Most of this blog’s readers would have heard about the Golden Ratio, supposedly first derived by Euclid. If one takes a straight line segment and divides into two such that the ratio of the longer part to the shorter is the same as the ratio of the whole line to the longer part, some simple algebra shows that this ratio works out to be about 1.618 to one.
Over time, this ratio has developed an almost mystical quality with people claiming that it represents an aesthetic standard of beauty and that it can be found occurring naturally in many naturally occurring objects that are supposedly pleasing to look at. People have looked for objects in nature whose height-to-width ratio is the Golden Ratio to suggest that there is some intrinsic value to this number.
Now the debunking has begun with people, often mathematicians, pointing out that many of the claims for it have no basis.
But the widespread belief that the golden ratio is the natural blueprint for beauty is pseudo-scientific “hocus-pocus” and a “myth that refuses to go away”, according to leading mathematicians.
The issue has flared up again, after one of the United States’s leading scientific organisations, the Smithsonian, promoted highly contentious claims about the ratio at the National Math Festival in Washington DC earlier this month.
…Eve Torrence, a professor at Randolph–Macon College in Virginia, said she was appalled to find a Smithsonian-branded stall which claimed the golden ratio is found in the human body. It offered visitors the chance to put their head through an oval, allegedly to demonstrate whether their face was in accordance with what is also known as the “divine proportion”.
“The idea that there’s this one rectangle [based on the golden ratio] that’s this perfect one… and is reflected in the human body, that’s one of the most silly things. Human beings are so different,” she said.
“There are lots of ratios and proportions in the human body, but they are not all the golden ratio and they are not all precisely the golden ratio. It’s a very loosey-goosey, pseudo-science kind of thing that they are promoting.
“There’s not this number that’s got this perfection in the way people think it does. It feels dirty to mathematicians. It’s hocus-pocus.”
…But Dr Keith Devlin, a Stanford University mathematician, said Euclid had never claimed the ratio had any aesthetic qualities, an idea largely invented by Gustav Theodor Fechner, a 19th-century German psychologist. More recently it appeared in a 1959 educational cartoon, Donald in Mathmagic Land, and Dan Brown’s The Da Vinci Code.
Dr Devlin, who campaigns against myths associated with the golden ratio, pointed to “considerable evidence” that people do not find golden rectangles more appealing than others. On the contrary, they tend to favour aspect ratios they are familiar with, such as an A4 piece of paper or a computer screen.
Of course, this will not deter true believers. There are always those who like to see mystical properties in the otherwise ordinary, maybe because it reassures them that there is some pattern and plan to the universe.
…this ratio works out to be about 1.618 to one.
Ha! Minus one point for not having the exact value!
(Sorry. I’m in the middle of grading exams.)
I’ve never bought into the claims of it being “perfect”, but it’s easy to see why the Greeks used in their architecture. It’s easy to make a right triangle with sides 1, 2 and √5, then work from there.
Two facts that I like about the golden ratio:
1) It can be found in Fibonacci numbers. (1, 2, 3, 5, 8, 13, 21, etc.). As the numbers get larger, the fractional ratio gets closer to ϕ (e.g. 144/89 = 1.617977, 233/144 = 1.618055). ϕ is irrational, but after a while the difference is negligible. The 100th and 101st Fibonacci numbers are accurate to ϕ by more than thirty decimal places.
2) It’s the easiest way to make accurate pentagons with only a straightedge and compass. A isosceles triangle with one side of n and two sides of ϕn has corners of 72, 72 and 36 degrees.
Dunno about aesthetics, but I’ve always been very fond of Stirling’s approximation. Very useful in calculating probabilities, etc, and it has e and π in it!
Re the golden ratio. Sadly, you can drag the poor thing into any problem which has a √5 in it.
I never even heard of the GR or Fibonacci numbers until many years out of school. Once I had, I did get the feeling that it was mostly honey to woo-inclined people, rather than something really significant.
I suppose the Pythagoreans would be happy to hear there are still people looking for mystical numbers.
The Pythagoreans wouldn’t be happy to hear that some people are looking for mystical irrational numbers…
Dang I liked the idea that body proportions were roughly based along the golden ration.
So I have to say thank you for pointing something that is wrong.
The ratio √2:1 — or 2:√2 — is also quite pleasing visually, and is observed in real life in paper sizes.
The long and short sides always have to be in the same ratio to one another, so you can always enlarge or reduce from one size to the next without fear of losing detail at the edges. At √2:1, cutting the sheet in half along the long edge leaves two pieces with ratios of 1:√2/2, which is the same as √2:1. This is the most economical way of making paper, since each size is exactly half the next size up with no wastage.
I thought this was due to artistic concepts like Da Vinci’s Vitruvian Man.
Of course, the fact that several proportions vary considerably between men and women makes this idea generally impossible.
“the golden ratio is the natural blueprint for beauty”
Omit the word natural and there is a strong point to make:
http://www.fiddletree.com/drawing_board/Fiddle%20Drawing-.JPG
http://www.goldennumber.net/acoustics/
After hearing something like this I don’t find it hard to associate the Golden Ratio with the divine.
mnb @9: Divine? I’ll give you divine.