Many of the moving myths we have heard may have begun with a true story, which really happened at a certain point in history, but in the praise of generations, it is covered with legends little by little, so that the truth is left behind little by little. Myths may always persist and evolve in people's minds because they carry some hidden truths that can meet certain needs of human beings in the mind.
Mathematics, known for its rigorous logic, seems to be an antithesis of mythology, but sometimes some mathematical truths enter the public consciousness in the form of "myths" under the repetition of lack of understanding. Today we are going to talk about a "myth" in mathematics - the golden ratio (golden ratio).
What is the Golden Ratio?
The golden ratio is a magical constant, and we usually use the Greek letter φ to represent it. It appears in many works of literature and art, such as the mysterious number mentioned in the novel and film The Da Vinci Code. It is mysterious because it does have more "myths" than many other concepts in mathematics: it has been described by many writers as the basis of all the beautiful patterns in nature, the sacred proportions; it is also known as the basis for the design of many works of art and buildings, such as the Parthenon in Greece and the pyramids in Egypt.
The golden ratio first appeared in Euclid's book The Primitives of Geometry, which Euclid defined as:
So, how much is φ equal? We know that a / b = φ, and (a + b) / a = φ, so the equation in the diagram above can become:
Solving this equation yields:
Since the φ must be greater than 1, we take φ = 1.61803... It is an irrational number, which is not difficult to understand, because the root number 5 is exactly an irrational number, that is, it is a number that cannot be written as the ratio of two integers. This is a very important property of the golden ratio.
The one-dimensional golden ratio can also be extended to the so-called golden rectangle, which we can draw according to the following steps:
1, first need to draw a square with an edge length of a; 2, and then take the midpoint of a side of the square (such as the bottom edge): take the midpoint as the center of the circle, and draw a circle with the distance from the midpoint to a vertex connected to the opposite edge as the radius; 3, extend the bottom edge, let it intersect with the arc, and the resulting intersection point is a corner of the golden rectangle.
In addition to the golden rectangle, there is another lovely geometric expression φ the golden ratio, which is the length of a diagonal line of a regular pentagon with a side length of 1. (Readers can try to test it with the cosine theorem!) ）
As shown in the figure above, the isosceles triangle BAD with a side length of 1, φ, φ formed by the diagonal and the bottom edge is called the golden triangle, which appears frequently in the study of five-fold symmetry, for example, the five-pointed star is composed of five golden triangles:
In fact, there are many ways to define the golden ratio, a very famous example is the Fibonacci sequence:
The next item in this sequence, the sum of the first two terms, is a method proposed by Fibonacci to understand rabbit population growth, which plays an important role in understanding population growth.
How does this sequence relate to the golden ratio? The first to discover this astonishing secret was Johannes Kepler, who noticed that if you take the ratio of two consecutive numbers in this sequence (the latter term is better than the previous one), the resulting ratio can form a sequence of numbers:
And this sequence will eventually converge to a familiar number - 1.618... The limit of this sequence is the golden ratio.
The irrationality of the golden ratio allows us to see in the golden rectangle the ratio of the Fibonacci sequence that can cycle indefinitely.
The golden ratio of "deification"
The golden ratio is an interesting number, it has a lot of peculiar properties, but also many useful applications. These peculiar properties have attracted the attention of some mathematicians, but for the public, these properties have unexpectedly been elevated to an inappropriate position.
In the eyes of mathematicians, there are many important constants, such as √2, which is the diagonal length of a square with a side length of 1, and the length-to-width ratio of an A4 paper. In fact, 1, √2, √3 appear much more frequently in geometry than φ.
Speaking of important constants, two other numbers that must be mentioned are π and e, which are of self-evident importance in both the mathematical world and the real world.
In geometry, pi π is the ratio of the circumference of the circle to the diameter, and its application extends far beyond geometry, which appears in all areas of mathematics, from calculus to number theory, from statistics to quantum mechanics. The number e is another constant that plays an equally important role in mathematics, it is an essential element of calculus, and it is related to anything about growth. Among the many important formulas of science and engineering, there are π and e, which are closely related to the universe.
In contrast, φ far fewer application scenarios. However, in popularizing mathematics, the mysterious color of φ makes it enjoy far more "glory" than the core numbers of these two universes. It's important to emphasize that this is not to say that φ is unimportant (we'll discuss the real magic of the golden ratio in Part 3), but rather that its role in mathematics and science is very different from the legend.
Why did φ gain such prominence in the mass media? Perhaps, like all myths, the reasons for deification have long been lost in the long river of history. But there are still some clues that can be followed to explore some of these stories.
The Golden Ratio hidden in nature?
The golden ratio appears in nature in many forms. As we mentioned earlier, the golden ratio is closely related to the Fibonacci sequence. The Fibonacci sequence is real in nature because it has to do with both the way populations grow and the way shapes can be combined.
For example, in the spiral of the sunflower we can see this sequence (left below), which are arranged together in an orderly manner that captures the most sunlight; for example, from the distribution of male and female bees in the hive (right below), we can also observe this ratio of nearly φ produced by the breeding pattern of bees.
However, there are many cases where the golden ratio is inappropriately linked. For example, many people say that perfect human body proportions and perfect face shape are related to φ. In fact, there are many possible ratios of the human body, most of which are between 1 and 2, and these "perfect" measures are not clearly defined, and if you think about it, the perfect human body proportions may be close to 1.6, 5/3, 3/2, √ 2, 21/13, and so on. And these are just some of the false correlations that the human brain feels. When we argue a point with a false correlation found in the data, this can actually be very dangerous, such as in a legal trial, where a false correlation can lead to false accusations or even wrong convictions.
Is the golden spiral a spiral?
The gold spiral shown in the figure above is perhaps the one most closely related to the gold ratio, which approximates a spiral. You only need to infinitely take the arcs in the smaller and smaller golden rectangles to get such a pattern.
In many places, this shape is applied to nature and art, such as the shape of the Nautilus, the shape of galaxies, the shape of hurricanes, and even the shape of ocean waves:
The problem, however, is that the golden spiral is not a spiral! It is a pattern made up of a series of circular arcs, transitioning from one arc to another, and the curvature of the spiral will jump, which is a jump that is unlikely to occur in any natural phenomenon. In the best case, the golden spiral can be approximated as a true spiral, which approximates an example of a logarithmic spiral, which is common in nature and can be expressed as follows by the polar coordinate equation:
In nature, such a spiral can be seen everywhere, and the value of b corresponds to a different reality, which holds true for any value of b, independent of the golden ratio. The b-value corresponding to the gold spiral is:
There is nothing special about this number. The shell of the Nautilus is a logarithmic spiral, and because this self-similarity allows it to grow without changing shape, its most common b-value is 0.18, which is far from the b-value of the golden spiral.
Proportions in art and architecture
It is believed that the golden ratio is more aesthetically pleasing to the eye, so in many works of art and architecture, the golden rectangle is also favored more than other rectangles. Admittedly, some artists and architects do incorporate the Golden Ratio into their work, but this is also an area where the concept of the Golden Ratio is overapplied.
Rationally speaking, the pleasurability of the golden rectangle is a statement of weak evidence. Psychological studies have shown that people's preferences for rectangles are wide-ranging, and various proportions have their audience groups, and the most popular of them is a rectangle with a length-to-width ratio of √ 2 to 1. Stanford mathematician and popular science writer Keith Devlin once wrote in a column of the American Mathematical Association that the relationship between the golden ratio and aesthetics is so deeply rooted in two people, the Italian mathematician Luca Pacioli and the German psychologist Adolf Zeising.
Luca Pacioli, a friend of Leonardo da Vinci, wrote a book in 1509 called The Magical Ratio, which, although titled on the Golden Ratio, did not assert any aesthetic theory based on the Golden Ratio. In addition, it is often said that Leonardo da Vinci used the golden ratio in his paintings, the most famous example is the painting "Vitruvius", but these ratios do not match the golden ratio, there is no direct evidence that Leonardo da Vinci used this ratio, he only mentioned the integer ratio in his work.
Zesin once described the golden ratio as "the beauty and integrity of the natural and artistic fields ... It is a supreme spiritual ideal that permeates all structures, forms and proportions, whether cosmic or personal, organic or inorganic, acoustic or optical. However, this claim went on to influence many others, laying the foundation for the modern myth of the "golden ratio."
There is also the view that the golden ratio is also important in music composition. However, as with art and architecture, there is little evidence to support this view. The number closely associated with the music is the root of 2 to the 12th power, not the golden ratio.
This exaggerated "myth" is actually very disturbing, and it can mislead many people and make people have a wrong understanding of the workings of mathematics. Those who believe in these myths discover that this is not the case, they may lose faith in the mathematical ability to explain the world.
The real magic of the Golden Ratio
If all we have said above is giving the golden ratio a "magical" hat, the next thing we are going to talk about is the real magic of the golden ratio.
There is no doubt that the golden ratio is a wonderful number in mathematics and science, and one of the important properties that really sets it apart from other numbers is its irrationality. Earlier we said that φ is an irrational number, that is, it cannot be represented as any fraction, but what is even more surprising is that it is the strongest irrational number in the irrationality. This means that not only can it not be accurately represented as a fraction, but it is even difficult to approximate it with fractions. This is a very special nature.
Why is φ the strongest number for irrationality? Mathematicians approximate an irrational number use fractions m/n consisting of two integers (m and n), and for any irrational number z, different n values correspond to different m values. To find the best approximation of z, it is necessary to find the absolute value that makes the difference between z and the approximate fraction, | z - m/n |, closest to 0 n, in other words, to find the n with the smallest approximation error.
Compared in the figure above is an approximate error plot of π (red) and φ (blue), the abscissa axis represents the value of n from 1 to 200, and the ordinate is the difference between the irrational number and the approximate value "Error = |z - m/n|". It can be seen that for π, when n=7 and n=113, a very good approximation of the π can be given. This is also known as π ≈ 22/7 and 355/113.
Compared to π, the approximate golden ratio φ is obviously not so clear. Its approximate error curve converges more slowly than the approximate error curves of other irrational numbers. The reason behind this is because φ has a special property - it can be represented as a "continuous score" so that φ can be written in this form:
It is a direct inference of the identity φ - 1 = 1/φ.
A key feature of the φ's fractional form is that each term has a 1, and the 1s contained in these denominators cause large errors, which makes the entire fraction converge slowly.
In contrast, the consecutive scores of π are as follows:
You can see that the numbers in its denominator are very large, such as 7, 15, 292, and so on. These large numbers will make the error of the continuous fraction much smaller.
However, this difficulty in approximating φ with fractions also makes it a very useful number for mathematicians and computer scientists when studying synchronization processes. It can be said that although the golden ratio is not as magical as the public imagines, when you understand what it really looks like, you may be even more amazed by the true charm of mathematics!
This article is excerpted and compiled from mathematician Chris Budd's lecture "Great Mathematical Myths" delivered at Gresham College on February 11, 2020, in which he also mentioned the famous three-door problem and the four-color theorem. Full-text links can be found in: https://www.gresham.ac.uk/lectures-and-events/great-maths-myths
Cover image from: mayeesherr. / Flickr
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