Visualization of Spherical Harmonics in Peirce’s Quincuncial Projection

Björn Malte Schäfer ()
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Björn Malte Schäfer: Zentrum für Astronomie der Universität Heidelberg, Heidelberg University, Astronomisches Rechen-Institut

A chapter in Handbook of Visual, Experimental and Computational Mathematics, 2026, pp 519-541 from Springer

Abstract: Abstract The spherical harmonics Y ℓm ( θ , φ ) $$Y_{\ell m}(\theta ,\varphi )$$ are complex-valued functions on the surface of a sphere and have found widespread application in physics and astronomy. Every physics student 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 chapter I aim to contrast their usual 3d visualization 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 visualization on Peirce’s 2d map, which might be useful for people challenged by processing 3d shapes, or which people might appreciate aesthetically.

Date: 2026
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DOI: 10.1007/978-3-032-16368-4_93

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