数学英语 14 How to Use the Golden Ratio

 

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by Jason Marshall

Despite many people’s assumption to the contrary, math is undeniably artistic¹. It takes a tremendous amount of creative muscle and artistry to devise mathematical solutions. And, as made evident by their frequent battles between elegant symmetry and rampant² chaos³, the traditional fine arts are chock-full of math. Suffice it to say that math and art are intimately related. Today we’re going to take a look at one of these happy relationships and see how the golden ratio that we talked about last time can make you a better photographer.
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 Golden Ratio
In the last article, we discovered a surprising connection between the Fibonacci sequence developed to model the growth of rabbit populations and the golden ratio used in art and architecture. By simply dividing each element in the Fibonacci sequence by the previous one, we obtained a brand new sequence which, amazingly enough, continually converged⁴ toward the value of the golden ratio—also known as “phi.” That may not seem all that amazing, but the strange thing is that phi and the shapes derived⁵ from it—likethe golden rectangle we talked about last time—had been used artistically⁶ for thousands of years before Fibonacci ever contemplated⁷ rabbits. So the fact that this number even shows up in the rabbit problem means there must be something fundamentally important about it...and that possibility is pretty amazing.
The Fibonacci Spiral
The most beautiful rectangle—the so called “golden rectangle”—has some very special properties. As we discussed last time, when a single line is drawn⁸ inside a golden rectangle to partition off a square, the leftover⁹ interior part is yet another golden rectangle. And this partitioning off and creating smaller and smaller pairs of squares and golden rectangles can continue forever. If you create such a figure, and then draw an arc sweeping¹⁰ diagonally from one corner of each newly formed square to the other, you will create a beautiful Fibonacci spiral.（见图一）

If you’re interested in seeing a video showing exactly how this is done, check out this week’s Math Dude “Video Extra!” episode on YouTube. Why do I call this spiral beautiful? Well, it certainly is subjective¹¹, but to me the symmetry is alluring¹². And it’s made even more so because this shape isn’t just some random¹³ mathematical construction—it frequently occurs in nature!
The Golden Ratio in Nature
Case 1: Compare the shape of the Fibonacci spiral to the shell of the chambered nautilus—which, if you’re curious, is an ocean-dwelling creature related to octopus¹⁴. The curves of the shell and the Fibonacci spiral trace out remarkably¹⁵ similar patterns. Case 2: Look at the seeds in the center of a sunflower—they’re arranged in spiral patterns curling in both directions. Though that by itself isn’t so strange, the fact that the number of spirals always equals a Fibonacci number is a little curious. It doesn’t matter which way you count the spirals—right or left—either. Both directions will give Fibonacci numbers. And they’ll be sequential Fibonacci numbers too—perhaps 34 in one direction and 55 in the other. That’s kinda weird¹⁶, right? Well, there are plenty more cases too. Fibonacci numbers and golden ratios show up in pinecones, pineapples, flower petals¹⁷, leaves on trees, and perhaps even make an appearance in the proportioning of the human body.
We should be a little careful here, however. Some of these facts could certainly be purely¹⁸ coincidental—in particular those about human anatomy¹⁹. But the Fibonacci numbers and Fibonacci spirals appear so frequently in nature that it seems these relationships must, at least in some cases, be a byproduct of some fundamental properties of biology—in other words, of life itself!
How to Use the Golden Ratio to Take Better Pictures
So, nature has bountifully embraced the golden ratio, artists have displayed its exquisite²⁰ proportions, and now you might be wondering: Can I use it too? Absolutely. Here’s a quick and dirty tip for improving the composition of your photographs using the golden ratio! First, when taking a picture, imagine placing the Fibonacci spiral on top of the scene you’re shooting. Then, the idea is to position the most important element of your shot—perhaps a person’s eyes—not at the overall center of the image, but at the off-centered eye of the Fibonacci spiral. It’s simple, but this technique really does make for more interesting pictures—search the web for examples and see for yourself.
（见图二）

This trick of using the golden ratio to lay out your image is related to the well-known “rule of thirds” you may have heard of. The idea here is to divide your image into thirds both horizontally and vertically²¹, and then to place important elements at the intersections²² of these lines. The rule of thirds is really just a simplified version of the golden ratio method—after all, it’s a lot easier to mentally picture dividing an image into thirds rather than a Fibonacci spiral. Either way, if you follow this rule when taking your pictures, your friends will soon be begging you to teach them your secrets—and now you know the math to do it!
Wrap Up
Okay, that’s all we have time for today. If you have a few minutes, I highly recommend watching the short video “Nature by Numbers” that’s recently been getting a lot of attention on YouTube. The video uses some stunning²³ computer graphics²⁴ to demonstrate the interplay between math and nature—including the Fibonacci spiral.
Also, Quick and Dirty Tips is happy to welcome the newest expert to our roster²⁵. Check out Get-Fit Guy's Quick And Dirty Tips to Slim Down and Shape Up. Whether you're a life-long couch potato looking to begin an exercise regimen for the first time or interested in training for a triathlon, professional trainer Ben Greenfield will give you the advice you need to build a stronger body and feel better. You can find his podcast wherever you download Quick and Dirty Tips podcasts or at QuickAndDirtyTips.com.
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.
Remember, you can get updates about the Math Dude and my day-to-day musings about math, science, and life in general by following me on Twitter, and join our growing community of social networking math fans by becoming a fan of the Math Dude on Facebook.
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!
 

1 artistic   

adj.艺术(家)的，美术(家)的；善于艺术创作的

参考例句：

• The picture on this screen is a good artistic work.这屏风上的画是件很好的艺术品。
• These artistic handicrafts are very popular with foreign friends.外国朋友很喜欢这些美术工艺品。

2 rampant   

adj.(植物)蔓生的；狂暴的，无约束的

参考例句：

• Sickness was rampant in the area.该地区疾病蔓延。
• You cannot allow children to rampant through the museum.你不能任由小孩子在博物馆里乱跑。

3 chaos   

n.混乱，无秩序

参考例句：

• After the failure of electricity supply the city was in chaos.停电后，城市一片混乱。
• The typhoon left chaos behind it.台风后一片混乱。

4 converged   

v.（线条、运动的物体等）会于一点( converge的过去式 )；（趋于）相似或相同；人或车辆汇集；聚集

参考例句：

• Thousands of supporters converged on London for the rally. 成千上万的支持者从四面八方汇聚伦敦举行集会。
• People converged on the political meeting from all parts of the city. 人们从城市的四面八方涌向这次政治集会。 来自《简明英汉词典》

5 derived   

vi.起源；由来；衍生；导出v.得到( derive的过去式和过去分词 )；（从…中）得到获得；源于；（从…中）提取

参考例句：

• Many English words are derived from Latin and Greek. 英语很多词源出于拉丁文和希腊文。 来自《简明英汉词典》
• He derived his enthusiasm for literature from his father. 他对文学的爱好是受他父亲的影响。 来自《简明英汉词典》

6 artistically   

adv.艺术性地

参考例句：

• The book is beautifully printed and artistically bound. 这本书印刷精美，装帧高雅。
• The room is artistically decorated. 房间布置得很美观。

7 contemplated   

adj. 预期的 动词contemplate的过去分词形式

参考例句：

• The doctor contemplated the difficult operation he had to perform. 医生仔细地考虑他所要做的棘手的手术。
• The government has contemplated reforming the entire tax system. 政府打算改革整个税收体制。

8 drawn   

v.拖，拉，拔出；adj.憔悴的，紧张的

参考例句：

• All the characters in the story are drawn from life.故事中的所有人物都取材于生活。
• Her gaze was drawn irresistibly to the scene outside.她的目光禁不住被外面的风景所吸引。

9 leftover   

n.剩货，残留物，剩饭；adj.残余的

参考例句：

• These narrow roads are a leftover from the days of horse-drawn carriages.这些小道是从马车时代沿用下来的。
• Wonder if that bakery lets us take leftover home.不知道那家糕饼店会不会让我们把卖剩的带回家。

10 sweeping   

adj.范围广大的，一扫无遗的

参考例句：

• The citizens voted for sweeping reforms.公民投票支持全面的改革。
• Can you hear the wind sweeping through the branches?你能听到风掠过树枝的声音吗？

11 subjective   

a.主观(上)的，个人的

参考例句：

• The way they interpreted their past was highly subjective. 他们解释其过去的方式太主观。
• A literary critic should not be too subjective in his approach. 文学评论家的看法不应太主观。

12 alluring   

adj.吸引人的，迷人的

参考例句：

• The life in a big city is alluring for the young people. 大都市的生活对年轻人颇具诱惑力。
• Lisette's large red mouth broke into a most alluring smile. 莉莎特的鲜红的大嘴露出了一副极为诱人的微笑。

13 random   

adj.随机的；任意的；n.偶然的(或随便的)行动

参考例句：

• The list is arranged in a random order.名单排列不分先后。
• On random inspection the meat was found to be bad.经抽查，发现肉变质了。

14 octopus   

n.章鱼

参考例句：

• He experienced nausea after eating octopus.吃了章鱼后他感到恶心。
• One octopus has eight tentacles.一条章鱼有八根触角。

15 remarkably   

ad.不同寻常地，相当地

参考例句：

• I thought she was remarkably restrained in the circumstances. 我认为她在那种情况下非常克制。
• He made a remarkably swift recovery. 他康复得相当快。

16 weird   

adj.古怪的，离奇的；怪诞的，神秘而可怕的

参考例句：

• From his weird behaviour,he seems a bit of an oddity.从他不寻常的行为看来，他好像有点怪。
• His weird clothes really gas me.他的怪衣裳简直笑死人。

17 petals   

n.花瓣( petal的名词复数 )

参考例句：

• white petals tinged with blue 略带蓝色的白花瓣
• The petals of many flowers expand in the sunshine. 许多花瓣在阳光下开放。 来自《现代英汉综合大词典》

18 purely   

adv.纯粹地，完全地

参考例句：

• I helped him purely and simply out of friendship.我帮他纯粹是出于友情。
• This disproves the theory that children are purely imitative.这证明认为儿童只会单纯地模仿的理论是站不住脚的。

19 anatomy   

n.解剖学，解剖；功能，结构，组织

参考例句：

• He found out a great deal about the anatomy of animals.在动物解剖学方面，他有过许多发现。
• The hurricane's anatomy was powerful and complex.对飓风的剖析是一项庞大而复杂的工作。

20 exquisite   

adj.精美的；敏锐的；剧烈的，感觉强烈的

参考例句：

• I was admiring the exquisite workmanship in the mosaic.我当时正在欣赏镶嵌画的精致做工。
• I still remember the exquisite pleasure I experienced in Bali.我依然记得在巴厘岛所经历的那种剧烈的快感。

21 vertically   

adv.垂直地

参考例句：

• Line the pages for the graph both horizontally and vertically.在这几页上同时画上横线和竖线，以便制作图表。
• The human brain is divided vertically down the middle into two hemispheres.人脑从中央垂直地分为两半球。

22 intersections   

n.横断( intersection的名词复数 )；交叉；交叉点；交集

参考例句：

• Traffic lights have been placed at all major intersections. 所有重要的交叉路口都安装了交通信号灯。
• Intersections are of the greatest importance in highway design. 在道路设计中，交叉口占有最重要的地位。 来自辞典例句

23 stunning   

adj.极好的；使人晕倒的

参考例句：

• 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。

24 graphics   

n.制图法，制图学；图形显示

参考例句：

• You've leveraged your graphics experience into the video area.你们把图形设计业务的经验运用到录像业务中去。
• Improved graphics took computer games into a new era.经改进的制图技术将电脑游戏带进了一个新时代。

25 roster   

n.值勤表，花名册

参考例句：

• The teacher checked the roster to see whom he would teach this year.老师查看花名册，想了解今年要教的学生。
• The next day he put himself first on the new roster for domestic chores.第二天，他把自己排在了新的家务值日表的第一位。