What
is the Golden Ratio? As with any ratio this term can, in simple terms, be
described as a proportion. Originally, the Golden Ratio was discovered by
Euclid of Alexandria, who used the example of a line broken into three segments
to derive the concept of an “extreme” and “mean” ratio. Mario Livio writes, “A
straight line is said to have been cut in extreme and mean ratio when, as the
whole line is to the greater segment, so is the greater to the lesser”, when a
segment is broken up into segments A,
B, and C as pictured below.
Though Euclid derived this ratio solely for geometry,
it was discovered to play a part in a myriad of things ranging from nature to
physics to astronomy, and even in the realm of arts and aesthetics. Similar to pi (
), the Golden Ratio is
one followed by a never-repeating, infinite decimal number. Conversely to pi, the Golden Ratio cannot be expressed
as a fraction – it is both irrational and incommensurable, which occurs when a
line has two lengths with no common measure. Originally the symbol for the
Golden Ratio was tau (τ), but at the
beginning of the twentieth century, an American mathematician gave the ratio
the name of phi (ϕ). The first six
numbers of its value are 1.61803, continuing infinitely.
), the Golden Ratio is
one followed by a never-repeating, infinite decimal number. Conversely to pi, the Golden Ratio cannot be expressed
as a fraction – it is both irrational and incommensurable, which occurs when a
line has two lengths with no common measure. Originally the symbol for the
Golden Ratio was tau (τ), but at the
beginning of the twentieth century, an American mathematician gave the ratio
the name of phi (ϕ). The first six
numbers of its value are 1.61803, continuing infinitely.
We as
somewhat simple human beings do not usually seek or recognize mathematics in
nature - or even anywhere outside of the classroom for that matter. However,
the Golden Ratio has been discovered in many facets of the natural world
ranging from the reproduction of rabbits, to the arrangement of leaves around
branches, to the optics of light rays through glass plates. One form in
particular that I find very interesting which incorporates the Golden Ratio is
the Nautilus pompilius, or chambered nautilus, which is the shell of a mollusk.
In pure mathematical construction, the nautilus is composed of "whirling
squares" (Livio, The Golden Ratio, p.119) which are formed by cutting off
squares from a Golden Rectangle, connecting the squares, and dividing the sides
by the Golden Ratio. This creates an image of a logarithmic spiral curling
inwards. It is also possible to devise a chambered nautilus from a Golden
Triangle.
Leonardo Fibonacci, discoverer of the famous Fibonacci sequence,
played an important role in the development and application of the Golden Ratio
in a variety of different areas including geometry, and nature as well. It was
he who studied the reproduction of rabbits and how they regenerate in the way
of the Fibonacci sequence. What he discovered was that when two rabbits of opposite
genders are paired together, they will produce one rabbit, the new sum of all
of the rabbits being three (0 + 1 = 1, 1 + 1 = 2, 2 + 1 = 3…). While this
sequence of rabbits may seem unrelated to the Golden Ratio itself, Fibonacci experiment
with the ratio by dividing the first number of the sequence by the second, the
third number by the second, the fourth number by the third, and so on (1/1,
2/1, 3/2, 5/3, 8/5…987/610 = 1.618033…). Eventually by going through these
prime numbers and dividing the latter by the former, one will reach the
approximate value of the Golden Ratio.
So how
is the ever-intriguing Golden Ratio and the absolutely enthralling Nautilus
pompilius, among other things, relevant to Leonardo da Vinci? In many of
his paintings, da Vinci focuses heavily on symmetry and geometrical balance.
For example, his painting “Leda and the Swan” portrays the chambered nautilus
in the arrangement of the woman’s hair. Da Vinci’s painting “Deluge” also imposes
these logarithmic spirals even further as he describes the scene:
The sudden waters rush into the pond that contains them,
striking the various obstacles with swirling eddies…The momentum of the
circular waves flying from the point of impact hurls them in the way of other
circular waves moving in the opposite direction.
The photo below of “Deluge” shows us even more clearly
the coils of the winds and floods of the storm. It’s incredible to me how an
artist, da Vinci in particular, plans their work of seemingly un-math-related
artistry, of nature or of humans for example, with such intricate underlying
sequences that most people would never be able to decipher without prior
knowledge of the patterns.
I, for one, would not have been able to tell whether
there was a method to the madness of this storm or not. Now that I have learned
about the concept of “whirling squares” and the connection of the vertices of
the Golden Triangles, I can more easily sort out the disparity of the spiraling
waters. Because of his prevalent use of the Golden Ratio, da Vinci has been
attributed by several authors the invention of the name “the Divine proportion”.
If it weren’t for the ratio, his well-known works “St. Jerome”, “Madonna on the
Rocks”, and the most famous “Mona Lisa” (to name a few), Leonardo da Vinci’s
paintings would not be the same. Thanks Euclid!!
Sources:
Leonardo, McCurdy, Edward. (1906) Leonardo da
Vinci's Note-books London, Duckworth & Co.
Livio, Mario. The Golden Ratio: The Story of Phi,
the World's Most Astonishing Number. New York: Broadway, 2002. Print.
Müntz, Eugène. (2006) Leonardo da Vinci: Artist,
Thinker, and Man of Science. New York : Parkstone Press
"The Golden Ratio: Phi, 1.618 - Golden Ratio,
Phi, 1.618, and Fibonacci in Math, Nature, Art, Design, Beauty and the
Face." PhiPoint Solutions, LLC, n.d. Web. 15 Nov. 2015.
The concept of the Golden Ratio is a little difficult to understand, even though it seems so simple. It is amazing that what seems like a normal line segment separated into three parts actually as a lot more to it. I never thought that it would have so many mathematical uses and have an infinite number value. I also never knew that da Vinci used such a concept in is paintings. Like Monica, I had believed that da Vinci had just painted based up likes, interests and experiences, yet as Monica explained he incorporated symmetry and geometry in his paintings. I now look at the paintings in a different way, not so much as a work of art, but as a planned out mathematical picture.
ReplyDeleteThe concept of the Golden Ratio is a little difficult to understand, even though it seems so simple. It is amazing that what seems like a normal line segment separated into three parts actually as a lot more to it. I never thought that it would have so many mathematical uses and have an infinite number value. I also never knew that da Vinci used such a concept in is paintings. Like Monica, I had believed that da Vinci had just painted based up likes, interests and experiences, yet as Monica explained he incorporated symmetry and geometry in his paintings. I now look at the paintings in a different way, not so much as a work of art, but as a planned out mathematical picture.
ReplyDeleteThe first time I had heard about the golden ratio was while reading Devin’s post over the Harriss Spiral; it is interesting to see how one mathematical concept is incorporated into so many different things, including topics beyond what it was created for. It’s rather intriguing that such a concept could be found in so many aspects of the world unintentionally. Euclid must have been a genius to come up with such a concept. I knew prior to reading this post that Leonardo da Vinci used symmetry in his artwork, but I was not aware that such a complex idea like the golden ratio was also incorporated into them. It definitely puts his works into a new perspective.
ReplyDeleteI have heard of and seen documentaries that have mentioned the Golden Ratio. But I still did not know exactly what it means. Monica did a good job of explaining the history and concept of the Golden Ratio into terms that we could understand. I do have to say that it’s not surprising that Leonardo Da Vinci used this concept in his paintings and his inventions because symmetry played a large factor in the design and artwork. I agree with Molly that it’s interesting that these concepts could be found in the world unintentionally. It does make you think of the world a little differently and opens your eyes to a whole new idea of something that you didn’t necessarily know was there.
ReplyDeleteI have never really heard of the golden ratio before, though I do find it very interesting that it shows up in historical art such as works done by Leonardo De Vinci. I find it a little strange that something as simple as a geometrical line can be connected to artwork. The golden rule is something that I might want to learn about in further detail because it is pretty difficult to understand. The idea that the golden ratio and a geometrical line is connected to a painting that looks something like a storm is pretty difficult to wrap my head around. Even though it is difficult to understand, like I said earlier, there is something interesting about it.
ReplyDeleteThe golden ratio was something I learned in art class in high school, but never really understood it beyond what I needed to know in art class. Monica did a great job of explaining where the history of the golden ratio and where/how it came about. I still do not understand where else this may be used, it seems like it does not relate much to the real world. Another interesting thing I found in this blog post was that Da Vinci was attributed with discovery. I never knew he was a main person in the Golden Ratio. Overall I found this post very interesting!
ReplyDeleteI find it very interesting how frequently this “Golden Ratio” appears in nature in the most arbitrary ways and places. I would never think to look for the Golden Ratio or think to apply it in situations like the arrangements of leaves on a tree, or the reproduction of rabbits. The part that talks about the optics of light rays passing through glass plates sounded like a very interesting subject, I’ve always been very interesting in the way light rays act and behave through glass or prisms I never would have thought that the Golden Ratio would have anything to with it.
ReplyDeleteThe topic that Monica chose for her final blog post was very interesting. I liked how the golden ration has been used so frequently over such a long span of time. If I were half as innovative or smart as Leonardo da Vinci, I would be very well off. It is absolutely amazing how he is able to weave mathematical constructs into his paintings. It reminds me of my topic, symmetry, and how a perfect ratio or symmetry can make something much more beautiful than it would be without. All in all, Monica’s post was full of little factoids that I thought were very fun to read.
ReplyDeleteThe Golden Ratio is a term I've been hearing for many years now but I''ve never really taken the time to try and understand it and its significance to the real world. So after reading the title of this post I was very interested in what the post would contain. A part of this post I enjoyed in particular was the part describing the Fibonacci sequence. This sequence is something that has always interested me since we talked about it so much in class and I've written many programs that involved it. It was also interesting to read about how Leonardo Da Vinci used the gold ratio in his paintings.
ReplyDeleteIv'e heard about the golden ratio many times with math projects. It's a very interesting concept and it's quite amazing how famous artists would use the golden ration in their art because it would be more appealing to the eye. It seems a lot of people find it hard to understand but it's actually quite simple. Maybe a picture showing the golden ration on a plant or painting would help people to visually see the golden ration in action. I like the example of the storm painting and explaining that there is actually a method the artist used to portray their painting. Overall, you did a great job of explaining the golden ratio.
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ReplyDeleteThis was an easy read for me because my blog post dealt a lot with the Golden Ratio, so I knew exactly what was being said because of my research I did. It is surprising to see how many things naturally that are involved with the golden ratio and not with just man made things. I didn't realize how involved Leonardo Fibonacci was involved with the f=golden ratio. I'm familiar with him just because of the famous Fibonacci sequence that I used I don't know how many times through out the years I took math. It was neat to see how Leonardo De Vinci was able to take the golden ratio and put them into his masterpieces that he painted. Although I knew exactly what the golden ratio was some of the things in the blog confused me, but it is not an easy topic to talk about but she did a good job explaining it
ReplyDeleteI love the Fibonacci sequence. I am a complete nerd as this was one of my favorite things to learn about when I was in Algebra. I think the sequence itself is extremely interesting and shows up in many different ways in math. One thing I didn’t know however was that the Golden Ratio appears frequently in nature and art. I’ve never really looked at art and thought about the different components that make up the painting. It is pretty amazing that two completely different subject like math and art coincide with one another.
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