We see, for example, the spheres of Euclid in atomic models, and we see the Cartesian coordinate system repeatedly in the whole of physics, such as in quantum models describing movement directions, in portraits of geological crystal types, and in countless equations relying explicitly on x, y, and z axes at 90 degrees.
In our view, one of the most bizarre instances of Cartesian thinking is in the current idea taught to students which is called orbital theory. In this theory, for example, the student is told to accept the idea that there are creatures called p orbitals (orbitals being a concept which has been vaguely defined, to say the least), and that these orbitals perfectly match the Cartesian axes at 90 degrees. Then the texts, such as "Chemistry The Easy Way" blithely go on to offer sentences such as the following: "Data have shown that bond angles in molecules with p orbitals in the outer shell do not conform to the expected 90 degree separation of an x,y,z, axis orientation." It leaves one to wonder who exactly it was that "expected" the angles to be the 90 degrees that are so erroneous. It certainly was no one at my office. This is one of many examples of popular thinking we challenge.
The connection between Euclidean geometry and the Cartesian coordinate system is described well enough by Estelle Delacy in her basic book on geometry and Euclid, wherein she details the way Descartes developed further the foundations established by Euclid, that I shall not tarry to dwell on this point here.