Independent Learning Assignment STEM

Can you Ever Truly Mix a Cuppa? – Spilling the Tea on Brouwer’s Fixed Point Theorem

This essay was written by upper-sixth former Ben Watkins, and a finalist for the 2020 Independent Learning Assignment. The following provides a short abstract to the full essay, which can be found at the bottom.

Estimated read time of abstract: 1 minute
Estimated read time of essay: 15 minutes

Is it possible that there are always two places on earth with the same temperature and pressure? How does the game show Blockbusters have any implications on algebraic topology? Can a general equilibrium ever be reached in an economy? Perhaps most crucially of all, can you ever truly mix a cup of tea? 

My ILA provides insight into Brouwer’s fixed point theorem, a theorem found in the field of algebraic topology. It uncovers how a remarkable and seemingly counterintuitive result in what is often considered to be an abstract field of mathematics can have such broad and pertinent results in the real world. However, this isn’t to say that this ILA doesn’t uncover the result of this theorem for the sake of the beauty of it as much as uncovering it for the sake of its applications. Indeed, Luitzen Egbertus Jan Brouwer himself (the discoverer of this theorem as well as often being called ‘the Father of Topology’) was very much an upholder of this mentality: that maths has great importance for the sake of maths itself. Philosophically, Brouwer was a neo-intuitionist, which means that he thought of mathematics as purely a mental phenomenon, the result of constructive mental activity rather than uncovering any principles of an objective reality. He is often quoted in saying that “The construction itself is an art, its application to the world an evil parasite.” 

To view Ben’s full article, follow this link below.

Independent Learning Assignment Law & Politics Social Sciences

Pot-Luck Politicians, A Parliamentary Chamber from Sortition

This essay was written by upper-sixth former Tom Welsh, and a finalist for the 2020 Independent Learning Assignment. The following provides a short abstract to the full essay, which can be found at the bottom.

Estimated read time of abstract: 2 minutes
Estimated read time of essay: 1 hour

Sortition is the random selection of individuals, and in this paper I sought to understand its political application in legislatures before providing a potential application in the United Kingdom via a third House of Parliament. The motive being sortition’s inherent equality and true representativeness. 

In order to do so, an investigation of sortition’s use in history was made, before its role in political theory was considered. I then briefly looked at its current application in both juries and citizens’ assemblies, before considering sortition’s hypothetical applications and existing use outside of the United Kingdom. 

Once the theory was covered, I then took to explaining the functioning of the existing UK government before looking at current UK political participation in both formal methods (elections and referenda) and informal (social movements and pressure groups). 

Having discussing the underlying theory, and the use-case it was being applied to, a substantial portion of my paper attempted to outline a comprehensive description of why I believe sortition’s best application would be as an addition to the existing Parliament given the important role that both the existing Houses of Parliament play. That is not to say that I felt the chamber from sortition would have a small role to play – far from it. 

I then ended the paper with an overview of some of the potential issues that such an implementation might entail, with an attempt also being made to explain how they might be dealt with and why if they cannot, on balance, that is of little concern in any case given the many positives associated with sortition. Furthermore, if you have an interest in either Plato or Jean-Jacques Rousseau, the absolute ending of the paper comes in the form of a synthesis of both, achievable, in my opinion, through sortition. 

If I gone some way to perhaps intriguing you about my, perhaps controversial, proposition then do consider giving my ILA a read. In fact, even if I haven’t – undoubtedly it is not easy in a short abstract to fully convey the true nature of a piece of work – maybe consider giving it a read in any case. A word of warning though, perhaps read the paper one chapter at a time, as I apologise it is not exactly short. Nonetheless, if you do choose to read the full work, thank you. Yet more importantly, I hope I cause you, even if you disagree with my conclusions, to reconsider your own political views – indeed if you don’t already have any, that is fine too! 

To view Tom’s full article, follow this link below.

Arts & Humanities Independent Learning Assignment Linguistics

Why have there been changes in the phonetics and phonology of Khmer since the existence of Proto-Austroasiatic?

This essay was written by upper-sixth former Tom Hemsworth, and shortlisted for the 2020 Independent Learning Assignment. The following provides a short abstract to the full essay, which can be found at the bottom.

Estimated read time of abstract: 4 minutes
Estimated read time of essay: 30 minutes

Khmer is the language spoken mainly in the Kingdom of Cambodia, in South-East Asia. Khmer is part of the Austroasiatic language family, which encompasses 168 languages spoken across South-East Asia. The origin of this family is the language Proto-Austroasiatic which is believed to have originated in southern China. From this proto-language, several other language branches began to evolve from around 5000 BC. Modern Khmer is derived from the Khmeric branch, which appeared around 2000 BC. This too is a proto-language, called Proto-Khmeric. 

Around 600, the first stone inscriptions were recorded at the time that a group of kingdoms called the Chenla Empire occupied what is now Cambodia. This marked the birth of a new period in the Khmeric branch: Old Khmer. After the fall of the Khmer Empire in 1431, the language once again shifted to Middle Khmer, which spanned until around 1800, when Modern Khmer emerged. 

Upon examining the phonetics of Proto-Austroasiatic and Proto-Khmeric, I began to notice that there were not too many differences between them. This was surprising, as when you inspect the Swadesh 100 list, a list of one hundred basic words used for historical and comparative linguistics, the transition between the two columns is visibly the largest between any two consecutive word lists. The question is: why do the phonetics differ only slightly whereas the vocabulary is in many cases quite dissimilar? 

When reconstructing pAA, Paul Sidwell and Felix Rau used 21 consonants which were all carried through to pK. On the other hand, there are several vowel changes including the appearance of the long vowel sound [*ɛː], a particularly odd case. On the Swadesh list, most pK words in which [*ɛː] is present are unrelated to their pAA predecessors, which leads me to think that these are either new terms coined by people as they migrated from the Austroasiatic homeland in China into Cambodia and other South-East Asian countries, or if they are related, merely slight differences in pronunciation, such as between the pAA *cgəj and the pK *ckɛː (‘dog’). This point about migration brings me back to the question I posed. As Figure 1 indicates, when the Austroasiatic language family broke off into branches, its speakers either travelled west towards India or south into Cambodia and its neighbours. Many general terms will have only gone in one direction. Examples of this can particularly be found in words relating to flora and fauna. Take the pAA word *draŋ (‘horn’), which does not relate to the pK *sneːŋ. Words that derive from the pAA term can be found in Munda languages, for example dereŋ in Santali, spoken in Northeast India. 

Next in the timeline is the transition out of proto-language to Old Khmer. The basic consonant phonemes do not change from pK to OK; however, the voiceless stops [t], [p], [c] and [k] also began to occur with aspiration in OK, purely because it is a lot easier to say a word like *pkaːj with aspiration as pʰkaːj. As well as the phoneme [ɗ] and its bilabial equivalent [ɓ] appearing due to the glottis in the vocal cords being lowered and narrowed, vowel sounds became simpler in Angkorian society. The diphthong [*ie] evolved into [iə] or [iː] and [*uo] underwent similar alterations: either to [uə] or [uː]. 

However, there are certainly more apparent evolutions in language from OK to the modern day. Firstly, devoicing occurred. This is the term describing when the voiced stops in a language become voiceless. The reasons behind devoicing can be discovered by examining other languages in South-East Asia, such as the Tai family. The Chinese linguist Li Fang-Kuei wrote that “voiced consonants have become devoiced in practically all dialects” of Tai languages. Since the fall of Angkor was at the hands of the Ayutthaya, a kingdom which was located in modern day Thailand, I have deduced that the influence of this kingdom is a key reason behind devoicing. Devoicing did not occur in Thai until after the 15th century, so I believe that this process started thereafter and thus influenced the Khmer language.  

Another change was registrogenesis, the development of separate ways of speaking in a language. Two levels of lexical register occur in Khmer: ‘Head Register’ for formal language and ‘Chest Register’ for informal language. Chest Register, characterised by a breathier voice caused by the lower pitch from the larynx, was present in OK and MK. The low voice characteristic remains in standard Khmer, but it has lost its breathiness. 

Devoicing and the registrogenesis of Khmer are the reasons why, between OK and modern Khmer, new vowel sounds developed. For the first time, the vowels [ɨ], [ɨː], [ɑ], and [ɑː] appeared regularly in Khmer, as well as new short diphthongs. There are three of these: [ĕə], [ŏə] and [ŭə] (the diacritic ˘ represents an “extra-short” vowel). The open front unrounded vowel [a] is unable to be lowered any further therefore when people speak with a lower pitch, short diphthongs are born. 

During the French control of Cambodia, Khmer picked up loan words, which included new phonemes which had never been part of the language before, including [f] as in tiːfoŋ (‘typhoon’), and [z] as in zaːmaːʔiːk (‘Jamaican’). After the departure of the French, the future of Khmer looked set: the language was in the safe hands of Buddhist monks, including Chuon Nath, who compiled the first Khmer Dictionary. They removed many French and Siamese words from Khmer, a reason why French is no longer used as much in Cambodia. The Khmer Rouge halted this ‘Khmerisation’ process, particularly in education, which was greatly suppressed. Chuon Nath’s dictionary was only brought back in 2009, but at this point, Cambodian society had been partially taken over by English – virtually every sign in Cambodia now has both Khmer and English writing on it. 

The future may seem menacing for Khmer, but despite the worry of Khmer being dominated by English, I see no reason that it will disappear entirely. The influence of the modern, business- and technology-rich world will surely produce many new phonetical and phonological changes in the future, and it will undoubtedly prove to be a fascinating area of study. 

(Appendices showing timeline, phonemic inventories and Swadesh 100 list can be found in the full version of the ILA)  


pAA – Proto-Austroasiatic 

pK – Proto-Khmeric 

OK – Old Khmer 

MK – Middle Khmer 

To view Tom’s full article, follow this link below.


Diffloth, G., 2005. The Contribution of Linguistic Paleontology to the Homeland of Austro-Asiatic. In: L. Sagart, R. Blench & A. Sanchez-Mazas, eds. The Peopling of East Asia. Oxford: RoutledgeCurzon. 

Ray, N. & Lee, J., 2016. Cambodia. 10th ed. Singapore: Lonely Planet Publications Pty Ltd. 

Sidwell, P., 2018. Austroasiatic Studies. Hsinchu, National Tsing Hua University. 

Ferlus, M., 1992. Essai de phonétique historique du khmer. Mon-Khmer Studies, Volume 20. (French). 

Li, F.-K., 1977. A handbook of comparative Tai. Manoa: University Press of Hawaii. 

Shorto, H. L., 1962. A Dictionary of Modern Spoken Mon. London: Oxford University Press. 

Henderson, E. J., 1952. The Main Features of Cambodian Pronunciation. Bulletin of the School of Oriental and African Studies, 14(1). 

Wayland, R. P. & Jongman, A., 2002. Registrogenesis in Khmer: A phonetic account. The Mon-Khmer Studies Journal, Volume 32. 

International Phonetic Association, 1999. Handbook of the International Phonetic Association. 1st ed. Cambridge: Cambridge University Press. 

Sasagawa, H., 2015. The Establishment of the National Language in Twentieth-Century Cambodia: Debates on Orthography and Coinage. Southeast Asian Studies. 

Sok, K., 1999. La khmérisation de l’enseignement et l’indépendance culturelle au Cambodge. Bulletin de l’École française d’Extrême-Orient, Volume 86. 

SEAlang, n.d. Khmer Dictionary and Dictionary of Old Khmer[Online] 

Headley, R. K., 1998. Cham Evidence for Khmer Sound Changes. Papers in Southeast Asian Linguistics, Volume 15. 

Jenner, P. N., 1974. The Development of the Registers in Standard Khmer. South-east Asian Linguistic Studies, Volume 1. 

Independent Learning Assignment STEM

Supramolecular Cages: Their Design, Chemistry and Applications

This essay was written by upper-sixth former Salvatore Nigrelli, and was the winner of the STEM category for the 2020 Independent Learning Assignment. The following provides a short abstract to the full essay, which can be found at the bottom.

Estimated read time of abstract: 4 minutes
Estimated read time of essay: 1 hour 15 minutes

Supramolecular chemistry is all about making functional molecular assemblies without chemically bonding the component molecules together. Take the reaction scheme below: 

Figure 1 The Formation of a Tennis Ball Capsule

In this scheme, two molecules are held together only using hydrogen bonds, to form a tennis ball shaped capsule. So it is the art of the supramolecular chemist to try and find innovative ways of making complex assemblies, with only a handful of intermolecular forces at their disposal. This is particularly shown in a type of structure called supramolecular cages. 

Cages are ubiquitous throughout the world of chemistry. The Buckyball (Figure 2, is a simple type of molecular cage, consisting of 60 carbon atoms in a spherical shell arrangement.  

Figure 2 The Structure of a Buckyball

Supramolecular cages take this idea one step further and ask the question: Can we design assemblies that allow us to put a small molecule ‘prisoner’ inside the cage?  

Take the Buckyball again. Using a type of complex reaction sequence called a molecular surgery reaction, it is possible to open the Buckyball up and place a water molecule inside, held in by the London forces it can form with the cage walls, transforming the Buckyball into an exciting supramolecular cage (Figure 3).  

Figure 3 A Buckyball with an Encapsulated Water Molecule Inside

But you may ask, is there actually any point in making these tiny molecular prisons, or is it purely to indulge a few curious supramolecular chemists? The answer is that, although the field is relatively new, it is becoming paramount that the applications of supramolecular cages are innumerable, from security to chemical analysis, and even cancer therapy.  

Cyclobutadiene is a pesky, annoying molecule – mainly because it reacts with itself extremely quickly in a dimerisation/isomerisation reaction to produce cyclooctatetraene: 

Figure 4 The Dimerisation/Isomerisation Reaction of Cyclobutadiene

This self-reacting property of cyclobutadiene makes it extremely difficult to probe its chemical structure. Until only a few years ago, the only way that it had been achieved was by holding the molecules in an argon matrix close to absolute zero. However, with the advent of supramolecular cages, all of this changed. If you make a single cyclobutadiene molecule inside a type of supramolecular cage called a carcerand (Figure 5), no other molecules can get to it, so it stays in its original, undimerised form, and can be analysed using NMR spectroscopy. This is a classic example of how supramolecular cages are already revolutionising the field of chemistry.  

Figure 5 An Example of a Carcerand

However, not all of the uses of supramolecular cages lie in a lab. The cage below is one such cage with extremely promising applications. In the presence of picric acid molecules, the cage can encapsulate one of them. Once the picric acid is inside the cage, it is close enough for a type of photochemical process called a Förster Resonance Energy Transfer to take place, which causes the cage to completely change colour. This is incredibly useful because picric acid is one of the most common explosives, so cages like these could be used in the next generation of fast, accurate explosives detectors and save countless lives.  

Figure 6 A Supramolecular Cage to be Used in Explosives Detection

Looking into the monumental applications of supramolecular cages got me thinking – could I use the skills in supramolecular chemistry that I picked up over the course of my ILA and design a novel type of supramolecular cage to solve a real world problem? I decided to try and solve the problem of fluorouracil as a chemotherapy drug. The essence of the issue is that fluorouracil is an extremely promising cancer drug, but it also readily attacks brain tissue, and so its use is limited to very extreme cases. This appealed to me as a problem to solve because of the great positive impact that it would have, and the fact that fluorouracil (Figure 7) has a number of structural features that make it very attractive to supramolecular chemists.  

Figure 7 The Structure of Fluorouracil

Fluorouracil can form three strong interactions with its fluorine and two nitrogen atoms that would allow it to be readily encapsulated. So if a cage with three parallel bars could be designed, it would strongly bind the fluorouracil molecule. However, finding a chemical arrangement that allows this to take place proved to be difficult, as it is a rare occurrence in chemistry. In the end, I managed to work out that if the cage used molybdenum centres with thiophene ligands, a trigonal prismatic arrangement around the molybdenum atom would be obtained (Figure 8) , making the three parallel bars possible. 

Figure 8 A Molybdenum-Thiophene Complex Showing a Trigonal Prismatic Geometry

The key principle behind the cage is that the fluorouracil molecule stays inside the cage, so it cannot react with anything, until it enters the cancer cells, where the cages are opened, thereby allowing the fluorouracil to kill only the intended cancer cells. This targeted opening is a rather unusual feature of molecules, and so in the end I used the fact that, if nitrophenyl ether groups (Figure 9) were placed on each bar, in the presence of a targeted beam of UV light, the cage could be successfully opened once inside the intended cancer cells.  

Figure 9 A Nitrophenyl Ether Group

My final cage design is shown below: 

Figure 10 My Final Cage Design

To summarise how my design works: Outside of the body, fluorouracil is encapsulated inside the cage, then a solution containing the encapsulated fluorouracil is injected into the patient’s bloodstream. Whilst inside the cage, no other molecules can get to the fluorouracil so it cannot react with anything and cause its bad side effects. Once the cage reaches the cancerous cells, using UV light, the cages in the cancer cells are opened, releasing the fluorouracil and killing only the intended cells. Therefore, this scheme allows for fluorouracil to be used to treat cancer patients, without causing any negative side effects.  

To view Salv’s full article, follow this link below.

Independent Learning Assignment Psychology Social Sciences

How does the inescapable conclusion of death influence people’s lives?

This essay was written by upper-sixth former Joseph Gibson, and shortlisted for the 2020 Independent Learning Assignment. The following provides a short abstract to the full essay, which can be found at the bottom.

Estimated read time of abstract: <1 minute
Estimated read time of essay: 10 minutes

In this essay, I seek to understand how knowledge of mortality impacts the way people lead their lives. From a psychological perspective, I engage with our biological, societal, cultural, social and professional response to this question. I look at when and why people come to grips with the reality of their inevitable conclusion and at what point in people’s lives they should begin to think about mortality and how to be able to do so in a constructive and healthy manner.

Mortality affects our mentality, behaviour, decision making and overall mental health in both positive and negative ways, but by teaching future generations how to approach contemplating this issue at an early and suitable age in an appropriate environment, we can prompt positive responses such as motivating people to accomplish their aspirations and leave a legacy and lasting impact on the world. Through referenced research, various studies and published psychological papers, it has become clear that living in denial of one’s own mortality is ultimately what leads to negative responses such as various forms of trauma, anxiety and depression.

To view Joe’s full article, follow this link below.