Golden Ratio in logo designs.
July 29, 2011
Beauty and aesthetics have been praised from time immemorial. But little did people know that the most effective, perfectly balanced, and visually compelling creations followed the tid-bits of mathematics. At least not until 1860, when German physicist and psychologist Gustav Theodor Fechner proposed that a simple ratio, an irrational number defines the balance in nature. The Golden Ratio! Fechner’s experiment was simple: ten rectangles varying in their length-to-width ratios were placed in front of a subject, who was asked to select the most pleasing one. The results showed that the most favored choice was the “Golden Rectangle” (with ratio 1.618).
Golden Ratio
Golden Mean, Golden Section, Divine Proportion are all common names for what is known as the Golden Ratio which is based off the number phi (φ) = 1.61803398874… discovered by Italian Mathematician Fibonacci. Phi (φ) is the ratio between the number sequence 1, 1, 2, 3, 5, 8, 13, 21 etc. where the next number in the sequence is derived by adding 2 numbers together. So, 1+1 = 2, and 1+2 = 3, 2+3 = 5 and so on. When we divide two sequential numbers i.e. 5/3 = 1.67 and 21/13 = 1.615 
the ratio between these numbers soon become very close to φ (1.618). Fibonacci’s 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian mathematics, by Brahmagupta in 598 almost a thousand years earlier.
What’s so amazing about this number? Some believe that it is the most efficient outcome, the result of natural forces. Some believe it is a universal constant of design, the signature of God. Whatever you believe, the pervasive appearance of φ in all we see and experience creates a sense of balance, harmony and beauty in the design of all we find in nature. It should be no surprise then that mankind would use this same proportion found in nature to achieve balance, harmony and beauty in its own creations of art, architecture, colors, design, composition, space and even music. From the Parthenon toMonalisa, from the Egyptian pyramids tocredit cards, φ has been there, always.
Logos with golden ratio
So, it was not surprising when I found the invasion of φ in logo designs. Let us have a look at some of the most popular brands which have used the golden ratio to induce the perfect harmony and balance in their logos.
National Geographic
Remember the yellow square in the National Geographic logo? Have you ever wondered why that simple logo appears to be so appealing? The answer is, as you might know, the Golden Ratio! The length and width of the square have a ratio of 1.61. It is quite fitting for an organization with a motto of “inspiring people to care about the planet” to have a logo based on the golden rectangle.
Pepsi
The new logo of Pepsi has been much simpler and effective, characterized by spare, pure design. It looks intriguing and beautiful. Almost like a laughing emoticon in red and blue. But did you know that the underlying backbone of the Pepsi logo follows the golden ratio? The Pepsi brand is created by intersecting circles with a set proportion to each other. And, the proportion: Golden Ratio (φ) !
Apple
Apple is one of those very few companies that do not have the company name in their logo. Yet, the Apple logo is one of the most recognized corporate symbols in the world. The logo is perfectly balanced, and the outlines that map the logo are circles with diameters proportionate to the Fibonacci series. Did Rob Janoff really considered the Fibonacci series while designing it, or is it a coincidence? Well, somebody needs to ask Mr. Janoff. Interestingly, in a different context, in an interview, Rob Janoff said, “… and years later you find out supposedly why you did certain things. And, they are all BS. It’s a wonderful urban legend.”
iCloud
Another product from Apple, and again a masterpiece of design. The ripples on the cloud are made up of circles whose diameters are proportional to the you-know-what number. Also the containing rectangle, as shown below, is a golden rectangle. In fact, most of the Apple products, ranging from ipods to iPhone are golden rectangles. These amazing Apple designers!
BP
BP is one of the world’s leading international oil and gas companies. They launched their new logo in 2000. What appears to be an attractive logo, however, turns out to be formed of concentric circles, again proportional to the Fibonacci sequence. Is it a mere co-incidence or a planned execution?
Toyota
The logo of Toyota consists of three ovals. “The two intersecting ellipses are intended to represent the customer and the product… and the importance of that relationship”, according to an e-mail from Mike Michels, VP of Communication at Toyota Motor Sales USA, Inc. “The outer ring represents the world and the global nature of our business.” On a closer look one can easily find a grid based on φ in their logo. The phi-grid is formed by gridlines at certain separation – the separations being in the ratio of the golden ratio φ.
Grupo Boticário
The logo of the Brazilian company Grupo Boticário was designed by the Brazil office ofFuturebrand. This logo uses a golden spiral. In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio. That is, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes. The golden spiral is very closely approximated by the Fibonacci spiral (shown above). The golden spiral is very common in nature, for example, the spiral galaxies and mollusc shells. Do you like the use of golden spiral in this logo?
Golden Ratio in nature – a short film
I sincerely do believe that any discussion on golden ratio or the Divine Proportion (a name more appropriate) remains incomplete without showing how accurately the number φ finds its way into a plethora of natural creations. I found this nice and short video on the interwebs to do the job easier for me.
从远古时代，美观与美学就开始受到人们的赞扬。但很少有人知道最有效、最平衡完美、最有视觉冲击力的创作往往和数学有着丝丝的联系。直到1860年，德国物理学家、心理学家Gustav Theodor Fechner提出一个简单比率，通过一个无理数来定义大自然中的平衡，即黄金分割率。Fechner的实验很简单：十个矩形具有不同的长宽比，请人们从中选出最美好的一个。结果显示，最受青睐的选择是具有“黄金分割率的矩形”（比例为1.618）。
黄金分割率
黄金分割率基于数字φ=1.61803398874……，该数字最早由意大利数学家Fibonacci提出。Φ是斐波那契数列1，1，2，3，5，8，13，21，……中从第二位起相邻两数之比，即2/3，3/5，5/8，8/13，13/21，...的近似值。在该数字序列中，下一个数字（从第三个开始）是前两个数字之和，即1+1=2，1+2=3，2+3=5，……。该序列中两个相邻数字相比，如5/3=1.67，21/13=1.615，所得的结果与φ(1.618)越来越接近。
这个数字了不起的地方在哪里呢？一些人认为它是最有效率的结果，自然力量的结果。一些人认为它是设计的普适常量，神的签名。无论你相信哪一种说法，我们在大自然中所发现的所有设计中，φ为其创造了平衡、和谐与美观的感觉。那么，人类在自己的艺术、架构、颜色、设计、作曲，甚至音乐创作中，利用这个在自然界中发现的比率以达到平衡、和谐、美观的目的，也就不足为奇了。从帕台农神庙到蒙娜丽莎，从埃及金字塔到信用卡，都应用了φ。
拥有黄金分割的Logo设计
Φ同样应用到了Logo设计中。让我们看看一些最著名的品牌，它们在自己Logo中应有到了黄金分割率，从而达到一种完美融洽与平衡的效果。
还记得美国国家地理的Logo中黄色的矩形框吗？你曾经是否很好奇为什么这个简单的Logo会如此吸引人？正如你所知道的那样，答案是黄金分割率。该矩形框的长和宽的比值为1.61。基于黄金矩形的Logo与该组织的座右铭（“激励人们去关心地球”）十分贴合。
百事的新Logo更简单、更有效、更具有空间感。它看起来有趣而漂亮，红蓝之间的图形像一个小笑脸。你知道Pepsi Logo最基本的框架符合黄金分割率吗？它由几个交叉的圆组成，彼此之间遵循一定的比率，该比率是：黄金分割率（φ）。
Apple的Logo上没有标识出公司名称，只有很少的几家公司会这样做。但Apple的Logo却成为全世界最著名的公司标识之一。该Logo具有完美的平衡，映射到Logo上的轮廓是在直径上遵循斐波那契数列的圆形。
iCloud是Apple的另一个产品，也是一个设计杰作。云边缘的“波纹”由一系列的圆组成，其直径遵循黄金分割率。同时该Logo还包含一个“黄金矩形”，如下图所示。实际上，Apple的大部分产品，从iPod到iPhone都包含黄金矩形。
BP是世界上最大的石油和石化集团公司之一。2000年，该公司发布了新Logo。该Logo之所以引人瞩目，原因在于它由一系列同心圆组成，在直径比例上同样遵循斐波那契数列。
Toyota的Logo由三个椭圆组成。美国丰田汽车销售公司首席发言人Mike Michels在邮件中表示，“两个交叉的椭圆意在代表客户与产品……及两者之间关系的重要性。最外面的椭圆代表业务的全世界及全球特性。”进一步观察这个Logo，我们会很容易地发现一个基于φ的网格。该网格的网格线间隔遵循黄金分割率φ。
巴西企业Grupo Boticário的Logo由Futurebrand的巴西分公司设计。该Logo用到了黄金螺旋。在几何上，黄金螺旋是一个对数螺旋线，其增长因子为φ——黄金分割率。即黄金螺旋会在每个四分之一处按照增长因子φ逐渐增宽。黄金螺旋十分近似于斐波那契螺线。黄金螺旋在大自然中十分普遍，例如螺旋星系与软体动物的壳。（编译：陈秋歌）
