## Academic Use of Magnet Spheres

### Teaching

Small rare-earth magnet spheres are used both in and out of the classroom to teach principles of mathematics, physics, chemistry, biology, and engineering. Shown below is a family of diagonal cubes built with Zen Magnets and illuminated by the setting sun. The magnets within each cube are arranged in a face-centered cubic lattice, which is the crystalline form of salt, aluminum, lead, copper, silver, and gold. Building these cubes assists students in understanding these lattices (Ref. 1). References 1-6 discuss many other educational uses of magnet spheres.

Diagonal cube family, side view

Diagonal cube family, top view

### Research

Small rare-earth magnet spheres have inspired ground-breaking scientific research into the nature of magnetic interactions. An essential component of many magnetic structures, including the diagonal cube, is the filled hexagon, which consists of a ring of six "perimeter" magnets and a seventh magnet at its center. Although all filled hexagons look the same visually, Ref. 8 shows that they come in two distinct magnetic configurations, a "circular state" for which the magnetic moment of the central magnet (the ray pointing from its south pole to its north pole) points toward the center of a perimeter magnet and a "misaligned state" for which this magnetic moment points toward the gap between two perimeter magnets. The first and third panels shown below are the theoretical predictions of the magnetic moments (red arrows) and magnetic fields (blue traces) for the circular and misaligned states, for a filled hexagon of seven magnets whose centers are marked by green dots. (Images made with Andrew Smith's Magnet Plotter) The second and fourth panels are experimental photographic observations of these states, with patterns of iron filings (black) showing the magnetic fields, with gray silhouettes showing the positions of the seven magnets, and with colored traces showing contours of constant magnetic field magnitude. References 7-16 discuss many other discoveries resulting from interest in magnet spheres.

Circular state, predicted

Circular state, observed

Misaligned state, predicted

Misaligned state, observed

### References

Edwards, B., (2014) Educational value of neodymium magnet spheres, in the matter of Zen Magnets, LLC, CPSC Docket No. 12-2, Item 189, Exhibit 4 (30 pp).

Richter, D., (2014). Teaching geometry with magnet sphere kits, in the matter of Zen Magnets, LLC, CPSC Docket No. 12-2, Item 175, Exhibit 3 (6 pp).

Segerman, H. and Zwier, R., (2017). Magnetic Sphere Constructions, in Bridges 2017 Conference Proceedings (8 pp).

Haugen, P., Edwards, B., (2020). Dynamics of two freely rotating dipoles. American Journal of Physics, 88:5, 365-370.

Edwards, B., Edwards, J., (2017). Dynamical interactions between two uniformly magnetized spheres. European Journal of Physics, 38:1, 015205 (25 pp).

Edwards, B., Riffe, D.M, Ji, J., Booth, W.A, (2017). Interactions between uniformly magnetized spheres. American Journal of Physics, 85:2, 130-134.

Schönke, J. and Fried, E. (2017). Stability of vertical magnetic chains. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 473 (2198), p. 20160703 (25 pp).

Smith, A., Haugen, P., Edwards, B., (2022). Hysteretic transition between states of a filled hexagonal magnetic dipole cluster. Journal of Magnetism and Magnetic Materials, 549, 168991 (11 pp).

Ji, J., Edwards, B., Spencer, J., Held, E., (2021). Potential, field, and interactions of multipole spheres: Coated spherical magnets. Journal of Magnetism and Magnetic Materials, 529, 167861 (7 pp).

Edwards, B., Johnson, B., Edwards, J., (2020). Periodic bouncing modes for two uniformly magnetized spheres I: Trajectories. Chaos: An Interdisciplinary Journal of Nonlinear Science, 30:1, 013146 (20 pp).

Edwards, B., Johnson, B., Edwards, J., (2020). Periodic bouncing modes for two uniformly magnetized spheres II: Scaling. Chaos: An Interdisciplinary Journal of Nonlinear Science, 30:1, 013131 (9 pp).

Edwards, B. and Edwards, J., (2017). Periodic nonlinear sliding modes for two uniformly magnetized spheres. Chaos: An Interdisciplinary Journal of Nonlinear Science, 27:5, 053107 (15 pp).

Inoue, J. (2014). 3-d Models for Hyperbolic Tessellation (19 pp).

Egri, S. and Bihari, G. (2018). Self-assembly of magnetic spheres: a new experimental method and related theory. Journal of Physics Communications, 2(10), 105003.

Asset, T., Chattot, R., Fontana, M., Mercier‐Guyon, B., Job, N., Dubau, L., and Maillard, F. (2018). Cover Feature: A Review on Recent Developments and Prospects for the Oxygen Reduction Reaction on Hollow Pt alloy Nanoparticles. ChemPhysChem, 19(13), 1549-1549.

Chattot, R., Martens, I., Scohy, M., Herranz, J., Drnec, J., Maillard, F., and Dubau, L. (2019). Disclosing Pt-bimetallic alloy nanoparticle surface lattice distortion with electrochemical probes. ACS Energy Letters, 5(1), 162-169.