|The distribution arrangement when six floating positively charged needles are put in the vicinity of the negatively-charged side of a large overhead hovering magnet, an experiment done in 1878 by American physicist Alfred Mayer. |
Gilbert | Loadstones
In circa 1590, English chemist-physicist William Gilbert conducted a “floating loadstones experiment”, in which spherical lodestones were floated on water in small wooden boats, something akin to the following toy version of the experiment: 
according to which he found that magnetic forces often produce circular motion (see also: turning tendencies). 
|A video of Libb Thims (2015) performing the Mayer "floating magnets experiment" to a group of kids.|
Mayer | Magnetized needles
In this experiment, Mayer took an assorted number of magnetized needles and stuck them through corks, floating in water, such that their north poles, i.e. positive ends, were all facing upwards at the same height above the water, all repelling each other equally. Mayer then held the south pole or negative end of a more powerful magnet some distance above the surface of the water to attract the needles simultaneously towards this center. The idea was to see what geometric equilibrium stability patterns formed for different numbers of needles under the influence of the similar attraction for the opposite pole as this attraction conflicted with the effects of mutual inter-magnet repulsion between the like (positive) poles.
Mayer found an interesting pattern: the like positive-poled needles liked to arrange themselves into concentric rings of stability. Three magnets formed a triangle, four a square, and a five a pentagon. When six magnets were used, however, one went to the center leaving the others to form a ringed pentagon. With eight magnets, two went to the center and six remained in an outer ring. As more magnets were added, up to eighteen, a separate inner system formed distinct from that of an outer ringed system. Beyond this, three systems formed an inner system, a middle ring, and an outer ring; and for a still larger number of magnets a four system ring pattern would emerge, and so on. Mayer's original diagrams for arrangements of 1-20 needles are shown below: 
|Geometric arrangements of 1-12 floating positively-changed needles; above which is held the positive end of a larger magnet.||Geometric arrangements of 13-20 floating positively-changed needles; above which is held the positive end of a larger magnet.|
To explain these curious equilibrium stability patterns, Mayer reasoned that the competing attracting and repulsing tendencies, between the like poles and oppositely charged central pole, respectively, resulted in the formation of geometric stability patterns, according to which the size of each pattern was determined by the relative strengths of the individual attractions and repulsions. The number of magnets, subsequently, determined the different stability patterns, in which each arrangement was the result of a local minimum energy configuration.
|In 1897, English physicist Joseph Thomson began to use the results of American physicist Alfred Mayer's floating magnets experiment to develop one of the first experimentally-based structural model of the atom.|
Thomson | Atomic structure
In 1897, English physicist Joseph Thomson (1856-1940) was aware of Mayer's experiment and in his “Cathode Rays” article began to cite him as follows: 
“If we regard the chemical atom as an aggregation of a number of primordial atoms, the problem of finding the configurations of stable equilibrium for a number of equal particles acting on each other according to some law of force—whether that of Boscovich, where the force between them is a repulsion when they are separated by less than a certain critical distance, and an attraction when they are separated by less than a certain critical distance, and an attraction when they are separated by a greater distance, or even the simpler case of a number of mutually repellent particles held together by a central force—is of great interest in connexion with the relation between the properties of an element and its atomic weight. Unfortunately the equations which determine the stability of such a collection of particles increase so rapidly in complexity with the number of particles that a general mathematical investigation is scarcely possible. We can, however, obtain a good deal of insight into the general laws which govern such configurations by the use of models, the simplest of which is the floating magnets of Professor Mayer. In this model the magnets arrange themselves in equilibrium under the mutual repulsions and a central attraction caused by the pole of a large magnet placed above the floating magnets.”
Thomson gives the following truncation of Mayer's results:
where, for example, 220.127.116.11 means an arrangement with one magnet in the middle, then a ring of six, then a ring of ten, and a ring of twelve outside, which corresponds to diagram #17 above.
In 1904, Thomson would go on to use this structural ordering insight, in his more famous “On the Structure of the Atom” article, to develop what soon came to be dubbed the "plum pudding model" of atomic theory, the key sentence of which is shown below, next to conceptual images of this atomic model: 
This model was later superseded by the Rutherford model (1911), developed by Ernest Rutherford, and then the Bohr model (1913), developed by Niels Bohr, as outlined below: 
“The atoms of the elements consist of a number of negatively electrified corpuscles enclosed in a sphere of uniform positive electrification.”
Plum pudding model (1904) Rutherford model (1911) Bohr model (1913)
Thims | Quantum electrodynamic evolution
In 2014, American electrochemical engineer Libb Thims redid the Gilbert-Mayer experiment using neodymium magnets on floating cork, according to which, to explain the turnover rate paradox, the electromagnetic field is what “holds” shapes of the positive and negative charges in regions of space, and that the geometric shape—like a person—will hold in form, if one charge is pushed into the pattern while another is removed:
As magnets are added, as Mayer famously diagrammed (up to 20 magnets), the 3D pyramidal structure will grow geometrically: a triangle base at 3 magnets, a square base at 4 magnets, a hexagon base at 5 magnets, a two ringed structure base at 10 magnets, a center magnetic surrounded by two ringed base at 15 magnets, and so on. In this sense, one could conceptually understand the "structure holding" paradox if one were to add a base magnet to a given geometry while simultaneous removing one, and do this for all the magnets of the structure, similar to the ship, river, or carriage variants of the paradox, and then ask if the resulting turned over or magnetic replaced 3D geometric structure is the same growing "thing" or a different growing thing?
5 magnets 6 magnets 7 magnets
This model, naturally enough, can be scaled up to the people viewed as powered atomic geometries or human molecules, whereby the positively charged (floating) magnets and positively (hovering) magnets become atoms (elements), with varying amounts of positive or negative charged electron valency, albeit the description becomes more complicated owing to the effect that quantum electrodynamic changes are involved owing to cyclical sunlight photon input (cause) and resulting electron orbital transitioning (effect), whereby molecular animation results, something that does not occur in the magnet, ship, carriage, or river water scenarios.
1. (a) Mayer, Alfred Marshall. "Experiments with Floating and Suspended Magnets, Illustrating the Action of Atomic Forces, the Molecular Structure of Matter, Allotropy, Isomerism, and the Kinetic Theory of Gases." Scientific American, supplement 5 [no. 129] (June 22, 1878): 2045-2047.
(b) Mayer, Alfred Marshall. "A Note on Experiments with Floating Magnets; Showing the Motions and Arrangements in a Plane of Freely Moving Bodies, Acted on by Forces of Attraction and Repulsion; and Serving in the Study of the Directions and Motions of the Lines of Magnetic Force." American Journal of Science, 3rd ser., 15 (1878): 276-277.
(c) Also: "Note on Floating Magnets [letter]." Ibid. 15 (1878): 477-478;
2. Mayer, Alfred M. (1878). "On the Morphological Laws of the Configurations Formed by Magnets Floating Vertically and Subjected to the Attraction of a Superposed Magnet; with Notes on Some of the Phenomena in Molecular Structure Which These Experiments May Serve to Explain and Illustrate." American Journal of Science, 16: 247-256.
3. Thomson, Joseph, J. (1897). “Cathode Rays.” Philosophical Magazine, 44, 293.
4. Thomson, Joseph, J. (1904). “On the Structure of the Atom: an Investigation of the Stability and Periods of Oscillation of a number of Corpuscles arranged at equal intervals around the Circumference of a circle; with application of the results to the Theory of Atomic Structure.” Philosophical Magazine, Series 6, Vol. 7, No. 39. March, pgs. 237-65.
5. Plum pudding model – Wikipedia.
6. Pumfrey, Stephen and Tilley, David. (2003). “William Gilbert: Forgotten Genius”, Physics World, Nov.
7. Magnetic fun and facts – EliWhitney.org.
● Thims, Libb. (2007). Human Chemistry (Volume One) (floating magnets experiment, pgs. 217-18, 248) (preview), (Google books). Morrisville, NC: LuLu.
● Thims, Libb. (2015). “Floating Magnets Experiment | Mayer (1878)” (Ѻ), Human Chemistry 101, YouTube, Mar 10.
● Fowler, Michael. (c.2010). “Models of the Atom”, Lectures, University of Virginia.