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So the story here has a little bit of history behind it: the naive quark model was invented, by Gell-Mann, Zweig and Nambu after many more baryons and mesons were known to exist than just the neutron, the proton and the three pions, having respectively spin 1/2, and spin 0. In fact a whole zoo of particle resonances was known that resembled the lowest lying - in energy - baryons and mesons, but which existed at somewhat higher masses and which had whole set of very complicated decays.All these new mesons and baryons were discovered once accelerator energies were high enough to produce the next higher lying analogs to the proton and neutron, the baryon octet, which includes [math]p,n,\Sigma^+, (\Sigma^0,\Lambda^0), \Sigma^-,\Xi^-,\Xi^0[/math] baryons, all having spin 1/2. Likewise there was a meson octet of spin 0, pseudoscalar mesons including the pions, but also the Kaons and the [math]\eta[/math] meson. Actually there really exists a meson nonet of rather low mass pseudoscalar mesons, but that is a complication that we can ignore for right now.In addition, a number of spin 3/2 baryons were known to exist, as well as spin 1/2 baryons. First discovered were the delta resonances, but others exist.Gell-Mann and Ne’eman first made an effort, in the early days, to organize all of these meson and baryon resonances and speak about their interactions and decays and of course also to calculate them, using representations of [math]SU(3),[/math] this group being strongly suggested by the occurrence of octets of clearly related mesons and baryons, and at first they were not thought of as necessarily being composites of quarks, or having substructure. Instead a lot of work was done to understand them using current algebra, which had already been invented. Generalizations of soft pion theorems were proved, and a lot of people gave up on field theory completely and started to think in terms of Regge theory, and the general symmetries of the S-matrix.However, if you had eventually come to believe, like Gell-Mann, Zweig and Nambu, that these particles could all be described as some sort of bound states of “smaller” particles, that they had some kind of substructure based on spin 1/2 particles called quarks, then to describe the octets, it was clear that [math]u[/math] and [math]d[/math] quarks would not be enough. You needed to have an [math]s[/math] quark also: to explain what was observed you needed at least three quarks falling into a fundamental representation of [math]SU(3).[/math]Nambu actually called the quarks [math]n,p,\Lambda[/math] in his papers. But the point is the same. The properties of the known particle resonances suggested that an approximate [math]SU(3)[/math] symmetry played a significant role. This all came under the rubric the “Eight Fold Way”.So if you believed in these quarks, that they were real, then some immediate predictions followed if there were three of them. In particular the spin 3/2 baryons were expected to fall into a decuplet representation. And in that decuplet was an undiscovered baryon called the [math]\Omega^-[/math] which had quark content [math]s,s,s[/math].So it consisted of three strange quarks. Now, you can easily see that the [math]\Omega^-[/math] presents a problem if there is no other quantum number for the quarks. All three quarks are identical, the state is maximally stretched in [math]SU(3)[/math],[math] [/math]and it is maximally stretched in spin as well. Its wave function will be totally symmetric, under exchange, if there are no orbital angular momentum or radial excitations of the quarks involved, which is certainly plausible for the lowest lying spin 3/2 baryons, and certainly the most natural model. This will be true for spin 3/2 baryons which consist of [math]u,u,u[/math] or [math]d,d,d[/math] as well.So the Pauli principle would forbid that baryon, if it has the simplest possible wave function, and it would appear very unnatural that it would be low lying if it didn’t have the simplest possible wave function. But it is nevertheless a prediction of the naive quark model with three flavors, and it is similar for the [math]\Delta[/math] resonances, though the requirements imposed on the quark wave function by the exchange symmetry required by the Pauli principle are a bit more difficult to write out in detail. They involve mixed exchange symmetry for spin and isospin.But in the case of the [math]\Omega^-[/math] it is obvious there is a problem, unless you allow for the quarks to have another degree of freedom, besides spin and isospin, since the most natural wave function is totally symmetric in spin and flavor. This third degree of freedom takes up all the anti symmetry under exchange, and if you include it then the entire spin 3/2 baryon decuplet also works out naturally. It is called color and it is a new [math]SU(3)[/math] symmetry. The antisymmetric color wavefunction is easily constructed as a determinant of the three fundamental colors.The [math]\Omega^-[/math] was eventually discovered and its mass was roughly predicted in advance of the discovery. It was discovered by a team led by Nick Samios at BNL, and that was an experiment that probably should have resulted in a Nobel Prize.This was the beginning of a return from a long foray into S-matrix theory and string theories of low energy strong interactions, to a theory of strong interactions based on quantum field theory.So the long answer is that there were many baryon and meson resonances in need of classification, not just the proton and neutron, and that you needed at least three quarks to begin to classify them. Technically, had it been only the proton and neutron, you could have gotten away without another degree of freedom in the naive quark model. But the existence of spin 3/2 baryons and eventually the proof of the existence of the [math]\Omega^-[/math] really started to clinch the case for another degree of freedom.These other low lying baryon states were quite unnatural to explain in the naive quark model, without the addition of a color degree of freedom and it turned out that the Pauli principle required at least three colors to most naturally explain the spin 3/2 baryon decuplet.There are other predictions, many other predictions that flow from the existence of three color degrees of freedom, but I will not go into those here. Those were explored and developed throughout the 1960’s and 1970’s.

Choose something you are passionate about.You will be working on your Math IA for a couple of months, it would be better if you chose something that actually interested you, than just for the sake of doing it. This is were the “Personal Engagement” criteria actually reflects in your IA. It is quite easy to catch, if your IA topic genuinely interests you or you are just completing you IA task. Therefore, I would highly recommend choosing something that interests you.Make sure language is clear, precise and concise. The IA must be written in such a manner that it is easily understood by your peers, anyone your age group must be able to read and understand it. Choosing a topic too complex can sometimes lead to a messy and hard to understand IA, so try avoid doing that. Also, avoid writing long stories on your personal experience. This is the “Communication” criteria of the math IA.Be sure to use proper mathematical notation/symbols throughout your IA report. Format all mathematical symbols properly, use MathType or a similar kind of software for mathematical expressions. This adds to clarity and will get you the easy points in the “Mathematical Presentation” criteria.Another thing, IB asks you to do in your IA is to reflect on your findings in the report. Comment (thoughtfully) but avoid paraphrasing the results, write about some of the insights you have gained out of the mathematical result of IA. The more in-depth your reflection, the more points you’ll earn in the “Reflection” criteria.Finally, I would strongly recommend that you choose a topic of some complexity but not completely foreign to you. High level of maths will earn you points in “Use of Mathematics” criteria. This might seem contradictory to second point, but try and maintain a balance between the two. Choose a topic you can explain easily yet makes use of high level mathematics.Following the above five points, should help you score highly in all the criterions of the Math IA rubric[1].Read my answers to these, if you need help finding topics.What should I do for my IB math exploration? I need a topic as soon as possible.What are some good ideas for a IB HL Math Internal Assessment?Good luck finding a topic!Happy exploring.Footnotes[1] https://mathteachme.files.wordpress.com/2013/08/hl-int-assessment-criteria-1gvshpj.pdf

Good question.It calls into question what mysticism is, and how it was/is represented throughout the world. It also asks what is meant by “the nature of reality”.There are many answers to this question. I will just consider Western mysticism in a very limited way. I will stay away from Ultimate Sources, and Grand Unification ideas. I do love those too!I will start with the nature of reality from Pythagorean thought. First of all, Pythagoras was known as a mystic. He spent maybe 15 years in Egypt learning (being initiated into) the secret mysteries. Most of what we think we know about Pythagoras comes from the Pythagorean school of thought, and some writings by Plato and Aristotle. There was a huge influence of this school (including on Plato and Aristotle) and the subsequent Neoplatonism.Now, I will become even more specific, and focus on math and geometry within this school of thought. Again this is an example to simplify, and by no means represent the vast area of mysticism or ontology. For more detailed information on Pythagoras and ideas, I refer the reader to cascred-texts.com, one of my favorite websites for source material. Here’s a start: Pythagorean Mathematics.It would appear that both numbers and geometry fell well within Mystical thought. That is, these things represented some principles of mysticism. I believe this to be accurate. All this can be googled. So, geometry and numbers represented reality. Such a view is commonplace in the modern world. Numbers and geometry are representations of some fundamental reality, to many. We are educated in school in this (mystical?) form of thought. They are representations and could be also considered as metaphysical. However, things like Newtonian mechanics and Euclidean geometry are extremely useful. Since their propositions can be tested, and predictions made, they then fall within the field of “science” which is a branch of philosophy (how our minds think of reality). Modern cosmology and quantum mechanics is hard to empirically test, therefore through the definition of an experimental scientist (me), much of these ideas have not and perhaps cannot be transferred to science.So, let’s jump to modern physics. Then, outside of that, what we call we “theoretical” math, which, through the symbolic creation of equalities (equations) is often used by physics for both the very small and the very large (like cosmology). Modern cosmology can be considered an ontology, for example the philosophy of the Big Bang (creation), or black holes (giant whirlpools in ancient times). Obviously theoretical math comes from our minds. I think it is great. The representation of time as a variable in physics is pure mysticism. Theoretical math presents an ontology of reality, and, it is always changing, always has throughout the ages. Yet, it seems to stay the same and only the vernacular of the times (words) seem to change for the same concepts represented in past times. Zeus is now called Electromagnetism; Time was once personified as Chronos (linear time). And so on.Through extensive reading, and given the ability to understand equations and how they came to be, I personally see little difference between much theoretical math and mysticism (I go so far as to call them “mathemagicians”). When such theoretical math is coupled to the ideas of “thought” physics,such as Einstein, what results is a new reality. Now we have principles of “uncertainty” and the fundamental importance of Light, both concepts existing within mysticism. In this modern interpretation of reality, assumptions must be made, and reality is based on these assumptions. Mystics also make “assumptions” based on their education and personal experience.If one considers some of theoretical math as mysticism (it has all the features), then we could say it can be traced back to its western mystical roots of Pythagoras. That is, the representation of reality through numbers and such. Thus, the ontology of mysticism is alive and well. Science at the forefront can be disorganized and messy. There is much rivalry and opposing views such as with black holes or the Big Bang. Their “real” existence is always under question and debate amongst astrophysicists, and, theoretical mathemagicians. All of this falls within the rubric of mystics and mysticism, and ontology. Although, there are always just a few mystic in the large field of physics. Most working in this field are trained to carry on a tradition of thought. Always has been this way.As I see it.Cheers