Wanna build a Death Star? While a quadratic is a polynomial formation, it is also something much more powerful. Parabolas belong to a special class of relations called conic sections. It gives the parabola the ability to intensify and focus incoming rays with enough strength to blow up a planet. 🙀
Dive into Marco's mind as you explore The Eye and the terrifying monster under the floorboards...
CONIC SECTIONS
While quadratics are polynomials they also belong to a powerful group of relations called conic sections. I suppose the story goes that one day in ancient Greece somebody wondered about the shapes one could make when slicing a cone.
They had no idea their findings would help Kepler describe planetary motion, or allow a doctor to dissolve a kidney stone without pain, or to create the Hubble Telescope–focusing rays of light to gather images of the universe. But alas, the exploration of conic sections has led to some amazing inventions.
There are four shapes that result from slicing a cone.
Circle
The mighty circle can be described as all the points equidistance from a given point. It's why to draw a circle you get a string of a certain length, tie one end to a pencil, and trace.
Hyperbola
A hyperbola is the set of all points whose distance from two fixed points is the same. These can be created with rational functions. Cool fact: lamps often cast hyperbolic shadows.
Ellipse
A bit more complicated, an ellipse is all the points whose combined distance from two points is some value. To draw an ellipse make two fixed points (with a nail or pushpin) and tie a string between the two. Push the pencil against the string to create your ellipse!
Parabola
Ah, the parabola. This is the conic section we are interested in. While the others are all about the points, a parabola is defined as all the points the same distance from a point (the focus) and a line (the directrix).
Try it Out!
Move the green point to see all the different ways a plane can slice a cone.
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FOCUS
FOCUS
FOCUS
To build a parabola, you simply need to pick your focal point and where to place your lever. Go ahead and give it a try!
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BUILDING A PARABOLA
All the good details are already in the book, so we won't go too much in depth here. The key idea is that a parabolic formation acts as an amplifier of light, lasers, sound waves, and more! This makes it super powerful and we'll want to know how to harness this strength. The farther away your focus and lever are from each other, the wider the resulting formation, allowing more energy to enter.
Just like a circle is all the points the same distance (radius) from the center point, a parabola is all the points on the plane that are equidistant from the focal point and the directrix. You could think of this like a soccer game. The focus is the goal and the directrix is the line of scrimmage. It makes sense to disperse your players at points which are the same distance from both. If you do this, you will have created a parabolic formation.
Focus
FINDING THE ORDER
If we know where we need our energy to focus, how can we find the order that will send our soldiers to the correct locations?
1
Let x be any soldier and y be their location. We need to first find the distance from this post, (x,y), and the focal point of our formation.
Since the focus aligns with the leader of the formation, –h, and is some distance, p, from the leader's location, the focus appears at (-h, k-p). We can use this to find the distance from any solider, (x,y), to our focal point.
2
Now we need to find the distance to our lever. Since this is a vertical line, this distance is just the difference of each station.
3
We know these two distances must be equal. So we make them equal! We can manipulate this equation into the order we are looking for.
That looks terrifying!
4
It looks horrible, but it turns out we have a lot of counterbalances on each side that will vanquish each other!
Hey! That looks just like vertex-form of a parabolic formation: y=a(x+h)²+k. It turns out a=-1/4p where p is the distance from the vertex to the lever. This means, if I know the order I can find the focal point!
Parabolic Art
All art is informed by mathematics. It may be ratios, scaling, perspective, but it all boils down to numbers and relationships. Try your hand at some parabolic art by moving the focus and directrix about the plane. Select new colors to build your masterpiece!
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Into Marco's Mind
THE EYE
"Without the terrifying layers of teeth, it reminded him of an eye. It was like he was controlling the pupil: as he pulled the directrix down it dilated, allowing more light to enter. When he brought the lever up impossibly close to the vertex, the pupil became tiny, an intimidating bouncer only letting select rays enter the party."
– page 243
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THE BLOB
"Marco imagined he was in Fredrick's factory. The floorboards split apart to reveal a colossal hole that housed a gigantic slimy monster. He couldn't even see the body of the beast as all that was visible was its jaws; rings and rings of teeth circling the dangling uvula. Looking to his left, he saw a lever attached to the wall. He pulled it down. The creature's jaw widened; mouth open to devour its meal. Pulling the lever up narrowed the opening. It became thinner and skinnier. Not in any danger, he taunted the beast. Up, down, up, down, open, closed, open, closed. The monster's jaw was snapping like the face exercises he'd seen singers and actors do when warming up. It exploded out of its hole revealing its long worm-like body. Gooey pink layers of chubby folds dripped as it focused its attention on Marco."
– page 235
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