**Author:** Lori-Anne Gardi **Date:** 2026-01-15 --- ## Abstract Physical systems as diverse as magnetic fields and atomic electron orbitals exhibit rich three-dimensional structures with lobes and nodes. Although these systems differ in physical nature (vector fields versus scalar wavefunctions), their angular patterns are governed by the same mathematical functions: spherical harmonics (Griffiths 2018; Jackson 1999). By examining magnetic field projections via Hall sensors and electron probability distributions via wavefunction squaring, we reveal a self-similar structure: both reduce complex three-dimensional information into scalar measurements that highlight relational alignments and cancellations. This conceptual and mathematical parallel provides insight into the recurring patterns in classical and quantum systems, illustrating how universal mathematical structures manifest across different physical domains. --- ## Introduction Understanding the spatial structure of physical fields often requires interpreting scalar measurements derived from richer three-dimensional information. In magnetism, Hall sensors measure a single component of a vector field (Ramsden 2014; Wikipedia 2025b), producing lobes and nodes that depend on the sensor orientation. In quantum mechanics, the probability density of an electron orbital is obtained by squaring its wavefunction, which removes phase information and reveals regions of high and low probability (Griffiths 2018). At first glance, these systems appear unrelated: one is classical and vectorial, the other quantum and scalar. However, both can be naturally described using spherical coordinates, with their angular structure determined by spherical harmonics Yₗᵐ(θ, φ) (Griffiths 2018; Jackson 1999). These functions encode the lobes, nodes, and symmetries observed in both magnetic projections and orbital shapes. By examining the role of projections, phase, and relational alignment in both systems, we uncover a self-similar pattern: the rules producing lobes and nodes are conceptually analogous, even if the underlying physics differs. This perspective allows us to connect classical and quantum systems through a shared mathematical framework, providing a deeper understanding of recurring patterns in nature and supporting a fractal-inspired view of physical structure. --- ## Magnetic Field Projections Consider a magnetic dipole oriented along the z-axis. The field at a point in space is a vector **B** = (Bx, By, Bz). A Hall sensor aligned along a particular axis measures the component of the field along that axis (Ramsden 2014): ![Eq1.png](https://files.peakd.com/file/peakd-hive/fractalwoman/23wgoMjaWBWWX1x2Gt6eBHqNcYA2KRQuf8rW2nnrAG55iTFYjgQMUYUyh9awRvkjbKnCc.png) where n̂ is the unit vector along the sensor direction. If the field is perpendicular to the sensor axis, the measured scalar is zero, creating a *measurement node*. Different sensor orientations produce qualitatively different scalar field maps, even though the underlying vector field remains unchanged. --- ## Electron Orbitals and Wavefunction Squaring An electron orbital is described by a wavefunction Ψ(r), which can have positive and negative regions as well as complex phase (Griffiths 2018). The probability density is obtained by squaring the wavefunction: ![Eq2.png](https://files.peakd.com/file/peakd-hive/fractalwoman/Eos7bKwdX2gp4K7AVwaAD28pNVCd1fN7kB9gHNMy4c1ZL7No2DdnNjL4bc8CUaPxc89.png) Squaring eliminates the phase, producing a scalar field that represents where the electron is likely to be found. Nodes in this context occur where the wavefunction crosses zero, analogous to points of destructive interference in overlapping waves. --- ## Analogy Between Systems The analogy arises when we consider that both systems have rich underlying structures and that the scalar measurement arises from a relational operation: - In magnets, the dot product of the field vector with the sensor axis determines the measured value (Ramsden 2014). - In orbitals, squaring the wavefunction removes the phase, yielding a scalar probability (Griffiths 2018). In both cases: 1. There is a "node" where the measured or projected scalar vanishes. 2. The node does not imply absence of the underlying structure. 3. The measurement outcome depends on the relational alignment (sensor direction or phase reference). Despite differences in geometry and rules (orthogonality vs. destructive interference), the two systems exhibit *self-similar* patterns: relational information in a higher-dimensional field manifests as scalar features in a lower-dimensional measurement. --- ## Mathematical Foundations and Spherical Harmonics Both magnetic fields and atomic orbitals are naturally described in spherical coordinates, where the angular dependence is elegantly captured by *spherical harmonics* Yₗᵐ(θ, φ) (Jackson 1999; Griffiths 2018). These functions define patterns on the surface of a sphere and provide a basis for expressing directional structure in three dimensions. ### Magnetic Fields For a magnetic multipole, the scalar potential outside sources can be expressed as: ![Eq3.png](https://files.peakd.com/file/peakd-hive/fractalwoman/23viTbagt4zJdGWyy9REPWUj5jGA5kDKX4HqjBbBPXFFkfUhs2qDpYrq8too6yYxaHNxU.png) The magnetic field is then obtained as the negative gradient: ![Eq4.png](https://files.peakd.com/file/peakd-hive/fractalwoman/23xAyLrK3PZmLLmH1xEfZALDPJ3sC5RtBNrpHQCP7kwFqTCZoujpp3xBSXnhxkcNXnetj.png) Measuring a single component of **B** (via a Hall sensor) gives a scalar projection that depends on the orientation, producing lobes and nodes analogous to orbital probability patterns. ### Electron Orbitals For an electron in a central potential, the wavefunction separates into radial and angular parts: ![Eq5.png](https://files.peakd.com/file/peakd-hive/fractalwoman/23x1agUjXhu3W4Fr5HSzKcqfxdcEYP1tYFByjw9AVVtX1Y6EDYPivX3bDtyHyoWCjtrDG.png) Squaring the wavefunction gives the probability density: ![Eq6.png](https://files.peakd.com/file/peakd-hive/fractalwoman/23x1agUjXhu3Ww8tMFbEupBchLqnZuRZ49UzRxAXXXNLrggX1kFRpqsUNwDK6yd7c5Psy.png) This produces the familiar orbital shapes. ### Bridging the Two Domains Spherical harmonics serve as the common mathematical bridge: - In magnetic fields, they define the angular pattern of the scalar potential, whose gradient produces the directional vector field. - In orbitals, they define the angular pattern of the wavefunction, whose squared magnitude produces the scalar probability distribution. Despite differences in physical meaning, the underlying angular structure is governed by the same mathematical functions. Lobes, nodes, and symmetries have clear analogues between the two domains. --- ## Discussion Including spherical harmonics as a unifying mathematical framework clarifies why magnetic field projections and electron orbital probabilities exhibit self-similar structures (Jackson 1999; Griffiths 2018). In both cases, the angular dependence of the underlying field is governed by the same family of functions, Yₗᵐ(θ, φ) Projecting the vector field onto a sensor axis reduces 3D information to a scalar map, while squaring the wavefunction removes phase information, producing a scalar probability distribution determined by the same spherical harmonics. This self-similarity highlights recurring mathematical structures underlying seemingly distinct physical systems. --- ## Conclusion By analyzing magnetic field projections and electron orbital probability distributions through spherical harmonics, we uncover a unifying pattern shared by classical and quantum systems (Jackson 1999; Griffiths 2018). This self-similarity reflects the recurring role of relational alignment across domains. Universal mathematical structures manifest in diverse physical contexts, supporting a fractal-inspired view of the universe in which similar patterns recur across scales and systems. --- ## References - Griffiths, David J. (2018). *Introduction to Quantum Mechanics*, 3rd ed. Pearson. - Jackson, John D. (1999). *Classical Electrodynamics*, 3rd ed. Wiley. - Ramsden, Edward. (2014). *Hall-Effect Sensors: Theory and Applications*, Elsevier. - Wikipedia contributors. (2025b). "Hall Effect." *Wikipedia*. [https://en.wikipedia.org/wiki/Hall_effect](https://en.wikipedia.org/wiki/Hall_effect) - Wikipedia contributors. (2025d). "Vector Spherical Harmonics." *Wikipedia*. [https://en.wikipedia.org/wiki/Vector_spherical_harmonics](https://en.wikipedia.org/wiki/Vector_spherical_harmonics)

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