This is a conceptual question. Does the Earth orbit the sun or does the sun orbit the Earth? I know this is silly of me to ask. After all, everyone learns at a young age the the Earth -obviously- orbits the sun.
We can represent the Earth orbiting the sun by defining the sun to be at the the origin of coordinates, (0,0). Then if the Earth's orbit were perfectly circular, its orbit can be described on a plane by
r = |r|(cos t, sin t)
Suppose we instead choose the Earth to be the center of coordinates. Furthermore, we allow the orientation to be preserved. For example, from the Earth, the positive y' direction is in the same direction as the original positive y direction, and the positive x' direction is in the same direction as the original positive x direction.
Consider the situation at t=0.
In the original setup, the Earth is on the x axis, |r| distance away from origin in the positive x direction. Under the transformation, the sun is on the x' axis, |r| distance away from the origin in the negative x' direction.
In the original setup, the Earth is moving in the positive y direction at t=0. Under the transformation, the sun is moving in the negative y' direction at t=0.
It suffices to show this for t=0. In every case, the transformation only reverses the sign, but everything else is preserved (the radius, the orbiting velocity, even the area that the orbit sweeps out). We can parametrize the sun's orbit relative to the Earth as
r' = |r|(cos (t-π), sin (t-π))
The sun is really (also) orbiting the Earth!
Generalizing this further, the shape doesn't have to be a circle. It can be an ellipse too. I haven't worked out the details, so I'm not going to claim other shapes (such as the hyperbola and the parabola).
Given that this transformation works and that the situations are equivalent, why is heliocentrism historically significant? Why did silly people (like Galileo) defend this notion with their lives? Why do most people learn about heliocentrism in elementary school, but not about the fact that heliocentrism is mathematically the same as geocentrism? Why is Galileo largely praised for defending heliocentrism, and why do most elementary school students learn about this like it's such an important fact (or at least, that's the impression I got)?
The Attempt at a Solution