Mad about Science: Spheres

By Brenden Bobby
Reader Columnist

In case you forgot, Pi Day is Friday, March 14 — a celebration of one of the most important numbers in our universe: 3.14159. 

Pi is the ratio of a circle’s circumference to its diameter, and is critical for calculating anything involving a circle, which is the basis for calculating anything involving a sphere. We’re all riding through space on a giant spheroid, so understanding how circles, spheres and spheroids work in relation to our environment is a big deal.

For instance, orbital dynamics explain how planets, moons and artificial objects travel through space. A spacecraft headed toward the moon does not travel in a straight line. Instead, it follows a circular or semi-circular arc. This is because objects are influenced by gravity, which perpetually pulls them toward the primary source of gravity. In the case of a rocket speeding to the moon, the Earth’s pull is the primary source of gravitational force influencing its trajectory.

If our hypothetical rocket is aimed well, it should intersect with the moon’s gravity in such a way as to either pull it into the moon or whip it around the moon to alter its speed. Pi is the key that engineers and scientists need in order to calculate how to make these precise intersections with other objects in space.

This circular motion predicted by pi is the reason there are so many spherical objects in the universe. Nature loves spirals, and the motion of a spiral influences the shape of objects with great mass. The sun is spherical because of the pull of gravity and the rotation of the star. Planets and moons are spheroids for the same reason. Early in planetary formation, large bodies of molten rock are drawn toward each other by gravity and smashed together, whipping around and rotating to create centrifugal force. 

Over a vast amount of time, impacts from other objects and the perpetual motion of spinning on an axis smooths the edges of these bodies to create a spherical shape. (You can observe this in real time while spinning a pottery wheel.)

Planets are seldom perfectly spherical, and are therefore technically not spheres but spheroids. A sphere is an object with every point on its surface being perfectly equal in distance from the center. The Earth bulges at the middle due to its rotation and is instead an oblate spheroid. The sun is also not perfectly spherical, but it’s the closest thing we’ve discovered to a sphere, with only a slight bulge at its equator.

There is a simple equation for calculating the volume of a sphere: 4/3*pi*radius cubed. You can demonstrate this by blowing up a balloon, whose optimal volumization is a sphere — though it may bulge in places because of how it was manufactured. A sphere is capable of containing the largest volume with the smallest surface area. Measuring the area of the sphere is done with the equation: 4*pi*radius squared.

The radius of a sphere is half the diameter. The diameter is the distance between the two farthest points straight through the middle, which means the radius is from the direct center to the edge. Since a perfect sphere has all points being equidistant from the center, anywhere you measure from edge to edge will be the diameter, and anywhere from the direct center to the edge will be the radius. 

Cleaving a sphere perfectly in half creates a hemisphere — a.k.a. a demisphere. When referring to the northern, southern, eastern or western hemispheres of Earth, you would effectively cut the Earth down the middle twice; but, because you’re referring to specific regions, you’re not actually creating quarters but halves. A quarter of a sphere is called a quadrant, though “quadrisphere” sounds really cool.

In geometry, spheres have no vertices (points) or edges (lines between points). This becomes very difficult to process when working with a 3-D design in a computer. 

A computer can’t calculate an infinite number of vertices for manipulating the surface of a sphere, so it needs to make approximations. A basic sphere in a 3-D design program has 64 vertices. Increasing the number of vertices increases the number of faces the object has, which allows for more detail but requires more computing power to process.

In the context of video games, fewer objects with fewer faces requires less processing power, which will allow for faster response times and less lag for the player. Fewer faces also reduces the amount of time that static objects need to render to create an image for artwork or movies. 

3-D design is a balancing act, where designers are choosing between giving the computer more power to process or creating less detailed objects for the user to view. Often, artistic license is applied to mask flaws in geometry. 

A good designer knows that flaws are the key to tricking the human eye into believing that something is real and not manufactured. Perfection is the enemy.

Stay curious, 7B.