Aristotle's wheel paradox
Aristotle's wheel paradox is a paradox or problem appearing in the Greek work Mechanica traditionally attributed to Aristotle. A wheel can be depicted in two dimensions using two circles. The larger circle is tangent to a horizontal surface (e.g. a road) that it can roll on. The smaller circle has the same center and is rigidly affixed to the larger one. The smaller circle could depict the bead of a tire, a rim the tire is mounted on, an axle, etc. Assume the larger circle rolls without slipping (or skidding) for a full revolution. The distances moved by both circles are the same length, as depicted by the blue and red dashed lines and the distance between the two black vertical lines. The distance for the larger circle equals its circumference, but the distance for the smaller circle is longer than its circumference: a paradox or problem.
The paradox is not limited to a wheel. Other things depicted in two dimensions show the same behavior. A roll of tape does. A typical round bottle or jar rolled on its side does -- the smaller circle depicting the mouth or neck of the bottle or jar.
There are a few things that would be depicted with the brown horizontal line in the image tangent to the smaller circle rather than the larger one. Examples are a typical train wheel, which has a flange, or a barbell straddling a bench. Drabkin called these Case II and the kind in the image Case I.  A similar but not identical analysis applies.
History of the paradox
In antiquity, the wheel problem was described in the Aristotelian Mechanica, as well as in the Mechanica of Hero of Alexandria. In the former it appears as "Problem 24", where the description of the wheel is given as follows.
For let there be a larger circle ΔZΓ a smaller EHB, and A at the centre of both; let ZI be the line which the greater unrolls on its own, and HK that which the smaller unrolls on its own, equal to ZΛ. When I move the smaller circle, I move the same centre, that is A; let the larger be attached to it. When AB becomes perpendicular to HK, at the same time AΓ becomes perpendicular to ZΛ, so that it will always have completed an equal distance, namely HK for the circumference HB, and ZΛ for ZΓ. If the quarter unrolls an equal distance, it is clear that the whole circle will unroll an equal distance to the whole circle, so that when the line BH comes to K, the circumference ZΓ will be ZΛ, and the whole circle will be unrolled. In the same way, when I move the large circle, fitting the small one to it, their centre being the same, AB will be perpendicular and at right angles simultaneously with AΓ, the latter to ZI, the former to HΘ. So that, when the one will have completed a line equal to HΘ, and the other to ZI, and ZA becomes again perpendicular to ZΛ, and HA to HK, so that they will be as in the beginning at Θ and I.
The problem is then stated:
Now since there is no stopping of the greater for the smaller so that it [the greater] remains for an interval of time at the same point, and since the smaller does not leap over any point, it is strange that the greater traverses a path equal to that of the smaller, and again that the smaller traverses a path equal to that of the larger. Furthermore, it is remarkable that, though in each case there is only one movement, the center that is moved in one case rolls a great distance and in the other a smaller distance.
In the Scientific Revolution
The mathematician Gerolamo Cardano discusses the problem of the wheel in his 1570 Opus novum de proportionibus numerorum, taking issue with the presumption the analysis of the problem in terms of motion. Mersenne further discussed the wheel in his 1623 Quaestiones Celeberrimae in Genesim, where he suggests that the problem can be analysed by a process of expansion and contraction of the two circles. But Mersenne remained unsatisfied with his understanding, writing,
Indeed I have never been able to discover, and I do not think any one else has been able to discover whether the smaller circle touches the same point twice, or proceeds by leaps and sliding.
In his Two New Sciences, Galileo uses the problem of the wheel to argue for a certain kind of atomism. Galileo begins his analysis by considering a pair of concentric hexagons, as opposed to a pair of circles. Imagining this hexagonal wheel "rolling" on a surface, Galileo notices that the inner hexagon "jumps" a little space, with each roll of the outer hexagon onto a new face. He then imagines what would happens in the limit as the number faces on the polygon becomes very large, and finds that the little space that is "jumped" by the inner polygon becomes smaller and smaller, writing:
Therefore a larger polygon having a thousand sides passes over and measures a straight line equal to its perimeter, while at the same time the smaller one passes an approximately equal line, but one interruptedly composed of a thousand little particles equal to its thousand sides with a thousand little void spaces interposed — for we may call these "void" in relation to the thousand linelets touched by the sides of the polygon.
Since the circle is just the limit in which the number of faces on the polygon becomes infinite, Galileo finds that Aristotle's wheel contains material that is filled with infinitesimal spaces or "voids", and that "the interposed voids are not quantified, but are infinitely many". This leads Galileo to conclude that a belief in atoms, in the sense that matter is "composed of infinitely many unquantifiable atoms" is sufficient to solve the problem of the wheel.
In the 19th century
Bernard Bolzano discussed Aristotle's wheel in The Paradoxes of the Infinite (1851), a book that influenced Georg Cantor and subsequent thinkers about the mathematics of infinity. Bolzano observes that there is a bijection between the points of any two similar arcs, which can be implemented by drawing a radius, remarking that the history of this apparently paradoxical fact goes back to Aristotle.
In the 20th century
The author of Mathematical Fallacies and Paradoxes uses a dime glued to a half-dollar with their centers aligned, both fixed to an axle, as a model for the paradox. The dime serves as the smaller circle and the half-dollar as the larger one. He writes:
This is the solution, then, or the key to it. Although you are careful not to let the half-dollar slip on the tabletop, the “point” tracing the line segment at the foot of the dime is both rotating and slipping all the time. It is slipping with respect to the tabletop. Since the dime does not touch the table top, you do not notice the slipping. If you can roll the half-dollar along the table and at the same time roll the dime (or better yet the axle) along a block of wood, you can actually observe the slipping. If you have ever parked too close to the curb, you have noticed the screech made by your hubcap as it slips (and rolls) on the curb while your tire merely rolls on the pavement. The smaller the small circle relative to the large circle, the more the small one slips. Of course the center of the two circles does not rotate at all, so it slides the whole way.
Alternatively, one can reject the assumption that the smaller circle is independent of the larger circle. Imagine a tire as the larger circle, and imagine the smaller circle as the interior circumference of the tire and not as the rim. The movement of the inner circle is dependent on the larger circle. Thus its movement from any point to another can be calculated by using an inverse of their ratio.
Analysis and solutions
The paradox is that the smaller inner circle moves 2πR, the circumference of the larger outer circle with radius R, rather than its own circumference. If the inner circle were rolled separately, it would move 2πr, its own circumference with radius r. The inner circle is not separate but rigidly connected to the larger. So 2πr is a red herring. The inner circle's center is relevant, but its circumference is not.
- First solution
If the smaller circle depends on the larger one (Case I), then the larger circle forces the smaller one to traverse the larger circle’s circumference. If the larger circle depends on the smaller one (Case II), then the smaller circle forces the larger one to traverse the smaller circle’s circumference. This is the simplest solution.
- Second solution
This solution considers the transition from starting to ending positions. Let Pb be a point on the bigger circle and Ps be a point on the smaller circle, both on the same radius. For convenience, assume they are both directly below the center, analogous to both hands of a clock pointing towards six. Both Pb and Ps travel a cycloid path as they roll together one revolution. The two paths are pictured here: http://mathworld.wolfram.com/Cycloid.html and http://mathworld.wolfram.com/CurtateCycloid.html
While each travels 2πR horizontally from start to end, Ps's cycloid path is shorter and more efficient than Pb's. Pb travels farther above and farther below the center's path -- the only straight one -- than does Ps. The nearby image shows the circles before and after rolling one revolution. It shows the motions of the center, Pb, and Ps, with Pb and Ps starting and ending at the top of their circles. The green dash line is the center's motion. The blue dash curve shows Pb's motion. The red dash curve shows Ps's motion. Ps's path is clearly shorter than Pb's. The closer Ps is to the center, the shorter, more direct, and closer to the green line its path is.
If Pb and Ps were anywhere else on their respective circles, the curved paths would be the same length. Summarizing, the smaller circle moves horizontally 2πR because any point on the smaller circle travels a shorter, more direct path than any point on the larger circle.
- Third solution
This solution only compares the starting and ending positions. The larger circle and the smaller circle have the same center. If said center is moved, both circles move the same distance, which is a necessary property of translation (geometry) and equals 2πR in the experiment. QED. Also, every other point on both circles has the same position relative to the center before and after rolling one revolution (or any other integer count of revolutions).
- Drabkin, Israel E. (1950). "Aristotle's Wheel: Notes on the History of a Paradox". Osiris. 9: 162–198. doi:10.1086/368528. JSTOR 301848.
- Leeuwen, Joyce van (2016-03-17). The Aristotelian Mechanics: Text and Diagrams. Springer. ISBN 9783319259253.
- Cardano, Geronimo (1570). Opus novum de proportionibus numerorum ...: Praeterea Artis magnae sive de regulis algebraicis liber unus ... Item De regula liber ...
- Mersenne, Marin (1623). Quaestiones celeberrimae in Genesim ... (in Latin).
- Galilei, Galileo; Drake, Stillman (2000). Two New Sciences: Including Centers of Gravity & Force of Percussion. Wall & Emerson. ISBN 9780921332503.
- Bunch, Bryan H. (1982). Mathematical Fallacies and Paradoxes. Van Nostrand Reinhold. pp. 3–9. ISBN 0-442-24905-5.