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Modelling the length of Silverstone Grand Prix Circuit

Modelling the length of Silverstone Grand Prix Circuit

Even within the esoteric world of mathematics, IB Higher Level Maths is a highly specialised subject. Maths is the hardest HL subject within the IB Diploma with only about 9% of IB Diploma students taking it each year. Taunton School has a history of gaining good grades in HL Maths and occasionally some students go on to HL Further Maths as a seventh subject.

This year all four projects from the HL Maths students were awarded high grades with two being awarded full marks, which has never happened before. Much of the mathematical content within the projects was an extension to the normal HL syllabus. Herewith follows a summary of the two excellent projects that achieved full marks:

David Blrtsyan – ‘Modelling the length of Silverstone Grand Prix Circuit’
David tackled this project by first teaching himself the mathematics behind finding the length of a curve using advanced integration. He then needed to obtain the co-ordinates of the actual Silverstone race track from a map of the circuit. He achieved this by using advanced online mapping software. Fitting accurate curves to the map was the next challenge. David taught himself hyperbolic functions in order to match some of the tighter corners on the circuit. He also used other complex compound functions such as lnsin(ax b). For accuracy he split the corners into sections, each with a different function. Finally he then had the onerous task of integrating every function in order to find the length of every piece of the circuit. His final result was within 0.13% of the actual length of the middle of the road around the actual 5.9km circuit.

 

Alessandro Rossi Polvara – ‘Analysing the tetration of the natural logarithm function, lnx’
Tetration is the fourth hyperoperation in mathematics after addition (a a a…….), multiplication (a x a x a……), exponentiation or powers of a (a, which is a1, a2, a3…………….an). Tetration is ‘a’ to the power of ‘a’ to the power of ‘a’ any number of times (a, aa………). It is used for the notation of extremely large numbers.

Not satisfied with investigating the tetration of just ‘x’, Alessandro investigated the tetration of the function lnx. He started his investigation by graphing the first four tetrations of lnx. In doing so he discovered an interesting fact that, whilst the odd number tetrations graphs were continually increasing functions, the odd number tetrations had minimum points that seemed to shift towards the point (1,1). He then decided to investigate these minima. In order to do so he had to carry some complex differentiation of the different tetration functions. In addition to the application of limit theory to define each tetration at the point where x=1, he found a pattern within the differentials and was able to provide the general term for the differential of the nth tetration of lnx.

The other two students in the group were Valentin Pechnikov and Giovanni Ometto. Valentin’s project was on the ‘Use of Elliptic Curves in Cryptography’. An elliptic curve is with the general equation ‘y2 = x3 ax2 b’. In modern cryptography the use of elliptic curves creates an almost impossible code to crack using private and public keys. Valentin gave examples of the use of the chord-tangent method to create a code. He was awarded a top grade 7.
Giovanni’s project looked at the ‘Use of Fourier Analysis Approximations in the Compression of Digital MP3 Music Files’. He first showed how Fourier analysis could be used to approximate compound polynomial curves. Having then represented a true sound wave from a piece of music as compound function of multiple sin(ax) and cos(bx) functions, he showed how the large files of the actual sound wave could be compressed into an MP3 file though careful approximations of the original functions with very little loss of clarity. His final graph overlapping the MP3 approximated function on top of the full function showed how similar the MP3 compressed file was to the actual sound wave. Giovanni was awarded a high grade 6 for his project.

The above work by these talented students gives an insight into how the IB Diploma course develops independent learning. This gives IB students a head start as they enter university and start studying for a degree.

Clive Large, Teacher of Mathematics