Visual Proofs - The highest realms of mathematics
[Maths of all levels in general, preferably Secondary - Visual Geometry]
You probably are tired of calculating the price of a lemon, or some shirt you'll never be able to wear from your Maths textbook. And nobody can fault you - Maths, for most of your primary and secondary years, involves concept drilling. To get used to the idea of adding two numbers together, or multiplying them to get a summarily large sum of how many oranges Peter is transporting from Town I-don't-care to Town Why-am-I-doing-this.
But, there comes a time where the things you learn in school are insufficient in solving the questions asked in your tests, your exams, and every once in a while, your head. Those questions you ask in your mind, nobody has an answer to, and it leaves your consciousness, substituted by the Saturday morning cartoon and never to be remembered until mentioned again much later on in life, like before your deathbed. There are indeed some questions which look deceptively simple at first glance, yet extensive effort to fully answer them may takes years upon years of research, some which extend far beyond a few hundred or thousand years.
[P vs NP problem] Would questions that can be easily identified and verified by computers (polynomial-time) be easily solvable using computers (non-polynomial time)?
[Navier Stokes existence and smoothness] We have mathematical models for calculating how fluids flow, but proper theoretical understanding of some of its elements are unknown, for example, turbulence. Is there a way to mathematically explain turbulence?
Some of the most difficult questions, in Mathematics included, are those without numbers to work from. Fundamentally, Maths is not about the numbers, but about what the numbers represent - The underlying patterns, the trends, the attributes that it subscribes to and the ones they do not.
Today I would like to share with you, an example of the highest realm of mathematics - Visual proofs.
Visual proofs, as you can see here, do not require numbers. There is no "price of 16 lemons - price of 14 pears = $2.40" Here, the ultimate understanding of mathematics is challenged - You have to play by the rules set before you, and from there work out an answer creatively to prove (or disprove) a particular claim. Heck, you have to understand the rules present before you work out whether it's effective or not.
In that sense, everything from Primary school onwards is simply to make you remember that the numbers obey simple rules that aren't too difficult to understand. 3 + 1 is no different from 1 + 3, yet 3 - 1 does not have the same answer as 1 - 3. You already recognise this rule: 1 - 3 would give a weird shortage of 2, and in secondary school we called this shortage a "negative number", -2.
Before you can move onward to the most beautiful parts of mathematics, you must first understand the underlying rules of mathematics. Much like how writing is the repetition of 26 letters and some digits in creative, clever ways to communicate thoughts, ideas and beliefs, you have to reach a point where your understanding of the rules is comfortable before you can begin to tackle its hardest questions creatively.