Aristotle's wheel

The wheel of Aristotle or Rota Aristotelis is a mechanical paradox that is described in the Quaestiones mechanicae . The work is attributed to Aristotle , but the actual author is not known.

Problem Description

After a complete revolution, the connecting sections between the points closest to the ground of both wheels are the same length.

The point of contact of a wheel with the ground is considered. With one complete revolution of the wheel, this point draws a distance in the direction of travel , the length of which is equal to the circumference of the wheel. If a second wheel with a smaller diameter is rigidly attached to the same axle and the point that is closest to the ground is also observed here, this point covers the same distance during the rotation. The obvious contradiction arises when it is concluded from the equality of the two routes that both wheels have the same circumference.

solution

The large hexagon leaves a continuous trail as it rolls, while the path of the smaller hexagon has gaps.

From a physical point of view, the distance covered by the point under consideration corresponds to the wheel circumference only if there is a real rolling movement . However, this is not possible for both bikes at the same time. If the large wheel rolls on the ground, the smaller wheel also performs a sliding movement along the imaginary line. Conversely, the large wheel spins partially ( slip ) when the smaller wheel rolls.

In 1638 Galileo Galilei presented a mathematical approach to the paradox in his work Discorsi e dimostrazioni matematiche . For this, the two wheels are initially assumed to be hexagons of different sizes . When you roll down the large hexagon you get a continuous line, the length of which is equal to the circumference of this hexagon. This also applies to the route that the small hexagon leaves behind; however, it is interrupted by several gaps. If the number of corners is now increased to infinity, so that the wheels are circular again, the infinitely many gaps become infinitely small and the route appears to be connected. Only the unrolled stretch of the big wheel is uninterrupted and therefore actually as long as the wheel circumference.

Individual evidence

↑ Thomas Nelson Winter: The Mechanical Problems in the Corpus of Aristotle (= Faculty Publications, Classics and Religious Studies Department. Paper 68). University of Nebraska-Lincoln, 2007, p. 26 ff. ( PDF; 352 KB ).
^ Stanley J. Farlow: Paradoxes in Mathematics . Dover Publications, Mineola (New York) 2014, ISBN 978-0-486-49716-7 , pp. 92 ff .
↑ Thomas Sonar : 3000 Years of Analysis: History - Cultures - People . 2nd Edition. Springer Verlag, Berlin 2016, ISBN 978-3-662-48917-8 , pp. 203 f .

[1] Thomas Nelson Winter: The Mechanical Problems in the Corpus of Aristotle (= Faculty Publications, Classics and Religious Studies Department. Paper 68). University of Nebraska-Lincoln, 2007, p. 26 ff. ( PDF; 352 KB ).

[2] Stanley J. Farlow: Paradoxes in Mathematics . Dover Publications, Mineola (New York) 2014, ISBN 978-0-486-49716-7 , pp. 92 ff .

[3] Thomas Sonar : 3000 Years of Analysis: History - Cultures - People . 2nd Edition. Springer Verlag, Berlin 2016, ISBN 978-3-662-48917-8 , pp. 203 f .