Do you remember plastic cat's eyes, light reflec­tors that we attach to the bicycle wheels? Attach one adjoining the border of the wheel and follow its trajec­tory. The curves we obtain belong to the family of cycloids.

The wheel is the cold the gener­ating disk (or circle) of the cycloid.

But let's return to our century and take use modern tech­nique. There is a stone on the bikes way that got stuck in the wheels tread. In what direc­tion will it fly after several turns when it leaves the tread? In the direc­tion of the bike or in the oppo­site?

As we know, the motion should start tangent to its old trajec­tory. A tangent to a cycloid has always the same direc­tion as the bike and passes through the top point of the gener­ating circle. It will fly in the same direc­tion.

Do you remember driving your bicycle over the puddles without a back wing in your child­hood? A wet stripe on your back was an every-day confir­ma­tion of this result.

The XVII century is the century of the cycloid. The best scien­tists were studying its prop­er­ties.

Which trajec­tory moves a body with grav­i­ta­tion acting on it for one point to another in the shortest time? This was one of the first prob­lems of the branch of science which is now called vari­a­tional calculus.

One can mini­mize (or maxi­mize) different para­me­ters: the length of a path, speed, time. The problem of brachis­tochrone is to mini­mize time (which is accen­tu­ated by the name: brachistos means the shortest, and chronos means time in Greek).

The first that comes to mind is a straight-line trajec­tory. Let's also consider a reversed cycloid with the cusp­idal point in the upper of two given points. And, following Galileo Galilei, a — quarter of a the circle connecting our points.

Build bobsleighing tracks of this shapes and watch which of the bobsleigh comes the first.

The history of bobsleighing rises to Switzer­land. In 1924 in a french town Shamonie the First Winter Olympics take place. The have already bobsleighing compe­ti­tions included for couples and quadru­ples. The only year when a bobsleighing team consisted of 5 persons was 1928. Since then only teams of two and four men compete. There is a lot of inter­esting in the bobsleighing rules. Of course, there is a limit on the bobsleigh weight, but there are even limi­ta­tions on the mate­rials one can use in bobsleigh blades (the front pair of them is connected to the wheel and the back pair is fixed). For example, one can not use radium to make his blades.

Let's start our quadru­ples. Which of the bobsleighs comes the first? The green one, spon­sored by Math­e­mat­ical etudes, on the cycloidal track wins!

Why Galileo Galilei consid­ered a quarter of a circle as the best trajec­tory in the sense of timing? He was inscribing poly­lines in it and mentioned that increasing the number of edges decreases the time of descent. This is how Galilei natu­rally came to a circle but made a wrong conclu­sion that this trajec­tory is the best among all the others. As we've seen, the best one is a cycloid.

One can draw a single cycloid passing through two give points such that the upper of them is a cusp­idal point of the cycloid. Even when the cycloid has to grow up in order to pass through the second point, it remains the curve of the shortest descent!

Another beau­tiful problem connected to the cycloid is the problem of tautochrone. In Greek tauto means equal, and as we already know, chronos means time.

Build three similar cycloid-shaped slides so that the end of the slide is the peak of the cycloid. Put three bobsleighs on different heights and let them go. An amazing fact is that all of them will finish simul­ta­ne­ously!

In winter you can build a slide of ice and check this prop­erty your­self!

The problem of tautochrone is to find such a curve that the time of descent doesn't depend on the starting point.

Chris­tian Huygens proved that the cycloid is the only tautochrone curve.

Of course, Huygens wasn't inter­ested in sliding. At his times scien­tists couldn't work just for the fun of it. The prob­lems they studied were rising from the demands of engi­neering. In the XVII century one does long sea voyages. Amaz­ingly, sailors could already deter­mine lati­tude quite accu­rately, but couldn't deter­mine their longi­tude at all. One of the proposed methods to deter­mine lati­tude was based on accu­rate chronome­ters.

The first one to think of an accu­rate pendulum clock was Galileo Galilei. However, at the moment he starts its construc­tion, he is already old, blind, and his last year of life isn't enough to build one. He entrusts this to his son but he also waits till his death and has no time to finish. The next key char­acter is Chris­tian Guygens.

He noticed that the period of swing of an ordi­nary pendulum, consid­ered by Galilei, depends on the starting point, i.e. the ampli­tude. Thinking of what the trajec­tory of the lead weight should be, so that the period doesn't depend on the ampli­tude, he solver the tautochrone curve problem. But how to make the weight move along a cycloid? Trans­lating theory into prac­tice, Guygens makes cheeks the sting winds on and solves a number of math­e­mat­ical prob­lems. He proves that the «cheeks» should also have a shape of cycloid showing that the evolute of a cycloid is a cycloid with the same para­me­ters.

Addi­tion­ally, the construc­tion of a cycloidal pendulum proposed by Guygens helps to deter­mine its length. If one deflects the blue string, whose length equals four times the radius of the gener­ating circle, it's end will be in the point of inter­sec­tion of the cheek cycloid and the trajec­tory, i.e. in the peak of the cheek cycloid. As it's a half of the cycloid's arc, the total length equals eight times the radius of the gener­ating circle.

Chris­tian Guygens constructed a pendulum and the such a clock was tested in sea voyages, but didn't become wide­spread. As well as the clock with an ordi­nary pendulum, by the way.

Why do we still have clock­works with an ordi­nary pendulum? If one looks hard, for small decli­na­tions the cheeks almost do not affect the string. Thus, the motions along a cycloid and a circle almost coin­cide for small decli­na­tions.