We show that a two-dimensional square lattice of magnets can be studied by placing small cylindrical neodymium magnets inside plastic spherical shells and floating them on water, leaving their magnetic moments free to re-orient within the plane. Experimentally, anti-correlated dipole orientations between nearest neighbors appear to be favored energetically. This motivates the construction of a simplified single-variable energy function for a 2D square lattice of magnetic dipoles. For odd numbers of spheres, this ansatz yields a continuum of dipole configurations with the same energies, matching the observed behavior that the orientation of the dipoles in these lattices can be rotated freely. The behavior of square lattices with even numbers of spheres is strikingly different, showing strongly preferred orientations. While the energy calculated in this simplified model is larger than that of the actual ground state for finite size clusters, its asymptotic value in the limit where the number of spheres goes to infinity is in good agreement with the literature value. Additionally, rectangular arrangements of magnetic spheres with and without a defect are analyzed within the class of the single variable energy function. Simple experimental demonstrations qualitatively reproduce several interesting results obtained from all these analyses.

1.
N.
Vandewalle
and
S.
Dorbolo
, “
Magnetic ghosts and monopoles
,”
New. J. Phys.
16
,
013050-1
(
2014
).
2.
R.
Messina
and
I.
Stanković
, “
Self-assembly of magnetic spheres in two dimensions: The relevance of onion-like structures
,”
EPL
110
,
46003-1
46003-5
(
2015
).
3.
D. S.
Borges
,
H. J.
Herrmann
,
H. A.
Carmona
,
J. S.
Andrade
, Jr.
, and
A. D.
Araújo
, “
Patterns formed by chains of magnetic beads
,”
EPJ Web Conf.
249
,
15004
(
2021
).
4.
N.
Vandewalle
and
A.
Wafflard
, “
Ground state of magnetocrystals
,”
Phys. Rev. E
103
,
032117
(
2021
).
5.
S.
Egri
and
G.
Bihari
, “
Self-assembly of magnetic spheres: A new experimental method and related theory
,”
J. Phys. Commun.
2
,
105003
(
2018
).
6.
D.
Vella
,
E.
du Pontavice
,
C. L.
Hall
, and
A.
Goriely
, “
The magneto-elastica: From self-buckling to self-assembly
,”
Proc. R. Soc. A
470
,
20130609
(
2013
).
7.
S.
Hidalgo-Caballero
,
Y. Y.
Escobar-Ortega
,
R. I.
Becerra-Deana
,
J. M.
Salazar
, and
F.
Pacheco- Vázquez
, “
Mechanical properties of macroscopic magnetocrystals
,”
J. Magn. Magn. Mater.
479
,
149
155
(
2019
).
8.
T. A. G.
Hageman
,
P. A.
Löthman
,
M.
Dirnberger
,
M.
Elwenspoek
,
A.
Manz
, and
L.
Abelmann
, “
Macroscopic equivalence for microscopic motion in a turbulence driven three-dimensional self-assembly reactor
,”
J. Appl. Phys.
123
,
024901
(
2018
).
9.
G.
Pál
,
F.
Kun
,
I.
Varga
,
D.
Sohler
, and
G.
Sun
, “
Attraction-driven aggregation of dipolar particles in an external magnetic field
,”
Phys. Rev. E
83
,
061504
(
2011
).
10.
B. F.
Edwards
,
D. M.
Riffe
,
J.-Y.
Ji
, and
W. A.
Booth
, “
Interactions between uniformly magnetized spheres
,”
Am. J. Phys.
85
,
130
134
(
2017
).
11.
F.
Deiβenbeck
,
H.
Löwen
, and
E. C.
Oğuz
, “
Ground state of dipolar hard spheres confined in channels
,”
Phys. Rev. E
97
,
052608
(
2018
).
12.
See supplementary material online for detailed descriptions regarding our theoretical approach and movies about experimental demonstrations.
13.
S.
Borgers
,
S.
Völkel
,
W.
Schöpf
, and
I.
Rehberg
, “
Exploring cogging free magnetic gears
,”
Am. J. Phys.
86
,
460–470
(
2018
).
14.
T. A.
Prokopieva
,
V. A.
Danilov
,
S. S.
Kantorovich
, and
C.
Holm
, “
Ground state structures in ferrofluid monolayers
,”
Phys. Rev. E
80
,
031404
(
2009
).
15.
J.
Schönke
,
W.
Schöpf
, and
I.
Rehberg
, “
Magnetkugeln − ein 10-Euro-Labor
,”
Phys. J.
15
(
4
),
31
37
(
2016
).
16.
A. J.
Petruska
and
J. J.
Abbott
, “
Optimal permanent-magnet geometries for dipole field approximation
,”
IEEE Trans. Magn.
49
,
811
819
(
2013
).
17.
J. D.
Jackson
,
Classical Electrodynamics
,
3rd ed.
(
Willy
,
New York
,
1999
).
18.
S.
Hartung
,
F.
Sommer
,
S.
Völkel
,
J.
Schönke
, and
I.
Rehberg
, “
Assembly of eight spherical magnets into a dotriacontapole configuration
,”
Phys. Rev. B
98
,
214424
(
2018
).
19.
P. I.
Belobrov
,
R. S.
Gekht
, and
V. A.
Ignatchenko
, “
Ground state in systems with dipole interaction
,”
Sov. Phys. JETP
57
,
636
642
(
1983
).
20.
K.
De'Bell
,
A. B.
MacIsaac
,
I. N.
Booth
, and
J. P.
Whitehead
, “
Dipolar-induced planar anisotropy in ultrathin magnetic films
,”
Phys. Rev. B
55
,
15108
15118
(
1997
).
21.
E. Y.
Vedmedenko
, “
Influence of the lattice discreteness on magnetic ordering in nanostructures and nanoarrays
,”
Phys. Status Solidi B
244
,
1133
1165
(
2007
).
22.
J.
Schönke
,
T. M.
Schneider
, and
I.
Rehberg
, “
Infinite geometric frustration in a cubic dipole cluster
,”
Phys. Rev. B
91
,
020410
(
2015
).
23.
T.
Kawai
,
D.
Matsunaga
,
F.
Meng
,
J. M.
Yeomans
, and
R.
Golestanian
, “
Degenerate states, emergent dynamics and fluid mixing by magnetic rotors
,”
Soft Matter
16
,
6484
6492
(
2020
).
24.
L.
Spiteri
and
R.
Messina
, “
Columnar aggregation of dipolar chains
,”
EPL
120
,
36001-1
36001-6
(
2017
).
25.
S. K.
Baek
,
P.
Minnhagen
, and
B. J.
Kim
, “
Kosterlitz-Thouless transition of magnetic dipoles on the two-dimensional plane
,”
Phys. Rev. E
83
,
184409
(
2011
).
26.
F.
Ebert
,
P.
Dillmann
,
G.
Maret
, and
P.
Keim
, “
The experimental realization of a two-dimensional colloidal model system
,”
Rev. Sci. Inst.
80
,
083902
(
2009
).
27.
H.
Massana-Cid
,
F.
Meng
,
D.
Matsunaga
,
R.
Golestanian
, and
P.
Tierno
, “
Tunable self-healing of magnetically propelling colloidal carpets
,”
Nat. Commun.
10
,
2444–2452
(
2019
).
28.
J. R.
Reitz
,
F. J.
Milford
, and
R. W.
Christy
,
Foundations of Electromagnetic Theory
,
4th ed
. (
Addison-Wesley
,
Reading, MA
,
1993
).

Supplementary Material

AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.