``
13 Nov 2020
Well, certainly I have finally got the time to write something.
Grant Sanderson, otherwise known as 3Blue1Brown, is a Youtuber that posts videos about mathematics. If you haven’t seen any of his work, do check out his website and channel.
For those who have watched 3Blue1Brown videos, you will know how he presents math concepts: with the use of very clean animations. Well, today’s topic isn’t about his channel, it’s about the animations.
Visualizing Mathematics
More often than not, mathematics requires a lot of formulae and graphs, and sometimes those things change over time; there’s even an entire field of math dedicated to studying change. Static figures often don’t do justice to these concepts, and with the widespread usage of computer screens, it seems counterintuitive to try understanding mathematics only through textbooks.
This problem was exactly what I encountered when trying to summarize Perelman’s proof of the Poincaré in 1250 words. Diagrams are very important and useful, but do they translate well if I were to present to an audience? Is there a better medium?
Mathematics Animation Engine
There is apparently a way to present mathematical concepts in the way 3Blue1Brown does it. In fact, the engine that he developed for his videos is available on GitHub. There even is a community-maintained version, which I have been told, is updated more regularly.
The engine differs from traditional animation engines in the sense that it is mathematics first, animation second. With Manim, you can create accurate graphs and objects, and paths are easily tracked and traced. Everything is mathematically defined, and therefore it is more convenient to scientific animations than vector drawing software, where every endpoint, every curve has to be tweaked and adjusted.
However, there are downsides, the biggest of which is that Manim does not have a user interface. It is a very powerful tool, yet it is not user-friendly at all. The user has to manually write code in Python, run sections of it on a terminal, before obtaining a single piece of animated math. For those who are not familiar with computer science, you will most likely have to learn the syntax of Python, the lexicon of Manim, and how to translate math into code (i.e. the formatting of LaTeX).
On the other hand, as a user with about a year of experience in Python and Java each, Manim was relatively easy to pick up once you understood its strengths and limitations, despite being as convoluted and complicated as it can be. Next time if you ever think about doing a math/science project, don’t hesitate to try using Manim to make your presentation look sleek as hell.
If you want to take a look at my animations, I have made a PowerPoint containing all of them. To see the code that made these animations, head over to my GitHub here.
``
16 Oct 2020
As I have written last week, my term 1 project involves writing about Perelman’s solution to the Poincaré conjecture. After some initial reading and a small discussion with my mentor, these are my first thoughts on the topic.
The Conjecture
The Poincaré conjecture, as conjectured by Henri Poincaré in 1904, states that:
Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.
This might look very confusing, and that is why we are going to break this down into parts and explain each concept briefly.
Firstly, a manifold. A manifold is defined as “a topological space that resembles Euclidean space at each point”. A 2-manifold means that object looks like a plane locally, but does not necessarily need to be a 2-dimensional object, like a piece of paper, a Möbius strip, or even the surface of the Earth. To expand this concept one dimension higher, a 3-manifold will look like 3-dimensional space locally, but might be bent in another dimension (time?). The fabric of space itself can be considered a 3-manifold, since it is a 3-dimensional space (duh), but gravitational forces can bend it in respect to time.
Secondly, “simply connected”. Simply connected (or 1-connected, since we all like numbers), means that any closed path (like a circle) on that manifold can be contracted into a point. Applying to 2-manifolds, it means that the surface cannot have any holes. The surface of a sphere is simply connected; the surface of a doughnut is not.
Thirdly, “closed”. A closed manifold has no boundary, and is “compact” (imagine a closed interval). A 1-dimensional example is the circumference of a circle, where there is no “end” to the line, and the distance that line travels before it meets itself is finite.
Next, “homeomorphic”. In layman’s terms, an object is homeomorphic to another when they can be transformed between each other by stretching and bending it without creating holes. A classic example would be a coffee mug is homeomorphic to a doughnut, since they are both objects with one hole.
Lastly, a 3-sphere is a hypersphere. A circle, or a 1-sphere, is constructed using all the points on a plane a fixed distance away from a single point. A sphere, or a 2-sphere, is constructed using all the points in space a fixed distance away from a single point. A 3-sphere, by analogy, is constructed using all the points in spacetime a fixed distance away from a single point.
So, the Poincaré conjecture can be understood as: space, if it is limited in volume, has no boundary, and has no “holes”, it is curved in another dimension, and is a hypersphere; an analogy for 2-manifolds would be how the surface of the Earth is limited in area, has no boundary, and isn’t a doughnut, therefore it must be spheric.
The Solution
Grigori Perelman, a Russian mathematician, provided the proof of the conjecture over 3 papers in 2002-2003. His proof utilized a method called Ricci flow with surgery, which successfully proved that those 3-manifolds can be reduced into 3-spheres. This problem regarding 3-manifolds is the last problem of its kind to be solved; all higher dimensions of the Generalized Poincaré conjecture have been proven true before the turn of the century.
Ricci flow, introduced by Richard Hamilton, is essentially a differential equation that changes a manifold based on its curvature over time. It tends to produce objects that are rounder in shape and reduced in volume, which is ideal in a problem that wants to reduce objects into a simpler one. However, as the differential equation proceeds through time, some objects might produce singularities, and since Ricci flow can only operate on smooth manifolds, not all 3-manifolds can be reduced to 3-spheres in this manner.
Perelman dealt with this scenario by introducing a “surgery” to cut, and then cap these singularities, and then perform Ricci flow again. He proved that all singularities can be dealt with this way without breaking homeomorphism.
A natural question to come up after this is the number of cuts needed. If singularities can be produced by a single Ricci flow process, it might happen again after cutting, potentially infinitely many times. Perelman spent most of his third paper proving that a finite number of cuts is enough for any object, and thus, Ricci flow can be used to reduce any simply connected, closed 3-manifold into a 3-sphere, and the Poincaré conjecture is proven true.
If you do have any insights on the Poincaré conjecture, please contact me on my discord server, I am in need of ideas to write this paper out.
``
03 Oct 2020
The mood of the week so far has been, well, “academic”. First off, I had to choose my term 1 project this Tuesday, and I willingly chose to study Perelman’s proof (parts 1 | 2 | 3) of the Poincaré Conjecture; secondly, I willingly used part of my Friday to learn about the Euler-Lagrange equation (albeit a version for dummies, thank you, Morgan); so it would only be normal for me to finish this week by talking about APs.
The Rationale for Taking APs
AP, as you might already know, stands for Advanced Placement. It was originally conceived as an idea for high school students to jump ahead and learn material from first-year university courses without leaving the comfort zone of high school, but currently, so many people take it that it is barely considered “advanced”.
The first tip I can give you is: never ever be peer pressured into taking APs. This was never a competition about the number of APs you are taking, nor should it ever be a race to get the most 5s. You should always be taking subjects that you like, or are useful for you to have a preliminary grasp of the material before re-learning it in university.
But you might be asking, “you took 12 APs, why are you telling us not to take that many APs?” There are a few reasons: for one, I am used to being constantly stressed; secondly, I am able to take those courses comfortably; and thirdly, I have experienced the pains of having an overwhelming number of APs, and I do not wish for you to repeat it.
The Pitfalls of Too Much Work
An AP course usually requires a lot more work than a regular high school course. Most often, only paying attention in class and doing your homework will not be enough.
As the material is university-level, sometimes the class will also be taught univeristy-style. Preparing for the class by doing pre-readings is almost mandatory, and taking notes will be integral to scoring well on your tests. In extreme cases, sometimes entire sections of the syllabus will not be covered in class, due to the lack of time.
If you want to be prepared for university, AP courses are definitely a more than appropriate way to experience the culture and the material that you will soon encounter, without too many consequences. It is certainly a good way to challenge both your knowledge and your time management.
However, too many APs and you will end up falling behind in every course that you are taking. Prioritize your work, and realize that time is limited. Enjoy high school while you still can, because university is going to get a lot harder once you get in.
I would also reccommend a limit of 5 AP courses per year. If university students are reccommended to cap their course loads at 5 courses per term, and 6 is already stretching it, I see no reason for a high school student to take more than 5 APs, since it is, by then, a less than accurate representation of both university life and your academic capabilities.
Taking APs as University Credit
Some universities allow students to get certain first-year credit if they have attained a good enough score in AP; some even mandate it. I will reccommend you to only take the credit for the courses that are not in your major (i.e. electives, requirements), and leave the credit that is related to your major. I am a firm believer, like many others, that to solidify your understanding, material should be learnt more than once. Moreover, courses in university are often taught in a different manner than AP courses, that it would be useful to get a different perspective of the same material.
However, if you need to get certain requirements out of the way, certainly. But as I said before, I do not reccommend taking AP courses just to escape a certain university requirements.
Course Overview
In this section, I would like to go through the 12 APs that I have taken, and provide a little bit of background information and a small description of my experience in those courses. They will be sorted by subject.
Note: after writing this, I realized it was way too long. But there’s nothing else after this section, so if you aren’t inclined to read all this, it’s fine!
English Language and Composition
Lang is a course about the practical uses of English language. For most of the class, you will be studying speeches, public letters, and other forms of writing that are very commonly used. This is still a good course to take even if you do not plan to become a language arts major, since it also enhances your skills in constructing solid arguments and writing coherent essays.
The exam consists of a multiple choice section and 3 essays (40 minutes each). The three essays are, in order, a synthesis essay, which requires you to construct an argument and take a stance based on the multiple sources that they provided; a rhetorical analysis, which asks you how does the author/speaker convince their audience of their point; and an argumentative essay, which asks you to take a stance on a topic and construct an argument without any given material.
I would personally say that the rhetorical analysis is the most difficult of the three, because it does not only ask “what”, but also “how”. However, if you are a slow reader, the synthesis essay might also be a source of trouble.
English Literature and Composition
Lit is the other branch of English in AP. It is mostly a study of prose and poetry, old and new. Unless you are pretty into the beauty of the language, I do not reccommend taking this course over Lang because of the high amounts of reading and workload associated.
The exam is similar to Lang, with a multiple choice section and 3 essays, 40 minutes each. The three essays are, in order, a poetry analysis, where you are given a poem and asked to analyze its literary merit; a short prose analysis, where you are given a short story or an excerpt from a drama and asked to analyze its literary techniques; and an analysis of a longer work, where you are not provided with any material asked to examine a theme and use any supporting texts of your choice.
The exam is just difficult, and that I believe a lot of preparation is required of the student. I clearly did not prepare enough, and rightfully deserve my 3.
Microeconomics
Micro is the branch of economics that is more like math/statistics than social sciences, as compared to Macro. The class is mostly about charts and graphs, and the theories that drive demand and supply.
The exam involves a multiple choice section and a written section of 3 free response questions, where you are asked to apply the theories on the data that you are given, draw graphs and calculate certain amounts (without a calculator).
For me, this is the easier of the two economics, since I feel like the graphs make the theories more clear-cut, that there are way less ambiguity than Macro.
Macroeconomics
Macro is the exact opposite. It studies a lot more of the societal impacts of fluctuating demands and supply, and because it is applied to the actual world, there are a lot more factors to consider, and things are less straightforward.
The exam is functionally identical to that of Micro, with multiple choice and 3 free responses, but this time with more writing and explanation of the effects that a single action has on the economy.
Again, I am good with numbers, not descriptions, and I did not do as well on Macro as I did on Micro. It feels very subjective to me, but I guess that is also a very subjective, biased opinion.
Calculus BC
Calculus is split into two branches: AB & BC. AB covers the standard differential and integral calculus and their applications; BC covers that and more. Venturing into series and their relationship with functions, you can see more of the syllabus here.
The exam is split into four sections: two of which are multiple choice and two of which are written responses, and each type of questions has a calulator and a non-calculator section.
Most people will think that BC is a really hard course, but I think it is only because of the pace. While AB gets to cover differentiation and integration over an entire year, BC students have to fit that in half a year, as to leave time for the BC-exclusive material.
Physics 1
There are a total of four physics courses, and the starting point for all of them is Physics 1. This course covers the fundamentals of high school physics, and serves as a great introduction to almost all areas of physics used in modern day, from Newtonian mechanics to circuits and waves.
The exam is a multiple choice section followed by 5 free response questions. There are way less calculations than you expect, and it involves a lot of explanation of phenomenon using the known laws of physics.
I personally prefer calculations and derivations, but the breadth of the course material makes up for the undesirable exam. In the end, Physics 1 is still the course that truly transformed me into hoping to become a physics major.
Physics 2
Physics 2, together with the aforementioned Physics 1, are the algebra-based physics courses that AP provides. This course covers the physics beyond Newtonian mechanics, and delves into fluid mechanics, thermodynamics, modern physics (special relativity), etc.
The exam is similar to Physics 1, in the sense that it involves a multiple choice and a written section, but this time the written section has one less question.
Honestly, this course does not cover enough material, in my opinion, as compared to Physics 1.
Physics C: Mechanics
The ultimate challenge in AP Physics is split into two courses. Mechanics, as its name suggests, deals with Newtonian mechanics, and covers everything from dynamics to rotational to energy. This course is calculus based, so it is reccommended to be taken after you have gained an understanding of differential and integral calculus.
The exam is half-length, at 1.5 hours, involving a multiple choice section and 3 written questions. Time will be very tight, as questions often require you to submit answers as a mathematical expression, accompanied with the occasional writing.
I would say that Physics C is the hardest course out of all the APs, due to the sheer amount of material that you have to know for a half-length exam.
Physics C: Electricity & Magnetism
E&M is the bane of most students’ existence. If Mechanics was considered difficult, then E&M is even worse. Covering from electrostatics, to circuits, to electromagnetism, E&M is a series of hard-to-visualize concepts with really specific rules with calculus slapped onto it.
The exam is identical to Mechanics, and is traditionally taken immediately after it. Your brain will mostly be fried after the two exams.
(For notes to Physics C, click here)
Chemistry
Chem is the most popular AP science, usually with at least twice the amount of students as all the Physics C and Biology combined. It is not as intensive as the other two (and if you are going into UK universities, you will have to study a lot more), but it does provide a good enough basis, covering a wide variety of chemistry areas.
The exam is also multiple choice and then written, with the written part requiring less detail than that of Bio (see below). The questions include quite the number of data interpretation and analysis, and also explaining known phenomena.
Biology
Bio is another candidate for being the most difficult AP course, due to the sheer amount of material that one has to study and memorize. The notes that I took for this course is unfortunately not measures in pages, or thickness, but rather by the number of binders I filled and the total weight of the notes.
The exam is again, multiple choice and written. The written section goes beyond regurgitating knowledge, but also requires data analysis, which links back to theories and mechanisms that you learnt in class. A lot of dots need to be connected, and the time limit surely also doesn’t help.
Although I did say it is more than regurgitation, I did, however, study 16 hours over the weekend before the exam on Monday morning, and successfully produced a 5.
Computer Science A
Comp Sci A is a course that teaches the basics of Java. Other than that, it teaches nothing. I am not too big of a fan of this course due to the lack of depth of the syllabus, but I will take what I can take, since it is, in the end, a course about programming.
The exam is, as I repeat for the twelfth time, part multiple choice and part written. The written section requires you to write code based on a context that includes way too many words. And when I say “writing code”, I mean physically writing code with pen and paper.
The exam format is honestly the biggest deterrent for me, since in no way in real life will I ever write code without a computer.
Conclusion
I came out of my 12 AP exams with a total of 6 5s, 5 4s, and a single 3, spread out over my final 3 years in high school. Is it helpful? Certainly. Is it really needed? Not at all. We will all one day learn that knowledge if we are devoted enough to that subject, AP or not.
If you have actually read all that stuff, I congratulate you, because even I fall asleep when I am asked to do long readings such as this article.
``
26 Sep 2020
So I realized, three weeks in, that university is a really big time sink, and I do not have that much time to play League of Legends anymore. Instead of queuing into an hour long game with no option to pause and potentially feeling down afterwards, I needed to find some sort of game that is easy to pick up, requires a little bit of thinking, and does not need huge amounts of dedication.
And look what I found.
Shapez.io
If you guessed that I would be doing a short game review, you would be right. Shapez.io is a Factorio styled game that is all about manufacturing different shapes from a variety of sources, and optimizing those pathways and processes. It does not require big brain activation to play the game, but in order to design your machines properly and compactly, some thinking and sketching is definitely reccommended.
The game starts off with a short interactive tutorial that introduces the aim of the game: to produce the required shapes and deliver them to the hub. It then introduces the machines that help you make different shapes; each has a different function, and together they form giant production chains that slowly assemble those quirky required shapes that over time, grow in complexity.
As you progress, machine upgrades and variants are introduced to streamline and ramp up your production to meet the ever-increasing demand. A painter that originally paints one shape at a time might have an option to paint two simultaneously. A stacker might be able to go from stacking an item per second to two. The map is also infinite in size, which means there is no need to worry about running out of resources. Keep expanding!
Review
I first found out this game from a medium-sized YouTube channel, aliensrock. He plays mostly puzzle and strategy games, which is right up my alley. I did not even click into the video when it was uploaded; the thumbnail alone was enough to tempt me into purchasing this.
The idea for this game never gets old, because honestly, expanding my factories Henry Ford-style without the fear of it all crashing down because of some energy crisis or demand fluctuation is quite satisfying. The content might be slightly limited if you are used to 30-hour long campaigns, since there are only 18 levels in this game, and I finished them in slightly under 16 hours, but frankly, I feel like I could do it quicker now that I have some basic ideas of how the game works and how stuff should be built. The dev, however, promised constant updates, and the next update apparently seems to expand endgame content with wires and logic gates and other cool stuff that I am never able to construct efficiently in computer science class.
The interface also looks clean and easy to understand. The required shapes are lines up nicely on the left side of the screen, while the machines are all shown in simple icons at the bottom. A nice, simple indicator at the top-right corner points towards the delivery hub, just in case you have forgotten where your origins are, after all that hectic expansion. Not to mention that this game has a light and a dark mode, catering to both casual gamers and hardcore edgy ones.
Overall, this is just nice game that burns up my extra time without demanding too much of it. I would certainly reccommend if you are looking for a casual game that requires a bit of thinking without wanting to get stuck on a level.
Other games
On the same vein, these are some games that I am either hoping to play, or currently playing but not enough to form a clear judgement. Most of them come from watching aliensrock videos.
Cosmic Express
Note: This game was bundled with Twitch Prime at some point. Check your Twitch account.
Cosmic Express is a game about building train tracks that move cute little aliens to their desired destinations. A few rules: the train has limited capacity, tracks cannot cross over each other, and the train travels one way.
5D Chess with Multiverse Time Travel
This is a massive brain game that requires a clear head and more to even understand the rules of the game. The name “5D Chess” is a bit of a misnomer, since you actually get only 4 different dimensions to play with; but that’s already 2 more than what we are familiar with. Play this game so that you can say to your friends “I’ve checkmated you in 10 turns ago” and “I’m literally four parallel universes ahead of you”.
Poly Bridge 2
Poly Bridge 2 is the sequel to the original bridge building game, Poly Bridge (duh). It is very simple: slam some roads across the chasm, build supports, and call it a bridge. Content never runs out, as there are also countless community-made levels.
Closing
I know this is a long post, but if you have any comments, pop into my discord and we can always talk about it!
``
23 Sep 2020
A brand new logo has been dropped! Yay me.
Complete with a new favicon, the new logo also appears on the sidebar, which you probably have seen. But anyways, this is what it looks like:
This logo takes inspiration from this small dome that I made back in grade 8, which you might have seen:
It would not hurt to tell a story today, right?
A Logo and a Lampshade
In grades 7 & 8, I had a class called Design and Technology. As its name suggests, there are two components: “design” and “technology”. The design portion of the course had us draw out architechtural ideas, saw out different bits of wood, learn about the aesthetics of different materials; the technology portion of the course introduced us to laser cutting with vector software, how to safely use table saws, how to operate a drill, and a little bit of robotics.
The first project in my DT class in grade 8 involved creating a lampshade out of mainly laser-cut plastic parts. We were specifically told not attempt to attach the plastic edges to anything, since the teacher was convinced that it would not hold. Being the rebellious kid I am, I proposed to my group that we should attach these plastic parts perpendicular to each other, directly defying the teacher’s orders.
Well, for once, my idea succeeded. The finished product is what you see: a carefully manufactured 3D dome, while all the other projects produced mainly 2D structures. As defying an adult never turned out too well for us kids, this surprising result made me very proud. I did not get to keep my dome, but this became my first tangible thing that I ever created and showed to the world.
Which brings us back to the logo itself. Since I designed all the parts for the lampshade with the vector drawing software Inkscape, why not create a vector art logo using Inkscape? And tada, here we are, with a cartoon dome on my website.
Small Update
Other than the new logo, there are also several small updates:
- The “About” page is now fully populated with my extended bio! I’m very glad that I can brag about my musical acheivements.
- The calculus notes are getting updated on a daily basis. I have decided not to make a separate document for first year calculus, since it covers mostly identical material as AP Calculus. Instead, I will be providing additional clarification with the knowledge that I have learnt this year.
That is all, I believe. Have a good night.
