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In order to solve this problem, famous mathematicians will not hesitate to…

Recently I learned a new curve-cycloid, come and see with me, you will also be surprised.

I think most of the shapes we recognize appear in everyday life from time to time, and it's hard to find new shapes. Since elementary school we have known squares, circles and triangles, and later learned hyperbolas, ellipses and sinusoids, but many people do not know this shape... that is, I recently discovered the amazing-cycloid. And I'm going to learn this new shape with you.

What is a cycloid? In Wikipedia, a cycloid is defined as "the trajectory of a point on the edge of a circle as it rolls along a straight line without sliding." "It might be more intuitive to use this animation:

The cycloid is the red trajectory that a point on the boundary travels as the circle rolls along this line. This is the cycloid? Simple, right? It's not true.

The cycloid is sometimes called the "Helen of Geometrists" because it provokes a lot of controversy among mathematicians, one of which is who discovered the shape.

One of the earliest candidates was Iambrijos, the biographer of Pythagoras.(Iamblichus, AD 245-AD 325). Other candidates include Nicholas Kusa of Germany. Nicholas of Cusa (AD 1401-1464), Charles de Bovelles (1475-1566), Galileo Galilei (1564-1642), Marin Mersenne (1588-1648), etc. But no one is sure who discovered the cycloid first.

Iambrios was an ancient Greek philosopher, trendsetter in robes, and (possibly) discoverer of cycloid, and apparently the fame of cycloid did not allow him to have his own marble bust. (Source) I think most people, including me, know only that Galileo was the first to study and name the cycloid, and that he even modeled it from metal plates to study the area under the cycloid. It would have been easier if calculus had existed. Evangelista Torricelli, by the way, invented the mercury barometer and was the one who finally solved the area under a single cycloid.

Over time, the cycloid attracted a large number of famous mathematicians, including Descartes, Fermat, Pascal, Newton, Leibniz, Lobida, Bernoulli, Euler, Lagrange, and so on.

They obviously enjoy creating contests and questions about cycloid lines, and then ending up attacking and abusing each other.

Blaise Pascal had previously created a contest to find the center of gravity, area, and volume of a cycloid, with Spanish gold coins as a prize. Unfortunately, the three judges decided that no one would win. Christopher Wren (1632-1723), the architect of St Paul's Cathedral in London, submitted a proof of calculating the length of a cycloid, which was not part of the competition but deserves praise. A judge claimed years later that he had solved the problem, but there was no written record of it, sparking a war of opinion. (At least Wren earned his reputation by publishing his own work.)

Unfortunately, Johann Bernoulli's challenge in 1696 also ended in failure, and I'll tell you about it later.

Now that we're familiar with the history of cycloids, you might have some of the same geometric questions as the great Galileo and Wren: What's the area under the cycloid? What is the length of the cycloid? What shape is a cycloid?

Fortunately, we have math and a well-developed network.

The following parametric equation can be expressed as a circle advances at a point above the time (t) changes in x, y coordinates represent the cycloid trajectory, x, y are independent of each other, so there are two equations:

x(t) = r(t−sin(t))

y(t) = r(1−cos(t))

To better understand these two equations, let t = π . In this case x (π) = r ( π − sin (π) ) = r ( π − 0 ) = πr . Since the circumference of the circle is 2πr, then the circle rolls half a revolution; the height of this point is y (π) = r ( 1 − cos (π) ) = r ( 1 + 1 ) = 2r, and twice the radius shows that this point on the circle has reached the highest point of one revolution.

With two equations, we can use calculus to calculate the length and area of the cycloid. With the help of the Internet and the recollection of earlier mathematical knowledge, I completed this elegant proof using pens of different colors:

As with all other problems about circles, this solution is very simple, the area under a single cycloid is 3πr². Amazingly, Galileo's calculation of the ratio of the area under the cycloid line (3πr²) to the area of the circle (πr²) was very close to 3:1, and this result was only achieved by very old-fashioned metal splicing methods. The length of the cycloid is 8r, which is consistent with Wren's calculation, and there is no shadow of pi.

This result can be said to be very beautiful.

Is the cycloid in physics just a charade? Is there a cycloid in nature? Although unlike their other geometric relatives, cycloids still exist in nature in some magical poses.

Let us return to Bernoulli's question to leading mathematicians in 1696:

I, Johann Bernoulli, to the most brilliant mathematicians of the world:

For intelligent people, there is nothing more attractive than a straightforward and challenging question, let alone solutions that may make them famous and immortal. Following the example given by Pascal, Fermat, and others, I hope I, too, can earn the gratitude of the academic community by posing a question that tests the skill and strength of the minds of the best mathematicians of today. If someone can give me a solution to my next question, I will praise him in public.

The man didn't think he was bluffing at all-although "public praise" didn't sound as attractive as Spanish gold. Then came his question:

In a vertical space with points A and B, there is a particle that is acted upon only by gravity. What curve does its trajectory take the shortest time to travel from A to B?

In other words, if a small ball were to move from a higher point A to a lower point B on a friction-free orbit, driven only by gravity (the AB line is not vertical), what trajectory would make the ball move in the shortest time?

But Bernoulli's "reward" is interesting considering that he derived the correct result the wrong way and copied it from his brother.

Bernoulli gave the public six months to submit a solution, but received no response. Leibniz proposed extending the deadline for submission to a year and a half, during which Newton fulfilled the challenge.

According to Newton, he received a letter from Johann Bernoulli on January 29, 1697 at 4:00 p.m. on his way home from the Royal Mint. He worked all night and anonymously mailed his correct solution the next day, but because it was too good, too Newtonian, Bernoulli immediately recognized "the lion who left this paw print."

Newton's solution time of one night beat Bernoulli's record of two weeks. Newton added to his letter some of the disdain that mathematicians of his time loved to express: "I don't like to be pestered and amused by foreigners in mathematics..." Newton was never very likable, and could be said to be unkind.

Newton, the most inhuman cycloid mathematician. (Source) This fastest path solved by Newton and Bernoulli is called the fastest descent curve (brachistochrone curve), which comes from the Greek word for "shortest time." According to the theme of this article, I believe everyone has guessed that this path is a section of the cycloid. The following animation shows this problem experimentally:

The fastest descent line in a dynamic graph is always the fastest descent path between two points at different altitudes. The steepest descent line is the middle one in the top image and the red curve in the bottom image. It's also interesting to recognize the nature of some of the shapes.

Another episode of cycloid is the tautochrone curve, which comes from the Greek for "same time." You can put a ball anywhere on this curve, and it takes the same time to reach the lowest point. This graph is derived from half a cycloid, and the following animation shows this curve:

Isochronal descent curves, another interesting form of cycloid. No matter where you place the ball on the curve, it takes the same amount of time for them to reach the bottom. There's also something called a cycloid pendulum, and the tip of this pendulum is at the intersection of two cycloids. The line of the pendulum bends along two cycloid lines, and the line swept by the pendulum turns out to be another cycloid line!

The cycloid pendulum creates another cycloid between two cycloids. We can also use cycloid to do many transformations. Also in the circle rolling forward along a straight line, the trajectory of a point inside or outside the circle can be changed into a more curved or flat curve, making a visual picture as shown below:

Different cycloid curves. (Source) Next we can see the cycloid family consisting of circles or other shapes rolling around certain shapes.

You can also create a cycloid by dropping an object from any height. The object's trajectory relative to the Earth is a vertical line, but since the Earth is a rotating circle, the trajectory will be a slight inverted cycloid (though really slight)!²

The cycloid in literature must have been somewhat famous for its occasional appearance in literature over the centuries, and although I cannot list all cases, here is a passage from Herman Melville's classic 1851 work Moby Dick:

In the left-hand cauldron of the Pequod, as the talc circled round and round, I became suddenly and indirectly aware for the first time of the fact that all objects sliding on the cycloid, my talc for example, for geometry, fall together wherever they were before.

The cycloid in architecture is really interesting to see, and I wonder if there are some cycloids missing in everyday life.

Architecture consists of a large number of geometric shapes. Many famous arches derive from circular (Roman arch), elliptical (semi-elliptical arch), parabolic (parabolic arch), and catenary (catenary arch). There are plenty of examples of each, and I've picked out a few very famous ones:

The Arc de Triomphe in Paris is a semicircular arch, also known as the Roman Arch.

Kew Bridge, which crosses the Thames River in London, has semi-elliptical arches that create wide spans for vehicles such as ships and trains.

Bixby Bridge on U.S. Route 1 in Big Sur, California has parabolic arches. Photo by Alamy.

The arch in St. Louis, Missouri, is a catenary arch, the strongest arch due to its uniform weight distribution. A cycloid looks similar to an arch, so is there a cycloid arch for architecture? According to the online search results, yes, but very few. Two examples recur in the introduction:

The first is the roof of the Kimbell Art Museum in Fort Worth, Texas, USA. The arches on this roof are composed of a series of spaced cycloid lines. This scroll wheel pattern gives it a smooth appearance and is very suitable for an art museum.

The trochoid arch at the kimbel museum of art, fort worth, texas. The second building with the cycloid arch was the arch on the facade of Hopkins Center at Dartmouth College, where I went as an undergraduate, and it got me thinking another way: Was it because I saw this building every day for four years that I was so fascinated by the cycloid?

A cycloid arch on the facade of Hopkins Center at Dartmouth College in Hanover, New Hampshire. Circles in art and entertainment may have been "played" with when you were a child. The rule is a toy based on a general cycloid called a hypocycloid, which, unlike a circle that rolls along a straight line, is "a special planar curve consisting of a locus of points attached to a small circle that rolls inside a large circle."

Ten thousand flower feet. There are two special forms of endocycloid: deltoid and astroid, which can be obtained by rolling a small circle along the inside of a great circle for three and four cycles, respectively. You may have seen star lines on some signs.

Two special types of endocycloid: triangular cycloid (left) and astroid (right).

The Pittsburgh Steelers football team logo contains three stars. If you find the lines comfortable, there are artists who create cycloid art by rolling circles in multiple size combinations:

A cycloid art installation on Pinterest.

Coilwheel art for sale on Kickstarter. Another form of cycloid in optics can be formed by tracing a point on a circle that rolls along the outside of a circle. A particular example is a cardioid, a graph formed by the trajectory of a point on a circle moving outside another circle of equal radius, as shown in the following figure. This shape happens to have a sharp corner similar to a heart, which is also the source of its name:

Cardioid is another type of cycloid. Heartlines are very common in nature and are particularly prone to caustics created by two circular surfaces. In optics, caustics defines a curve or surface "the envelope of light rays produced by irregularities or reflections on an object surface, or the projection of the envelope of light rays from other surfaces," on which every ray is tangent, and where these rays are concentrated is the boundary of the envelope of light rays.

We can see the heart line in the caustics produced by many circular objects, from coffee cups to watches.

Next time you have tea in the morning, be sure to keep your eyes open and look at the graphics in the teacup! The boundary of the central region of the Mandelbrot set, the framework of fractal geometry and chaos theory, is also an exact cardioid, although I don't know why, but it is still another manifestation of cardioid.

The central region of the first stage of the Mandelbrot set is enclosed by a perfect heart line. Cyclogon shapes are not limited to circles. You can scroll along a straight line to a non-circular shape and discover a completely new shape-a polygon cyclogon. Here are triangles and squares:

The arc of a revolution formed by an equilateral triangle rolling along a straight line without sliding. (Source)

The arc of a circle formed by a square rolling along a straight line without sliding. The cycloid of the universe is not just a pattern on everyday scales such as wheels, watches, teacups or spirals, it can even reach planetary scales. Jupiter's moon Europa (a small circle) orbits the giant Jupiter (a large circle), and its gravitational pull (a straight line) forms a cycloid on the moon, as can be seen in the cracks in the ice in the Europa satellite image. This fracture is consistent with the gravitational pressure on the satellite's orbit.

The spiral shape of the surface of Jupiter's moon Europa. (Source)

The cycloid lines on Europa's surface form. (Source) Summary I hope you also learn some new graphics from this article, after all, cycloid is a group of very interesting graphics, after I saw a series of cycloid, more want to go to understand the universe around...

References:

¹ Eli, Maor and Eugen Jost. "Twisted Math and Beautiful Geometry. " American Scientist.

² Lynch, Peter. "The curved history of cycloids, from Galileo to cycle gears. " The Irish Times. 17-Sep-2015.

This article comes from Weixin Official Accounts: Institute of Physics, Chinese Academy of Sciences (ID: cas-iop), author: Ry Sullivan, translation: zhenni, revision: Nothing

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