinci himself, based his famous Vituvian Man sketch on it. Credit cards and business cards tend to have this ratio of sides.
The brain appears to be hard wired to accept this ratio as a perfection of beauty. Beautiful faces have been defined with golden ratio proportions.
While all of this is likely to be true and may be based on biological processes being additive, we are bordering on the edge of pseudo science. There are many different spiral shapes, based on other mathematical proceses (eg logarithmic spirals). These mathematical spirals often match nature’s shapes better – including the Nautilus spiral. Similarly the pyramid shapes are only approximately golden ratio in dimensions. We do not know if they were designed with divine proportions in mind.
Mersenne numbers are another series, often used in facebook tests of “are you a genius?”. This series has each number one less than the next power of 2 ie 1,3,7,15,31… ( from 2,4,8,16,32…). In binary (where we only use 0 and 1) the series is trivial (1,11,111,1111,11111…). If you know a few basic series these tests become trivial as well!
Series and Limits
Back about 450 BC a guy by the name of Zeno came up with his Achilles and the tortoise paradox. The tortoise essentially bet Achilles that he could not beat him in a race if the tortoise had a head start of ten cubits. The tortoise argued that by the time Achilles had run 10 cubits, the tortoise would have progressed 1 cubit and still be in front, and by the time Achilles ran that 1 cubit the tortoise would have run 0.1 of a cubit ad infinitum. Sadly the tortoise was wrong. Achilles would catch up in 11 and 1/9 cubits – or the limit of the series 10+1+.1+.01…. = 11.111111….
Similarly dividing anything in halves over and over reaches a limit such that 1=1/2+1/4+1/8….
In mathematical terms we could write this as or in words – the sum of the numbers created in order, by varying 1 to an infinite limit (infinity sign is sideways 8) of 1/2n (1/2+1/4+1/8… etc) is 1. Again, that is pretty cool and thankyou Zeno for thinking this up.
It so happens that many numbers can be written as an infinite series. We saw initially that our transcendental number e is the limit of a series where
e = 1+1/1+1/(2*1)+1/(3*2*1)+…. = 1+1/1! +1/2! +1/3! +…. =
The latter is a more succinct ways of writing the same thing. The exclamation mark (or bang) is called “factorial” in maths so that 5! = 5*4*3*2*1 (use * instead of x to avoid confusion when multiplying).
Other important “series expansions” include
We can deduce the cos (x) expansion from the sin (x) expansion because we know that the derivative of sin (x) is cos(x) and the derivative of .
These expansions also lead us to Euler’s formula
= cos (x) + i sin(x)
A lot of pretty serious maths!
If you have followed it (again don’t sweat the details) we are now starting to see the relationship between e and pi and i and have taken in some insights into the mathematics of series and limits. We have not actually proved the relationships yet. We probably need to look at basic trigonometry first so we understand what sin (sine) and cos (cosine) are.
These “series expansions” are known as Taylor series (or Maclaurin series if centred on zero for the pedantic mathematicians out there). They are the basis for signal processing and describing waves (of any kind). Taylor series are infinite series. Their expansions are infinite sums from negative infinity to positive infinity.
A similar series expansion can be done for periodic series, known as the Fourier series. We get a periodic series when the series periodically gets back to zero, for example a sinusoidal wave. These are used to create very powerful mathematical tools for working with waves, known as Fourier Transforms.
We need a bit of trigonometry about here, and in reality there is not a lot to it. We define the three sides and 3 angles of a right angle triangle and give the three ratios names sine (abbreviated to sin), cosine (cos) and tangent (tan). The ratios relate to a particular angle called theta (ϑ) and the other angles are 90 degrees and 90-ϑ, giving 180 degrees in total.
Sin ϑ = opposite / hypotenuse = a/c
Cos ϑ = adjacent / hypotenuse = b/c
Tan ϑ = opposite / adjacent = a/b
Just remember SOHCATOA for sin = opposite/hypotenuse etc.
Sin, cos and tan are inter related and define a circle in four quadrants. Think of ϑ varying from 0 degrees to 90 degrees to define the first quadrant, then flipping over to the right defining the quadrant from 90 to 180 degrees, then down for 180 to 270 and bottom left for 270 to 360 (or in other words back to zero). If you move this motion along (eg in time) you will trace out a sin (or cos or tan curve).
If we rotate the diameter line anti clockwise and move steadily then the red and blue lines define harmonic sin and cos curves (of ϑ and 90-ϑ). The green line simply has a lesser amplitude. Note that sin and cos curves are 90 degrees out of phase to each other. Opposite sides of the diameter are completely out of phase and separated by 180 degrees. If you add two 180 degree out of phase waves together you get zero!
Complex signals (waves) can be separated into their various component waves and manipulated (eg remove noise). In our digital world these are defined by sampling at small intervals to create a digital representation of a wave and manipulated using FFTs (Fast Fourier Transforms). The data can be transformed into different domains (eg time / space domain to frequency / wave number domain) and filtered by deleting specific frequency ranges. Domains will be the subject of another blog as they are a very useful concept.
Tan ϑ is also harmonic but is a little more problematic as it creates “infinities” at each 90 degree mark.
These infinities are called singularities, just like a black hole. In circular motion negative and positive infinity are the same thing. All gets seriesly complex!!