Visualisation of spherical harmonics in Peirce's quincuncial projection

The spherical harmonics $Y_{\ell m}(\theta,\varphi)$ are complex-valued functions on the surface of a sphere, and have found widespread application in physics and astronomy. Every physics students knows them from quantum mechanics and electromagnetic theory, where they form the basis of hydrogen orbitals and of the multipole expansion, respectively. More advanced applications include the physics of the cosmic microwave background, gravitational lensing, and gravitational waves. In this paper I aim to contrast their usual $3d$ visualisation with Peirce's quincuncial projection, a conformal projection of the sphere onto a $2d$ unfolded square dihedron, where the projection respects the fundamental rotational symmetries and preserves angles. With this mapping, I guide the reader through the properties of the spherical harmonics in a pedagogical way and show that many of their mathematical relations have an intuitive visualisation on Peirce's $2d$ map, which might be useful for people challenged by processing $3d$ shapes, or which people might appreciate aesthetically.