Part of my week was spend abroad at MIT! It is insanely beautiful there--despite being very cold. Fortunately, the blizzard didn't phase me much. Also shout-out to the dining hall chefs of Next House for making some great stir fry!
The following picture is proof that I did at least some work (that is, while I wasn't partying it up with the gracious residents at Next House over study breaks with some tasty, tasty cheese).
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| Picture taken in the middle of finding more out about Stereographic Projection |
Anyway, while I was taking an Uber with my awesome cousin Jonathan to a entertaining Improv show and the world renown Mike's Pastry, I realized just how much symmetry surrounds us, and I thought this would be a good opportunity to blog about the basics of symmetry.
Take MIT's mascot, Tim the Beaver.
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| Get it? Beaver, like nature's engineers? Whatever, engineering is too applied anyway. |
His name TIM is MIT spelled backwards, which is an example of bilateral symmetry, that is, symmetry under reflection. Imagine Tim the Beaver now with a vertical plane separating him in half. If we had some sort of magical machine that would copy every molecule of his left half and move it horizontally and by the same distance to the right side, then that would reflect the left half of him to his right half and give us bilateral symmetry.
Tim's name is even bilaterally clever: TIM | MIT. Each letter, M, I, and T, is equidistant from the line in the middle, and when reflected MIT gives TIM, and vice versa.
This is the type of symmetry you probably think of when you imagine something symmetric, as we as people love bilateral symmetry. Vehicles like planes, boats, and cars are usually bilaterally symmetric, and people almost universally think other with bilaterally symmetric faces are, ceterus peribus, prettier.
But there are two main other kinds of symmetry, too!
The second kind is translational symmetry. This is the kind of symmetry you get when you move picture
s over some distance. It's essentially just repeating some abstract motion over some object. An object is said to have translational symmetry if when physically translated somewhere, the the view of an object as a whole remains constant even though the parts are translated.
For the simplest analogy, imagine being in the middle of the desert and there's only a straight road which winds from horizon to horizon, until it bends out of sight. If you were to take each point which makes up the road and translate it a foot downwards, then though each part of the road is moving, the overall object remains the same.
If that wasn't clear, here is a picture from Mumford's Indra's Pearls of Vision, the wonderful book which this project is based out of, and which you can find a link to in the side bar (caption from the book as well).
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| Two rows of flowers moved along by the same translations. The flowers are quite different but the symmetry of the two rows is the same |
Another interesting aspect of translational symmetry is wallpaper groups or crystallographic restriction. These are the different patterns which can tessellate the plan without overlap. I don't know too much about this, but perhaps it will be of some study later. For more information, check out wikipedia (really use wikipedia as a first hand resource for anything you're unsure of in this blog--that's why I included a wikipedia widget in the side bar).
Finally, the third kind is rotational symmetry. This is the symmetry you get if you have an object that can be rotated around some point but still look the same.
Take a square and rotate it about its middle 90, 180, 270, or 360 degrees, you'll get the same image back. If you have a pentagram, you'll have 5 rotational symmetries. An octagon, 8. If you keep going, you'll see that a circle has an infinite amount of rotational symmetry.
Rotational symmetry is why the complex plane is so useful for geometric interpretations of groups, and is why I will be using it extensively. A complex number is determined by an argument (angle) and a modulus (length)--together z = r * e i*θ--and multiplying complex numbers not only increases their modulus, but it also increases their argument.
Take z = e i*θ. Its square, z2, is e -2θ. Obviously the argument of z2 is twice that of z, so though it's been rotated, but it is still overall the same. This is similar to other complex properties like roots of unities which I won't go into here.
All together, these help build some symmetry or dihedral groups, as the rotations correspond to different ρ's (pronounced "rho") and the reflections to τ's (pronounced "tao"). But I will post on symmetry groups later.
Why symmetry is important is that it allows us to better understand transformations (functions or maps both in the same geometry) on the plane. Symmetries can be thought of as pattern-preserving transformations.
Let T be a transformation on the plane R2 (so that T sends every point in R2 to a point in R2).
If T is a reflection of the form T(x,y) = (-x, y), then every point in R2 has is x coordinate flipped over the y axis. Obviously this is not every kind of reflection on the plane, as the transformation depends over what line it is being reflected, but I'll just provide reflections over the y-axis here, and leave the rest as an exercise for the reader.
If T is a translation of the form T(x,y) = (x + a, y + b), then every point in R2 has its x coordinate mapped to the number a ahead of it and its y coordinate mapped to the number b ahead of it. If for example, our particular translation is T(x,y) = (x +1, y), pretty much the graph is shifted over 1 unit to the right.
As a quick aside for the mathy people, the set of translations in R, T = {a | a in SR, x -y = xa - ya}, x,y in SR, forms a subgroup of the symmetry group of R.
Proof: i) identity is pretty obvious.
ii) inverse: let a be in T, so (xa-1) - (ya-1) = x(a-1a) - y(a-1a) = x - y,
iii) closure: let a, b be in T, so xab -yab = xa - ya = x -y.
If T is a rotation of the form T(x,y) = (x cosθ - y sinθ, x sinθ + y cosθ), then every point is being rotated anticlockwise around the origin by the angle θ.
Though it looks more complicated, if we plug in some numbers we
intuitively know, we see this makes sense: T(0,1) by 90 degrees should
map to (-1,0), and if we plug it in with θ=90, we see T(0,1) = (-sin90, cos90) = (-1,0). T(1,0) by 90 degrees should map to (0,0), and if you plug it in, it checks out. Same with T(0,-1) mapping to (1,0). Thus we can conjecture (correctly I might add, but you can prove it yourself) that when using θ = 90, then T(x,y) = (-y,x).
Okay this went on longer than I thought it would, and though I still have material to blog about (composition, groups, conjugation, etc.), I can just make a new post in a few days. I should be able to post in detail about fractals within the month.
See you soon,
Connor
I really appreciate the pictures that you use. They help with visualizing what you're trying to explain. For symmetry though, does it apply to more dimensions than two? And how would that work?
ReplyDeleteRemember that symmetry is just preserving some pattern under a transformation, so that is independent of any dimension. You can easily have symmetry in higher dimensions, think about in 3-D, maybe a rollercoaster that twists and turns and corkscrews. That has the same principles of symmetry and is in 3-D. It's just like in vector calc.
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