数学英语 26 What is the Golden Ratio
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by Jason Marshall
Some math is functional 1. Some math is fun. And some math is simply stunning 2. If that last description sounds improbable to you, then today just might change your mind. Because now that we’ve covered enough ground, we’re going to take a look at some of the surprising, elegant, and downright mysterious ways that the Fibonacci sequence shows up in the world around you.
But first, the podcast edition of this tip was sponsored by Go To Meeting. Save time and money by hosting your meetings online. Visit GoToMeeting.com/podcast and sign up for a free 45 day trial of their web conferencing solution.
Recap of the Fibonacci Sequence
In the last article, we talked about how a seemingly innocent question about the growth of rabbit populations led Fibonacci to the sequence of numbers that now bears his name—the Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21, ...
Each successive number in this sequence is obtained by adding the two previous numbers together. And, save a few complicating 3 details like the fact that rabbits eventually grow old and die, this sequence does an admirable job at modeling how populations grow. But the numbers in Fibonacci’s sequence have a life far beyond rabbits, and show up in the most unexpected places.
What is the Golden Ratio?
One such place is particularly fascinating: the golden ratio. So, what is this golden ratio? Well, it’s a number that’s equal to approximately 1.618. This number is now often known as “phi” and is expressed in writing using the symbol for the letter phi from the Greek alphabet. Phi isn’t equal to precisely 4 1.618 since, like its famous cousin pi, phi is an irrational 5 number—which means that its decimal digits 6 carry on forever without repeating a pattern. If you’re interested in seeing how the actual value of phi is obtained, check out this week’s Math Dude “Video Extra!” episode on YouTube. But how did this number come to be of such importance? Oddly, it started as a question of aesthetics 7.
The Golden Rectangle
What’s the most beautiful rectangle? More specifically: What’s the ratio of this “most beautiful” rectangle’s height to its width? This question seems strange, but it isn’t crazy. We won’t go into the details right now, but there is evidence that people tend to perceive one particular shape of rectangle as being most pleasing to the eye. Of course, the Greeks knew this long before modern psychologists tested it, which is why they used golden rectangles, as well as other golden shapes and proportions adhering to the golden ratio, in their architecture and art.
For example, almost 2500 years ago, a Greek sculptor 8 and architect named Phidias is thought to have used the golden ratio to design the statues he sculpted 9 for the Parthenon (note the word “phi” in Phidias’ name—that isn’t a coincidence and actually inspired the naming of the number in the 20th century). And since Phidias’ time, numerous painters and musicians have incorporated the golden ratio into their work too—Leonardo da Vinci, Salvador Dalí, and Claude Debussy, among many others.
But back to the problem of figuring out the shape of the most pleasing rectangle. If you simply draw what you believe to be the most beautiful rectangle, then measure the lengths of each side, and finally divide the longest length by the shortest, you’ll probably find that the ratio is somewhere around 1.6—which is the golden ratio, phi, rounded to the nearest tenth. It won’t be exactly 1.6, but it should be pretty close. Besides being “beautiful,” the resulting shape has an intriguing 10 characteristic: If you draw a golden rectangle, and then draw a line inside it to divide that rectangle into a square and another smaller rectangle, that smaller rectangle will amazingly be another golden rectangle! You can do this again with this new golden rectangle, and you’ll once again get a square and yet another golden rectangle.
（见图）
Connection Between the Golden Ratio and the Fibonacci Sequence
Okay, but what about the Fibonacci sequence? How does that figure into this? I know it might seem totally unrelated, but check this out. Let’s create a new sequence of numbers by dividing each number in the Fibonacci sequence by the previous number in the sequence. Remember, the sequence is
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...
So, dividing each number by the previous number gives: 1 / 1 = 1, 2 / 1 = 2, 3 / 2 = 1.5, and so on up to 144 / 89 = 1.6179…. The resulting sequence is:
1, 2, 1.5, 1.666..., 1.6, 1.625, 1.615…, 1.619…, 1.6176…, 1.6181…, 1.6179…
But do you notice anything about those numbers? Perhaps the fact that they keep oscillating around and getting tantalizingly 11 closer and closer to 1.618?—the value of phi: the golden ratio! Indeed, completely unbeknownst to Fibonacci, his solution to the rabbit population growth problem has a deep underlying 12 connection to the golden ratio that artists and architects have used for thousands of years!
Applications of the Golden Ratio
But the golden ratio isn’t just for mathematicians 13, Greek sculptors 14, and Renaissance 15 painters—you can use it in your life too. In fact, in the next article we’ll talk about how you can use the golden ratio to help you take better pictures. And there’s even more. Not only do these pleasing shapes show up in human art, they also show up in the “art” of the natural world—in everything from shells to sunflowers! We’ll talk about all that next time too.
Wrap Up
That’s all the math we have time for today. Thanks again to our sponsor this week, Go To Meeting. Visit GoToMeeting.com/podcast and sign up for a free 45 day trial of their online conferencing service.
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Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
- The telephone was out of order,but is functional now.电话刚才坏了，但现在可以用了。
- The furniture is not fancy,just functional.这些家具不是摆着好看的，只是为了实用。
- His plays are distinguished only by their stunning mediocrity.他的戏剧与众不同之处就是平凡得出奇。
- The finished effect was absolutely stunning.完工后的效果非常美。
- High spiking fever with chills is suggestive of a complicating pylephlebitis. 伴有寒战的高热，暗示合并门静脉炎。
- In America these actions become executive puberty rites, complicating relationships that are already complicated enough. 在美国，这些行动成了行政青春期的惯例，使本来已经够复杂的关系变得更复杂了。
- It's precisely that sort of slick sales-talk that I mistrust.我不相信的正是那种油腔滑调的推销宣传。
- The man adjusted very precisely.那个人调得很准。
- After taking the drug she became completely irrational.她在吸毒后变得完全失去了理性。
- There are also signs of irrational exuberance among some investors.在某些投资者中是存在非理性繁荣的征象的。
- The number 1000 contains four digits. 1000是四位数。 来自《简明英汉词典》
- The number 410 contains three digits. 数字 410 中包括三个数目字。 来自《现代英汉综合大词典》
- Sometimes, of course, our markings may be simply a matter of aesthetics. 当然，有时我们的标点符号也许只是个审美的问题。 来自名作英译部分
- The field of aesthetics presents an especially difficult problem to the historian. 美学领域向历史学家提出了一个格外困难的问题。
- A sculptor forms her material.雕塑家把材料塑造成雕塑品。
- The sculptor rounded the clay into a sphere.那位雕塑家把黏土做成了一个球状。
- a display of animals sculpted in ice 冰雕动物展
- The ladies had their hair sculpted by the leading coiffeur of the day. 女士们的发型都是当代有名的理发师做的。
- These discoveries raise intriguing questions. 这些发现带来了非常有趣的问题。
- It all sounds very intriguing. 这些听起来都很有趣。 来自《简明英汉词典》
- A band of caribou passed by, twenty and odd animals, tantalizingly within rifle range. 一群驯鹿走了过去，大约有二十多头，都呆在可望而不可即的来福枪的射程以内。 来自英汉文学 - 热爱生命
- She smiled at him tantalizingly. 她引诱性地对他笑着。 来自互联网
- The underlying theme of the novel is very serious.小说隐含的主题是十分严肃的。
- This word has its underlying meaning.这个单词有它潜在的含义。
- Do you suppose our mathematicians are unequal to that? 你以为我们的数学家做不到这一点吗？ 来自英汉文学
- Mathematicians can solve problems with two variables. 数学家们可以用两个变数来解决问题。 来自哲学部分
- He is one of Britain's best-known sculptors. 他是英国最有名的雕塑家之一。
- Painters and sculptors are indexed separately. 画家和雕刻家被分开，分别做了索引。
- The Renaissance was an epoch of unparalleled cultural achievement.文艺复兴是一个文化上取得空前成就的时代。
- The theme of the conference is renaissance Europe.大会的主题是文艺复兴时期的欧洲。
- A group of enthusiasts have undertaken the reconstruction of a steam locomotive. 一群火车迷已担负起重造蒸汽机车的任务。 来自《简明英汉词典》
- Now a group of enthusiasts are going to have the plane restored. 一群热心人计划修复这架飞机。 来自新概念英语第二册