I started to like math when I was a kid. I remember being the first in my third grade class being able to do the hard long division class my terrifying teacher, Mrs. Jackson, would give me. How little I actually knew back then about the language of math that most people find disturbing, disgusting, and devilish. Even now, I am often in awe of how little I actually know about the subject. I have touched on basic calculus and differential equations, and yet I am still just at the tip of the iceberg. Set theory, linear algebra (without the dumbed down examples of Dr. Dydak's class), PDE's, statistics, topology and non-Euclidean geometry, the vast array of fractals, and higher dimensional maths leave me dumbfounded.
I have come so far from basic algebra. When a first grade teacher asks her students what 10 + 1 is, my younger self would have innocently said "Eleven, duh!" Now I find myself jokingly asking, "In what base system?" I guess it's just the nerd in me, but honestly, I love it. I often daydream about sitting with my future children around age 7 (not babies, as you will read later) around the kitchen table helping them with their math and science homework and teaching them some side lessons that I hope they will eventually thank me for. I personally take offense when I hear some lost person utter, "I hate math!" I think, oh how you are missing out on some of the most amazing ideas humans have ever come across!
I hate how school children are being taught math nowadays. I guess I am one of the lucky ones who felt it necessary to ask deeper questions other than whether or not an answer or right. Why is the most amazing question you can ask, and I may actually write a post about this.
On the subject of science, without math, the modern science would be unrecognizable. Seriously. Qualitative analysis is okay to an extent, but quantitative measurements (made only through mathematics) are what separate a good idea from a verifiable theory. I won't say much on the subject, as I have yet to talk about some of my favorite things in math.
- Purity of Math - By far the most pure subject of the sciences, math is almost certainly in a class all it's own. As random and chaotic the world can be, the opposite is mostly true with mathematics. By setting a few, short constraints, one can navigate through hundreds of years worth of hard work by very great men. Very little needs to be said about this to understand the concept. It is one thing that math has above all other subjects. Ten plus one in base 10 system is always 11. It is so wonderful that it pains me to see people disregard such a useful subject.
- Euler's Formula - One of the most famous formulas in mathematics, and one of my favorites to boot, Euler's formula (shown above) combines many, seemingly random areas of math into one beautiful formula. The formula says that when "e" or Euler's number is raised to the pi times the imaginary number (square root of negative one) and then added to one, the result is zero. How seemingly boring this is to others behooves me. The proof lies in the properties of complex numbers, and is a little tedious (I'll leave this as an exercise to the reader - Bazinga!), but when I first read about this, I was overwhelmed with curiosity!
- Phi - Starting at zero and one, sum the two previous numbers to get the number. The resulting sequence of numbers is given the name the Fibonacci sequence, and there are many interesting properties associated with it (I read books about the subject to give some size). Ad infinitum, the ratio between the last number and it's predecessor becomes a special ratio, phi. If you graph boxes with sides equal to the Fibonacci numbers, and connect the corners, you get a spiral seen all throughout nature (in sunflowers, shells, etc.) and another ratio gives you phi, and this is called the Golden Spiral. To think, a shape could be considered "golden!" Pentagrams and pentagons have properties associated with phi. Ratios in Golden pentagrams and Golden pentagons (again with golden) will lead to phi. Amazing!
- Fractals - If you are looking for some of the most aesthetically pleasing math topics, not just theoretical amusements, you'd be hard pressed to find something more beautiful than a fractal. Fractals are shapes that each reduction in size leaves the same shape! They occur somewhat imperfectly in nature (snowflakes), and are absolutely beautiful. Do yourself a favor (if you are still here, that is) and Google image some fractals. You will not be disappointed.
Below is the Julia set, a type of fractal:
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Reads like a post I would've written! Although, I might argue the title of the post. In a way, math is totally separate from nature. There are very few whole numbers in reality, for example. In my head, math is much more an art than a science. Now, it's slightly more useful in some of its forms at approximating reality than, say, a modern dance, but at its core are some axioms that may or may not have any actual connection to reality.
ReplyDeleteThere are a couple of ways to get interesting math theorems: 1. Do whatever you want and then figure out what are the most relaxed constraints in which that thing can happen. or 2. Given a set of restraints, what are some of the wildest things that can (or have to) happen in that system.
Both of those ways depend on a set of constraints/axioms that a person creates. Now, maybe you've chosen option 1 and "whatever you want" is to mimic some pattern in life (the relationship between force and acceleration, perhaps), but ultimately, the conditions necessary for F = ma may or may not actually be true. It may be purely coincidental that real life tends to follow that pattern as well.
From the same example, we now believe that quantum mechanics are more "realistic" in the mimicry of reality and that the F = ma formula comes from a deeper formula under certain constraints and/or assumptions, but who's to say that the Schrodinger equation isn't also just some simplification of an even deeper formula (maybe in higher dimensions or fractional ones or whatever)? Even science is giving a best guess model for reality, but who can ever know if it's "right" or just happens to match closely enough through "current values of t" or under some other set of special circumstances that may factor in, but are unobservable.
Anyhow, good post. I'm glad to see some of my own admiration of the subject has passed on to students--whether you got it from me or elsewhere.
--==--petersen
I'm glad you liked it. I actually got a lot of my enjoyment of math from your class, so thanks for that! Also, I completely agree with you; I should have rethought the title a bit some. I guess the way that I meant it was that without mathematics, our understanding of nature would be so minimal. Anywho, thanks again for reading.
ReplyDeletePhi can also be seen in the Parthenon in Athens.
ReplyDelete