If you’ve ever hung out with physics or exact sciences students, you’ve probably heard a joke about a "spherical cow" at some point. While it sounds crazy, this concept is one of the fundamental pillars for understanding how scientists model the universe. At byprofesolmar, we wanted to pay homage to this inside joke with our new magnetic bookmarks, and what better way to do it than by combining Profe Solmar's beautiful art with a bit of scientific rigor.
The Famous Spherical Cow
The classic joke tells of a farmer who asks a team of scientists for help increasing his cows' milk production. After weeks of calculations, the team's physicist announces he has the perfect solution, but with one small condition: "it only works with spherical cows in a vacuum." This joke illustrates how scientists simplify extremely complex real-world problems to solve them mathematically. By removing legs, ears, and air friction, we are left with a pure and manageable shape.
Once we have our simplified cow, we need a way to study it and position it in space. This is where coordinate systems come into play, those invisible grids that allow us to describe the physical world using numbers.
Tools for Locating Points
A coordinate system is, in essence, a mathematical map that allows us to find any point in space. It works like your phone's GPS, giving you an exact address based on references and distances from an origin point. Without these tools, it would be impossible to calculate anything from a rocket's trajectory to an atom's structure.
Depending on the problem we want to solve (or the shape of our cow), physicists and mathematicians choose a different type of system. It's not always practical to use straight lines, so throughout history we have developed different mathematical "molds" that adapt to the geometry of what we are studying.
Three Systems, Three Coordecows
To celebrate the diversity of shapes in physics, we have designed "Coordecow" bookmarks, each representing a different spatial system with its own geometric illustration:
The Cartesian Coordinate System
The first of our bookmarks honors the most classic and easy-to-visualize system: the Cartesian. In this version, our cow has the peculiar shape of a perfectly square cube. This mathematical system is named after René Descartes, who, according to legend, conceived it while observing a fly's flight and trying to describe its exact position relative to the walls and ceiling of his room. To achieve this, the Cartesian system uses three straight axes that intersect at ninety-degree angles, which we mathematically know as .
Navigating a Cartesian system is exactly like walking through a modern city with grid-patterned streets. To reach any point in space, you first move a certain distance forward or backward along the -axis. Then, you move left or right along the -axis, and finally you move up or down using the -axis. It's a positioning method based on straight lines, ideal for moving a robotic arm or for a 3D printer to function.
In physics, we use this "cubic cow" when the problem we are studying has flat faces, linear trajectories, or rectangular geometry. It is the favorite system of architects, civil engineers, and anyone studying blocks sliding down ramps. If the universe of your experiment looks like a box, Cartesian coordinates are your best tool to avoid getting lost.
The Cylindrical Coordinate System
Sometimes, forcing nature to fit into straight lines is a true mathematical nightmare, and that's why our second cow is tube-shaped. The cylindrical coordinate system comes to the rescue when we need to describe objects or movements that rotate around a central axis. Instead of using three straight lines, this model combines radii and angles with a straight line, expressed with the variables .
To understand it, imagine you are cutting a circular multi-tiered cake. First, you need to know how far from the center your slice is, which is the radius . Then, you must choose the cutting angle around the cake, represented by , and finally, you decide which floor level the portion belongs to using the vertical axis . With these three instructions, you can locate any point within a perfect cylinder.
Physicists and engineers love this tubular coordecow for analyzing any system with rotational symmetry. It is indispensable for calculating how water flows inside a pipe, how a motor rotates, or how the magnetic field behaves around a straight wire. When science revolves around a central pillar, this system amazingly simplifies calculations.
The Spherical Coordinate System
Finally, we come to the jewel in the crown and the inspiration for the original joke: the spherical cow. This system is used when everything in our problem radiates or orbits from a single central point, as if it were an explosion frozen in time. Here we abandon straight lines almost entirely to embrace curves, locating points by a distance and two angles that we denote as .
It works very similarly to how we locate ourselves on planet Earth using a globe. To find a city, you don't measure straight-line distances by piercing the Earth's crust, but rather use latitude and longitude (our two angles and ). You only need to add the distance from the planet's core to the surface (the radius ) to have your exact location in three-dimensional space.
The universe is fascinated by spheres, so this coordecow is the best friend of astrophysics and quantum physics. Stars, planets, black holes, and even the electron cloud in an atom have spherical geometries. Whenever you need to calculate the gravitational force of a celestial body or how sound expands from a small speaker, the spherical coordinate system will save you from filling an entire blackboard with painful equations.
Next time you open your physics book, math textbook, or your favorite science fiction novel, let one of these friendly coordecows mark your page. They are not only a small magnetic work of art by Profe Solmar, but also a reminder that science, however rigorous it may seem, always has room for humor and creativity.