Recently I learned a new curve, the cycloid (also known as the cycloid), come and see it with me, you will be very surprised.
I think most of the shapes we know appear in everyday life from time to time, and it's hard to find new shapes. We have known squares, circles and triangles since elementary school, and later learned hyperbolic, elliptic and sinusoidal curves, but many people do not know this shape... That's what I've recently discovered amazingly – the spiral line. Next I will learn this new shape with you.
What is a rotary wheel line?
In Wikipedia, a revolver line is defined as "a trajectory of a little motion on its sides when a circle rolls in a straight line without sliding." The following GIF may be more intuitive:
The spiral wheel line is the red trajectory of a point on the boundary of the circle as it rolls along this straight line. Is this the rotary wheel line? It's simple, right? Not really.
The history of the spiral wheel line
The spiral line is sometimes referred to as the "Helen of the Geometrists" because of the many disputes it provokes among mathematicians, and one of the disputes is who discovered the shape.
One of the earliest candidates was Iamblichus (245-325 BC), a biographer of Pythagoras, and other candidates included Nicholas of Cusa (1401-1464 AD), Charles de Bovelles (1475-1566), the Italian Galileo Galilei, 1564-1642), the Frenchman Marin Mersenne (1588-1648) and many other learned men. But no one can be sure who was the first to discover the spiral line.
Ian Blixus was an ancient Greek philosopher, a trendsetter in robes, and (possibly) a discoverer of the wheel line, and it was clear that the fame that the wheel line brought could not allow him to have a marble statue of himself. (Source)
I think most people, including me, just know that Galileo was the first to study the wheel line and name it, and he even made a model of the wheel line out of metal plates to study the area under the wheel line. If there had been calculus back then, it might have been easier. By the way, Evangelista Torricelli, who invented the mercury barometer, was the one who finally solved the area under the line of a single spiral wheel.
Over time, the spiral line attracted a large number of famous mathematicians, including Descartes, Fermat, Pascal, Newton, Leibniz, Lobida, Bernoulli, Euler, Lagrange, and so on.
They obviously enjoyed creating some races and questions about the wheels, which ended with mutual attacks and verbal abuse.
Blaise Pascal had earlier created a race to solve the center of gravity, area, and volume of the spiral line, with Spanish gold coins as prize money. Unfortunately, the three judges decided that no one had won. Christopher Wren (1632-1723), the designer of St. Paul's Cathedral in London, submitted a proof of calculating the length of the spiral, which, while not part of the competition, was laudable. One judge claimed years later that he had solved the problem but had never documented it, sparking a war of opinion. (At least Wren earned his reputation through his published work.) ）
Unfortunately, the challenge posed by Johann Bernoulli in 1696 also ended in failure, and I will introduce it to you later.
Use mathematics to gain a deeper understanding of spiral wheel lines
Now that we are familiar with the history of the wheel line, you may have some geometric questions like the great men Galileo Galileo and Rennes: What is the area under the wheel line? What is the length of the rotary wheel line? What exactly is the shape of the rotary wheel line?
Fortunately we have math and a well-developed network.
The following parametric equation can represent the spiral line trajectory with x,y coordinates over time (t) as a circle advances, 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 make t = π . At this point x(π) = r ( π sin(π) ) = r ( π 0 ) = πr . Because the circumference of the circle is 2πr, the circle rolls in half a circle; the height of this point is y(π) = r ( 1 cos(π) = r ( 1 + 1 ) = 2r , and twice the radius can be seen that this point on the circle reaches the highest point of the rolling week.
With two equations, we can use calculus to calculate the length and area of the spiral wheel. Using the help of the web and memories of earlier mathematical knowledge, I completed this elegant proof with different colored pens:
Like other questions about circles, this solution is very concise, with the area under the line of a single spiral wheel being 3πr . Surprisingly, Galileo's calculation of the ratio of the area (3πr) and the area of the circle (πr) under the wheel line was very close to 3:1, and this result was only done using a very old-school metal splicing method. The length of the spiral line is 8r, which is consistent with Wren's long-term calculations, and there is no shadow of π in it.
This result can be said to be very beautiful.
Spiral wheel lines in physics
Is the rotary wheel line just useless? Is there a spiral wheel line in nature? Although not like other geometric relatives, the spiral line still exists in nature in some magical gestures.
Let's go back to the question Bernoulli asked the top mathematicians in 1696:
“
I, Johann Bernoulli, to the world's brightest mathematicians:
For smart people, there's nothing more appealing than a straightforward and challenging problem, not to mention that these solutions might make them famous and go down a lifetime. Based on the examples given by Pascal, Fermat and others, I hope that I will also be able to get the gratitude of the academic community by asking a question about the skills and strength of the brains of the top mathematicians today. If someone can give me a solution to my next question, then I will express my compliments to him in public.
The man didn't think he was talking big at all – though "public praise" didn't sound as attractive as Spanish gold coins. Then came his question:
In a vertical space with a little A and point B, there is a point of mass that is only affected by gravity from A to B, and what curve does its trajectory take the shortest time?
In other words, if a ball is subject only to the gravitational field, moving from a higher A point to a lower B point in a frictionless orbit (the AB line is not vertical), what trajectory can make the ball move for the shortest time?
But considering that Bernou used the wrong method to derive the correct result and copied the correct derivation from his brother, his "reward" became a lot more interesting.
Bernoulli gave the public six months to submit answers, but received no response. Leibniz proposed extending the deadline for submissions to one and a half years, during which Newton fulfilled the challenge.
According to Newton, he received a letter from John Bernoulli on his return home from the Royal Mint on January 29, 1967, at 4:00 p.m. He worked all night and anonymously mailed out his correct answer the next day, but because it was too good, too "Newton," Bernoulli recognized "the lion who left this paw print."
Newton's one-night settlement broke Bernoulli's record of two weeks. Newton added some of the disdain that mathematicians at the time liked to express in his letters: "I don't like to be pestered and entertained by foreigners in mathematics..." Newton was never very likable, it can be said that it was unkind.
Newton, the most impersonal spiral mathematician. (Source)
The fastest path solved by Newton and Bernoulli is called the brachistochrone curve, derived from the Greek "shortest time", according to the theme of this article, I believe everyone guessed, this path is a section of the spiral wheel line, the following GIF uses experiments to show this problem:
The fastest descent line in the dynamic graph is always the path that drops the fastest by gravity between two points at different heights. The steepest descent line is the middle one in the upper figure and the red curve in the lower figure.
It's also too interesting to recognize the characteristics of some of the graphics in nature.
Another episode about the spiral line is the tautochrone curve, derived from the Greek "same time", where you can place a ball anywhere in this curve and reach the lowest point in equal time. This graph is derived from a half-spinning wheel line, and the following GIF shows this curve:
Isochronous landing curve, another interesting form of spiral wheel line. No matter which colored ball you place on the curve, they take equal time to reach the bottom.
There is also something called the spiral line pendulum, and the top of this pendulum is at the intersection of the two spiral lines. The line of this pendulum will bend along the two spiral lines, and the line swept by this pendulum is actually another spiral line!
The spiral wheel line is placed between the two wheel lines to create another wheel line.
We can also use the circular wheel line to do a lot of transformations. Also in a circle that rolls forward along a straight line, the trajectory of a point inside or outside the circle can become a more curved or flat curve, and the visualization is shown in the following figure:
Different spiral line curves. (Source)
Next we can see the family of spiral lines composed of scrolling circles or other shapes wrapped around certain shapes.
You can also create a spiral line by dropping an object from any height, which is a vertical line relative to the Earth's falling trajectory, but since the Earth is a rotating circle, this falling trajectory will be a slight inverted wheel line (although it is really slight)!
The spiral line in literature
The spiral line, which has occasionally appeared in literature over the centuries, must have been a little famous, and while I can't list all of them, here's a passage from Herman Melville's 1851 classic Moby Dick:
In the cauldron on the left-hand side of the Pekod, as the talc kept circling around, I suddenly and indirectly realized for the first time the fact that all the objects sliding on the spiral line, in the case of my talc, for geometry, no matter what point they had before, would fall together after that.
Spiral wheel lines in buildings
I can see that the spiral wheel line is really interesting, and I wonder if I have missed some wheel lines in my daily life.
The building consists of a large number of geometric figures. Many of the famous arches are derived from circular (Roman arches), ovals (semi-elliptical arches), parabolic (parabolic arches), and catenary chains (catenary arches). There are a large number of examples of each, from which I have selected a few very famous:
The Arc de Triomphe in Paris is a semicircular arch ticket, also known as the Roman Arch.
The Kew Bridge, which crosses the River Thames in London, has semi-elliptical arches that create a wider span for vehicles such as boats and trains.
The Bixby Bridge on U.S. Highway One in Big Sur, California, has parabolic arches. Photograph by Alamy.
The arch in St. Louis, Missouri is a catenary arch that is the strongest arch due to its even weight distribution.
The spiral wheel line looks very similar to the arch, so is there a building with a spiral wheel arch? According to online search results, there are, just very few. There are two examples that come up repeatedly in the introduction:
The first is the roof of the Kimbell Art Museum in Fort Worth, Texas, USA, where multiple arches are made up of a series of spaced spiral wheels, and the shape of this roller gives it a smooth appearance, which is ideal for an art museum.
Spiral arch of the Kingbell Art Museum in Fort Worth, Texas.
The second building with the Spiral Arch is the arch on the front of the Hopkins Center in Dartmouth College, which is the school I attended as an undergraduate, which makes me think differently: Was it because I saw this building every day for four years that I was so fascinated by the Spiral Line?
The Hopkins Center at Dartmouth College in Hanover, New Hampshire, is fronted by a spiral arch.
Spiral lines in art and entertainment
Maybe you've been "playing" with the spindle line since you were younger. The kaleidoscope (multiplication curve gauge) is based on a general rotary wheel line called the hypocycloid, which, unlike a circle that rolls with a straight line, is a "special planar curve composed of a fixed point of trajectory attached to a small circle rolling inside a large circle".
Kaleidoscope. (Source)
The inner spiral wheel line has two special forms of triangular rotation line (deltoid) and astroid , which can be obtained by rolling three and four times along the inner part of the large circle by a specific small circle. You may have seen a star line on some signs.
Two special internal rotation wheel lines: the triangle wheel line (left) and the star line (right).
The Pittsburgh Steelers football team logo contains 3 star lines.
If you find this line comfortable, there are artists who create the art of spiraling lines using multiple combinations of rolling circles of different sizes:
Spinning wheel line art installation on Pinterest.
Spinning wheel art sold on Kickstarter.
Spiral wheel lines in optics
Another form of spiral wheel line can be constructed by a fixed-point trajectory on a circle that rolls along the outside of a circle. A particular example is the cardioid, a figure of a circle moving along the trajectory of another circle of equal radius outside its upper point, as shown in the following figure, a shape that has just a sharp angle similar to a heart, which is also the source of its name:
Heart line is another type of rotary wheel line.
Heart lines are very common in nature and are particularly prone to appear in caustics created by two circular surfaces. In optics, caustics defines a curve or surface that is "a light envelope due to unevenness or reflection on the surface of an object, or a projection of the envelope line on another surface," in which each ray is tangent to the boundary of the light envelope at the location where these rays are concentrated.
In the caustics produced by multiple round objects, from coffee cups to watches, we can see the heart line.
The next time you drink tea in the morning, be sure to widen your eyes 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 a precise heart line, and although I don't know the specific reason, it is still another manifestation of the heart line.
The central area of the first stage of the Mandelbro collection is surrounded by a perfect heart line.
The shape of the spiral line is not limited to circles, you can also scroll a non-circle along a straight line and discover a completely new shape - the polygonal trace (cyclogon), below which is the trace of the triangle and square scrolling:
An arc of a grommetal arc formed by an equilateral triangle rolling in a straight line without sliding. (Source)
A circular arc of a square rolling along a straight line without sliding. (Source)
Spiral wheel lines in the universe
The spiral line is not just a graph on everyday scales such as rollers, watches, teacups or spirals, it can even reach the planetary scale. As Jupiter's moon Europa (small circle) orbits the giant Jupiter (the great circle), the gravitational pull (a straight line) forms a spiral line on the moon, which can be seen from the cracks in the ice on Europa's satellite image. This rift is consistent with the gravitational pressure on the satellite's orbit.
Jupiter's moon Europa has a spiral wheel on the surface. (Source)
Spiral wheel lines on the surface of Europa are formed. (Source)
summary
I hope you also learned some new graphics from this article, after all, the spiral wheel line is a group of very interesting graphics, after I watched a series of spiral wheel lines, I want to know the universe around me in depth...
bibliography:
Eli, Maor and Eugen Jost. “Twisted Math and Beautiful Geometry.” American Scientist.
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
was
good fortune
li
sharp
shi
time
jian
space
The translated content represents the author's views only
Does not represent the position of the Institute of Physics, Chinese Academy of Sciences
Edit: zhenni