Summary Relativists invoke planes to introduce the uninitiated into the world of General Relativity. Specifically, they use planes to contrast our 3-D universe against Abbott’s world of Flatland. The flat world of planes helps the neophyte visualize and contrast dimensional aspects of planes versus solids. So what is a plane? Is a plane an idealized figure? Is it a figure? A real object? An abstract object? Do planes exist? Is the plane the same thing as a surface? Does a plane have two dimensions? If not, how many dimensions does it have? Do we have any use for planes in Physics? It turns out that a mathematician talks about the 2-D world of Flatland, yet the scenarios he describes are without exception 3-D. A mathematician also constructs his circles dynamically and claims that a triangle has curved angles. He also tells you that he builds a cube by scanning a square. And so on. How can the mathematicians assert that our Universe is four-dimensional when they don't even understand the two- dimensional world of Flatland? The mathematicians never learned the first lesson of planes: there is no such thing as a standalone plane. A 2-D geometric figure is necessarily part of a solid! The misconceived definition of the word plane conduces the mathematicians to irrational conclusions. Here I explore the infamous plane of Mathematics to get to the bottom of perhaps the most misunderstood category of figures in Geometry. Specifically, I investigate the two planes that cause so much confusion to the mathematicians: the triangle and the circle.