In order to solve this problem, the famous mathematician did not hesitate to...

Recently I learned a new curve - the cycloid. Come and take a look at it with me, you will also be amazed.

I think most of the shapes we know appear from time to time in our daily lives, and it is difficult to discover new shapes. Since elementary school, we have known squares, circles, and triangles, and later we learned hyperbolas, ellipses, and sine curves, but many people don’t know this shape... that is the amazing shape I discovered recently - the cycloid. Next, I will learn this new shape with you.

What is a trochoid?

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 slipping." This may be more intuitive with the following animated image:

The cycloid is the red path that a point on the boundary of the circle travels when it rolls along this straight line. This is the cycloid? Pretty simple, right? Not really.

History of the trochoid

The trochoid is sometimes called the "Helen of geometers" because it stirs up much controversy among mathematicians, including over who discovered the shape.

One of the earliest candidates was Iamblichus (245-325 BC), the biographer of Pythagoras, and other candidates included Nicholas of Cusa (1401-1464 AD), Charles de Bovelles (1475-1566), Galileo Galilei (1564-1642), Marin Mersenne (1588-1648), and many other learned men. But no one is sure who was the first to discover the cycloid.

Iambrichos was an ancient Greek philosopher, toga tastemaker, and (possibly) discoverer of the trochoid, though the fame of the trochoid apparently didn’t allow him to have his own marble bust.

(Source: https://en.wikipedia.org/wiki/Iamblichus)

I think most people, including me, only know that Galileo was the first person to study the cycloid and give it a name. He even made a model of the cycloid with a metal plate to study the area under the cycloid. If calculus had existed at that time, it might have been easier. By the way, Evangelista Torricelli, who invented the mercury barometer, was the one who finally solved the area under a single cycloid.

Over time, cycloids attracted a large number of famous mathematicians, including Descartes, Fermat, Pascal, Newton, Leibniz, L'Hôpital, Bernoulli, Euler, Lagrange, and many more names I can name off the top of my head.

They apparently enjoy creating competitions and questions about the spinning wheel, which then end with mutual attacks and insults.

Blaise Pascal had earlier created a competition to find the center of gravity, area, and volume of a cycloid, with Spanish gold coins as prize money. Unfortunately, the three judges thought no one had won. Christopher Wren (1632-1723), the designer of St. Paul's Cathedral in London, submitted a proof for calculating the length of a cycloid, which was not part of the competition, but still worthy of praise. One of the judges claimed years later that he had solved the problem but had never written it down, which triggered a war of public opinion. (At least Wren earned his reputation through his published results.)

Unfortunately, the challenge proposed by Johann Bernoulli in 1696 also ended in failure, which I will introduce to you later.

Using mathematics to gain a deeper understanding of cycloids

Now that we are familiar with the history of the cycloid, you may have some of the same geometry questions as the greats Galileo and Wren: What is the area under the cycloid? What is the length of the cycloid? What is the shape of the cycloid?

Fortunately, we have mathematics and a developed network.

The following parametric equations can express the x and y coordinates of a point on a circle as it moves forward over time (t). The x and y coordinates represent the trajectory of the cycloid. x and 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, we let t = π. At this time, x(π) = r ( π − sin(π) ) = r ( π − 0 ) = πr . Because the circumference of the circle is 2πr , the circle has rolled half a circle at this time; the height of this point is y(π) = r ( 1 − cos(π) ) = r ( 1 + 1 ) = 2r , and the double radius shows that this point on the circle reaches the highest point of one rolling circle.

With these two equations, we can use calculus to calculate the length and area of ​​the cycloid. With the help of the Internet and the memory of my earlier math knowledge, I used different colored pens to complete this elegant proof:

As with all the other problems about circles, the solution is pretty neat: the area under a single cycloid is 3πr². Amazingly, Galileo’s calculation of the ratio of the area under the cycloid (3πr²) to the area of ​​the circle (πr²) was pretty close to 3:1, and this was done using old-school metal splicing. The length of the cycloid is 8r, which is exactly what Lane had calculated long ago, and there’s no trace of π in it.

This result can be said to be very beautiful.

Cycloids in physics

Are cycloids just for show? Do cycloids exist in nature? Although not like their geometric relatives, cycloids still exist in nature in some magical ways.

Let’s go back to the question that Bernoulli posed to leading mathematicians in 1696:

“

I, Johann Bernoulli, to the world's brightest mathematicians:

Nothing is more appealing to intelligent people than a straightforward and challenging problem, not to mention the possibility that the solution will bring them fame and immortality. Following the examples of Pascal, Fermat, and others, I hope to earn the gratitude of the academic community by proposing a problem that will test the skills and power of the minds of the current top mathematicians. If anyone can give a solution to my next problem, I will express his praise in public.

This guy doesn't think he's bragging at all - although "public praise" doesn't sound as attractive as Spanish gold coins. Then there's his question:

“

There are points A and B in a vertical space. A particle is only affected by gravity and moves from A to B. What curve does its trajectory take the shortest time?

In other words, if there is a small ball that is only affected by the gravitational field and moves from a higher point A to a lower point B on a frictionless track (the line AB is not vertical), then what trajectory can make the ball move in the shortest time?

But considering that Bernoulli derived the correct result using an incorrect method and copied the correct derivation from his brother, his "reward" becomes a lot more interesting.

Bernoulli gave the public six months to submit an answer, but received no response. Leibniz proposed extending the deadline to one and a half years, during which Newton completed the challenge.

According to Newton, he received John Bernoulli's letter when he returned home from the Royal Mint at 4:00 pm on January 29, 1967. He worked all night and mailed his correct solution anonymously the next day, but because this solution was too good and too "Newtonian", Bernoulli immediately recognized "the lion who left this paw print".

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

Newton, the most unkind of cycloid mathematicians.

(Source: https://whatculture.com/offbeat/10-times-well-loved-scientists-were-total-jerks?page=10)

The fastest path solved by Newton and Bernoulli is called the brachistochrone curve, which comes from the Greek word for "shortest time". As you can guess from the topic of this article, this path is a section of the cycloid. The following animation uses an experiment to demonstrate this problem:

The brachistocentre line in the dynamic graph is always the fastest path of descent due to gravity between two points at different heights. The brachistocentre line is the middle line in the figure above and the red curve in the figure below.

It is also very interesting to recognize the characteristics of some shapes in nature.

Another episode about the cycloid is the tautochrone curve, which comes from the Greek word "same time". You can put a small ball at any position on this curve, and the time it takes to reach the lowest point is equal. This graph comes from half a cycloid. The following animation shows this curve:

The isochronous drop curve is another interesting form of the cycloid curve. No matter where you put the ball on the curve, the time it takes for them to reach the bottom is the same.

There is also a thing called a cycloidal pendulum, the top of which is at the intersection of two cycloidal lines. The line of this pendulum bends along the two cycloidal lines, and the line swept by this pendulum is actually another cycloidal line!

A cycloid pendulum creates another cycloid between two cycloids.

We can also use the rolling wheel curve to make many changes. In the same circle rolling forward along a straight line, the trajectory inside or outside the circle can become a more curved or flat curve, as shown in the following visual picture:

Different cycloid curves.

(Source: https://www.researchgate.net/figure/Cycloidal-motion-and-examples-of-cycloids-Cycloid-blue-prolate-cycloid-red-curtate_fig12_304707433)

Next we can see the family of cycloids, which are composed of rolling circles or other figures around certain figures.

You can also create a cycloid by dropping an object from any height. The object's falling path will be a vertical line relative to the Earth, but since the Earth is a spinning circle, the falling path will be a slightly inverted cycloid (although very slightly so)!²

Cycloids in Literature

The trochoids that occasionally appear in literature over the past few centuries are certainly notable, and while I cannot list them all, here is one from Herman Melville’s 1851 classic, Moby Dick:

“

As the soapstone circled round and round in the cauldron on the left-hand side of the Pequod, I suddenly became conscious for the first time indirectly of the fact that all bodies sliding on a cycloidal line, in my case the soapstone, are geometrically bound to fall together wherever they were before.

Cyclops in Architecture

It can be seen that cycloids are really interesting, and I wonder if there are some cycloids that we are missing in our daily lives.

Architecture is made up of a lot of geometric shapes. Many famous arches are derived from the circle (Roman arch), the ellipse (semi-elliptical arch), the parabola (parabolic arch), and the catenary (catenary arch). There are many examples of each, but I have selected a few of the most famous ones:

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

Kew Bridge, which spans the River Thames in London, has a semi-elliptical arch that creates a wider span for traffic such as ships and trains.

The Bixby Bridge on US Highway 1 in Big Sur, California, features a parabolic arch. Photo: Alamy.

The Gateway Arch in St. Louis, Missouri, is a catenary arch, which is the strongest type of arch because of its even distribution of weight.

A trochoid looks very similar to an arch, so are there any buildings that use trochoid arches? According to online search results, there are, but they are rare. There are two examples that appear repeatedly in the introduction:

The first is the roof of the Kimbell Art Museum in Fort Worth, Texas, USA. The multiple arches on this roof are composed of a series of spaced roller lines. The pattern formed by this roller gives it a smooth appearance, which is very suitable for an art museum.

Trochanteric arch at the Kimbell Art Museum in Fort Worth, Texas.

The second building with a cycloid arch is the arch on the front of the Johns Hopkins Center at Dartmouth College, where I studied for my undergraduate degree. This made me think in another way: Is it because I saw this building every day for four years that I was so fascinated by the cycloid?

Trochoidal arches on the facade of Dartmouth College's John Hopkins Center in Hanover, New Hampshire.

Trombones in Arts and Entertainment

You may have played with cycloids as a child. The floret ruler is based on a common cycloid called a hypocycloid, which is a "special plane curve consisting of the trajectory of a certain point on a small circle attached to a large circle that rolls inside a large circle," unlike a circle that rolls along a straight line.

Kaleidoscope.

(Source: https://en.wikipedia.org/wiki/Spirograph)

There are two special forms of hypotrochoids: deltoid and astroid, which can be obtained by rolling a small circle three times and four times inside a large circle, respectively. You may have seen astroids on some signs.

Two special hypotrochoids: the trigonometric hypotrochoid (left) and the stellate hypotrochoid (right).

The Pittsburgh Steelers football team has three stars in its logo.

If you find this kind of line art comforting, some artists use multiple circles of different sizes combined to create cycloid art:

Rotating wheel line art installation on Pinterest.

Cyclone artwork sold on Kickstarter.

Trochoids in optics

Another cycloid form can be formed by the path of a point on a circle rolling along the outside of a circle. A special example is the cardioid, which is a figure formed by the path of a point on a circle moving along the outside of another circle of equal radius, as shown below. This shape happens to have a sharp corner that resembles a heart, which is where its name comes from:

The cardioid line is another type of trochoid line.

Cardioids are very common in nature, especially in caustics created by two circular surfaces. In optics, caustics are defined as a curve or surface "the envelope of light caused by surface irregularities or reflections, or the projection of the envelope of light on other surfaces". On this line or surface, every ray of light is tangent to it, and the location where these rays are concentrated is the boundary of the light envelope.

We can see the heart line in the caustics created by multiple round objects, from a coffee cup to a watch.

Next time you drink your morning tea, be sure to open your eyes and look at the pattern in the teacup! ☕️

The boundary of the central region of the Mandelbrot set, a framework for fractal geometry and chaos theory, is also an exact cardioid, although I don't know the specific reasons, but it is still another form of cardioid.

The central area of ​​the first stage of the Mandelbrot set is bounded by a perfect heart line.

The shape of the cycloid is not limited to circles. You can also roll a non-circular shape along a straight line and discover a new shape - a polygonal cyclogon. Here are the cyclogons of triangles and squares:

The arc of rotation formed by an equilateral triangle rolling along a straight line without sliding.

(Source: https://en.wikipedia.org/wiki/Cyclogon)

The arc of a square rolling along a straight line without sliding.

(Source: https://en.wikipedia.org/wiki/Cyclogon)

Cycloids in the Universe

Cycloids are not just patterns on everyday scales like wheels, watches, teacups, or gyroscopes, they can even reach planetary scales. As Jupiter's moon Europa (the small circle) orbits the giant Jupiter (the big circle), the gravitational force (a straight line) forms cycloids on the moon, which can be seen in the cracks in the ice on the Europa satellite image. This crack is consistent with the gravitational pressure on the moon's orbit.

Cycloids on the surface of Jupiter's moon Europa.

(Source: https://www.science.org/doi/10.1126/science.1248879)

Cycloid formation on Europa's surface.

(Source: https://www.researchgate.net/figure/Model-of-cycloidal-crack-formation-on-Europa-by-Hoppa-et-al-The-arrows-represent-the_fig6_13779479)

Summarize

I hope you also learned some new graphics from this article. After all, cycloids are a very interesting group of graphics. After seeing a series of cycloids, I want to know more about the universe around me...

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.

By Ry Sullivan

Translation: zhenni

Reviewer: Nothing

Original link:

https://medium.com/@rysullivan/celebrating-the-cycloid-be4350ff187b

The translated content only represents the author's views

This article does not represent the views of the Institute of Physics, Chinese Academy of Sciences.

Editor: zhenni