## 5 Fun Facts of Mathematics

Indeed, a lot of interesting facts about real life can be derived from different fields of mathematics. Stock markets, natural phenomena and science are examples of how algebra and calculus together can explain everyday concepts.

- 1. Golden Ratio
- 2. Fibonacci Sequence
- 3. Plot a “heart” figure using Desmos
- 4. Seven Bridges of Königsberg
- 5. Trigonometry & Musical Notes

The Golden Ratio has been heralded as the most beautiful ratio in art and architecture for centuries. However, do you know what is the exact value of the golden ratio? For IB students who have already mastered quadratic equation, the value of the golden ratio is actually the positive numerical solution to the equation x2-x-1=0 and is approximately given by x=1.618034… The Mona Lisa painting by Leonardo Da Vinci , for instance, is drawn according to this golden ratio of 1 : 0.618… and is said to be aesthetically pleasing to her admiring audiences . In fact, you can even measure how close your face is to that golden number.

The spiral shapes of sunflowers and other patterns in nature follow a Fibonacci sequence, where adding the two preceding numbers in the sequence gives you the next (1, 1, 2, 3, 5, 8, 13, 21, etc.)

Most IB and advanced math students regularly use a GDC (Graphical – Displaying Calculator) to plot a graph out when doing exercises. Yet, do you know that you can also sketch a heart figure with different orientation on a graphing software like GDC or Desmos?

Let’s try this little experiment at home now!

Can you generalise the effect of these numbers to the “heart” that we have plotted?

The city of Königsberg in Prussia (now Kaliningrad, Russia) was set on both sides of the Pregel River, and included two large islands —Kneiphof and Lomse —which were connected to each other, or to the two mainland portions of the city, by seven bridges.

Now, repeat the same experiment again for the following bridge problem. Is it possible to find one particular path now?

Try to create your own version of the bridge problem with a different number of edges and vertices. How does the number of vertices and edges affect the possibility of finding a particular path that passes through all the bridges (edges) once and only once? Discuss any finding(s) with your friends!

I know that it’s not very plausible to say that music theory on different instruments and musical notes can be explained by maths (as well as physics). However, the maths knowledge required is quite advanced – even beyond the scope of IB higher level! Of course, I am not going to write you a bunch of scary formulae and ask you to “understand” them. Instead, I will introduce some really easy principles of music that are connected to maths.

A particular musical note (e.g. middle C) when played on a piano and a violin sound similar in some ways, yet different in others. Two different notes played on the violin also have similarities and differences. How do you hear which note is being played, and how do you know what instrument you’re listening to?

When you play different instruments with different notes, we can always record the air pressure produced by the sound against time and obtain a graph similar to the one below:

If you use different instruments to play different notes, we will obtain a unique and yet different graph every single time (which can be likened to a signature of the instrument).

I will try to convince you that the trace produced is indeed a function that consists of a bunch of trigonometric functions (e.g. sine, cosine functions) with the following “unpleasant” presentation.