The Joy of Mathematics

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First annotation on Sep 11th, 2025.

Some of the properties of the cycloid discovered during the 17th century are:1. Its length is four times the diameter of the rotating circle. It is especially interesting to find that its length is a rational number independent of π. 2. The area under the arch is three times the area of the rotating circle. 3. The point on the circle that is tracing the cycloid takes on different speeds—in fact at one place, P5, it is even at rest. 4. When marbles are released from different points of a cycloid shaped container, they arrive at the bottom simultaneously.
There are many fascinating paradoxes tied to the cycloid. The train paradox is especially intriguing. At any instant a moving train never moves entirely in the direction the engine is pulling it. There is always part of the train moving in the opposite direction than that in which the train is moving.

Penrose has been responsible for the theory of twistors. Although twistors are not visible, Penrose believes that space and time are interwoven by the interaction of twistors.