Visualisation of spherical harmonics in Peirce's quincuncial projection

TL;DR

This work introduces visualizing spherical harmonics $Y_{\ell m}(\theta,\varphi)$ on Peirce's quincuncial projection, mapping the sphere conformally onto a 2D square to reveal full angular structure. It reviews the theory of $Y_{\ell m}$, including their eigenvalue equation $\Delta Y_{\ell m} = -\ell(\ell+1) Y_{\ell m}$, explicit form, and spectral completeness, and demonstrates how key properties such as Hermiticity, parity, orthonormality, and completeness are manifested on the 2D map. The paper also discusses advanced relations (Herglotz generating function, Rayleigh expansion, Wigner $3j$-symbols), small-angle Fourier limits, and unitary time evolution, all visualized within Peirce's projection to provide intuitive understanding and pedagogical value. This approach offers a visually intuitive, geometry-preserving tool for studying spherical functions in physics, geophysics, and cosmology, potentially aiding education and data analysis by exposing spherical structure that is obscured in conventional 3D plots.

Abstract

The spherical harmonics $Y_{\ell m}(θ,\varphi)$ 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.

Figures (15)

• Figure 1: Spherical harmonic $Y_{43}(\theta,\varphi)$ as an absolute value $\left|Y_{43}(\theta,\varphi)\right|$ (top), real value $\mathrm{Re}\:Y_{43}(\theta,\varphi)$ (centre) and on the unit sphere (bottom), all with the phase $\exp(\mathrm{i} m\varphi)$ as colouring.
• Figure 2: The mapping of meridians and circles of latitude on a $6^\circ$-graticule in azimuth and a $15^\circ$-graticule in polar angle in Peirce's projection onto a square. The black diamond-shaped line indicates the equator (top). The inverse mapping of a Cartesian grid on Peirce's domain back onto the sphere. There are four points on the equator (two are visible in the front and two on the far side of the sphere) where the conformal factor becomes zero (bottom).
• Figure 3: The Earth in Peirce's quincuncial projection, generated as a topographical map from the superposition of spherical harmonics, up to $\ell_\mathrm{max} = 100$. The yellow diamond shape is the equator in Peirce's projection. The map is smoothed with a Gaussian filter, for suppressing diffraction-like artefacts.
• Figure 4: Zonal spherical harmonic $Y_{40}(\theta,\varphi)$, for $\ell = 4$ and $m = 0$, in a representation with $3d$ orbitals (right) and in comparison with Peirce's projection (left).
• Figure 5: Sectoral spherical harmonic $Y_{44}(\theta,\varphi)$, for $\ell = m = 4$, in a representation with $3d$ orbitals (right) and in comparison with Peirce's projection (left). In both cases, colouring indicates complex phase.