Humanity has always been in search of patterns.
They make us feel comfortable. They give us meaning. Whether they be in the deepest, most
conceptually difficult topics like string theory and quantum mechanics, or even in simple
things like the behaviour of the person we are talking to, we love to seek patterns, and
do so sometimes against our better judgement. Nature and mathematics are no exceptions to
that list when it comes to pattern-seeking. In fact, it is at the forefront of it. We have evolved to notice patterns and be
alerted when something isn’t right. One of these so-called patterns that has
fascinated mathematicians and individuals alike for centuries is the golden ratio.
Also known by the Greek letter Phi, it can be defined by taking a line and
breaking it into two separate pieces. If the ratio between these two new portions is
the same as the ratio between the original line and the now larger piece, the portions
are said to satisfy the golden ratio. The value that satisfies this equation is
roughly 1.618… It’s an irrational number, meaning we don’t know how to represent it using
the ratio of 2 whole numbers. In fact, we can’t even write the number fully, it’s unending.
It’s effectively the same as pi in that regard, and so we instead use the dots to represent
the non-terminating nature of this number. But what makes this ratio so golden? Well, I
should point out, the ‘golden’ part of the name is rather modern. The more original name that was
given to this ratio was ‘the divine proportion.’ And right away, you can see that the
term implies some sort of divinity - an extraordinary property that people must have
noticed when they were dealing with this number. To further explain this fascination, we should begin by dividing the fascination
into two parts - mathematics and aesthetics. The golden ratio and the fibonacci sequence
appear in nature every day, and arguably the strongest evidence of “goldenness” in the golden
ratio is in the floral arrangement of seeds. Take this for example: if you were a sunflower,
how much of a turn would you make before you make a new seed? If you don’t turn at all, well you
just continue making a straight line of seeds, kinda boring. If you make half a turn, or a
180 degree turn each time, well now you have a line of seeds, but in opposite directions. 120
degrees gives you 3 lines, 144 degrees gives you 5 lines. There should exist some angle, some
number of turns that, if properly executed, produces a pattern of seeds that is closely packed
together with no gaps between them. Something like this, this seems more natural to nature, right?
The number of “turns” needed to produce a spiral design like this is… well, the golden ratio.
One seed placed every 1.618 turns, or every 137.5 degrees. This is known as the golden
angle, and it is seen all throughout nature. The idea is to arrange seeds in a way that can
maximize the sunlight and rain that they receive, so that the genetic material can successfully be
passed on to the next generation. If you don’t do this efficiently, evolution won’t be very kind to
you. If you pack in the seeds too tightly, all the seeds won’t get the nutrients they need. If you
pack them too sparsely, you’re just wasting space. This happens with not only sunflower seeds, but in
plant leaves, tree branches, and more. In fact, it goes even deeper. If you were to count the number
of spiral arms in both directions, left and right, you’ll find that they aren’t equal; however,
they will both always be fibonacci numbers. The higher and higher the numbers go, the closer
and closer the ratio between them approaches Phi. The beautiful spirals that result are purely a
creation of nature. These spirals are consistent across different flower types, and even
the numbers of petals seem to be related to the golden ratio. The sunflower example is
particularly interesting because it actually ties the aesthetic element of the golden ratio to the
mathematics behind it; there is a reason why the golden ratio is used in flowers, and any other
value doesn’t seem to work very well. But what does that really tell us about the underlying,
and possibly grand design of the universe? We are all somewhat familiar with the
aesthetic elements of the divine proportion. The idea is that visually pleasing faces tend
to have proportions that are somewhat close to the golden ratio. The closer you are to
it, the more beautiful your face should be. This inclination of harmony extends to the rest
of the body as well, with many claiming that Da Vinci’s ‘Vitruvian Man’ demonstrates that the
human body owes its beauty to the golden ratio. The phenomenon also has its place in architecture,
with claims thrown around that structures as old as the pyramids of Egypt are in some way shape
or form inspired by the golden ratio... except for the fact that there’s hardly any evidence
that the Egyptians even knew this ratio existed. All of these beliefs contributed to the idea of
the golden ratio and made it what it is today. The mathematical beauty is undeniable, it’s proven;
however, it’s generally hard to verify any of these claims regarding it’s aesthetic purpose in
our lives. For one, it is not sufficient for a structure to just have the golden ratio somewhere.
It must be meaningfully incorporated and be the result of someone’s conscious or subconscious
choice. What does meaningful incorporation entail? Why would you choose the ratio of, say, this
pillar with that one, and not with another? Then there are questions of measurement. Oftentimes,
you see pictures thrown around the internet with a famous celebrity’s face with random lines drawn on
them that claim to satisfy the divine proportion. Long story short, they’re reaching pretty hard.
Without proper definitions of the facial features they are referring to, like what a cheek
bone is, where it starts and ends and so on, there is basically no scientifically
verifiable structure to what they’re claiming. Even if you do find that a certain
arrangement and placement of measurements do wind up representing the golden ratio
very well, it’s never quite exactly that. Phi, the golden ratio value, is irrational.
It’s unending. You can come close, but you can never truly achieve it
just by drawing some lines. Despite these claims, however, one could still
argue about the idea of facial beauty and the importance of the golden ratio in it. All
that matters is our perception of something. If the ratio of some facial feature
is close enough to the golden ratio, so much so that we’re unable to distinguish it
from the actual ratio itself, and we find it pleasing, wouldn’t you say that the aesthetic
element of the golden ratio is justified? To investigate this point, experiments have been
conducted relatively recently using a geometric object that has the golden ratio written
all over it. It’s the golden rectangle. It’s any rectangle that has sides a, b
that satisfies the relationship I just described at the start of the video, except
this time it’s a shape and not just lines. The experiments include showing participants many
different rectangles of different proportions, with one or more of them being golden rectangles. They were then asked to choose which rectangle
they felt represented the ratio best. If the hypothesis that things with the golden
ratio embedded in them are more attractive to us is indeed true, you would expect to see a
significant number of participants choosing the golden rectangles. But, turns out, the
evidence is not very great on this one. Instead of choosing the golden rectangle, participants seem to prefer rectangles
around that number, like 1.5 or 1.75. It’s close, but it already somewhat disproves the
idea of the golden ratio. If a rough value around the golden ratio is so good, then the golden ratio
itself isn’t so golden after all. But even if, say, the preferences do seem to be around that
golden ratio, there are theories as to why that might be. People who are more aware of the golden
ratio might be predisposed to having a preference toward it because of the place it holds in history
and in mathematics. Besides, if you spend a lot of time staring at a screen with aspect ratios
relatively close to that of golden ratio, something like a 16:9 monitor, you will
unintentionally find them more attractive. But now it would be a fair question to
ask, well what started this golden trend? There does seem to be a sincere focus on the
golden ratio in the modern design language, especially with UI/UX design. While no designer in
their right mind would swear by one single rule, the golden ratio has seemingly become a
mainstay in graphic design. For example, a lot of websites tend to have headings that
are 1.6 times the font size of the general text. Website layouts also incorporate golden
rectangles. Again, once something like the golden ratio has been generally accepted as something
divine, it’s very hard to distinguish whether it’s desirability stems simply from its familiarity
in our lives or from some deeper meaning. But it just so turns out that there is some
deeper meaning, deeper inside us. There’s a body of research that focuses instead on deeper
biology and how the golden ratio plays a crucial role in it. The cardiovascular system is of
keen interest in this regard. This is because it is one of those biological systems that has
very small differences across ethnic groups. So, if the golden ratio were to be found in such
a system, we could claim that it is indeed a more universal phenomenon, rather than existing
simply due to some ethnic variation or bias. The cardiovascular system is
deep in us, it runs our lives. There is no element of fabrication down
there. Evidence found here would much better support the idea that there is some
natural significance to the golden ratio, and in fact, there is. Your heart has many moving
parts that help keep your body functioning. One of them being the left ventricle, it’s what
helps pump oxygen throughout your body. Now, the average ratio of left ventricle horizontal and
vertical lengths of over 1,400 Swedish and Chinese subjects was found to be 1.618 – awfully close
to the golden ratio. Patients with left ventricle ratios that were further away from this value
were deemed to have poorer functioning hearts, and they were. And as we age, our hearts tend to
lose some of their power. Over time, your blood begins to pump slower and slower throughout your
body. Even the age-related cardiovascular decline was related to the golden ratio, with an early
diastolic velocity decline of 1.6 cm/s per decade. On the other side of the spectrum, the early
diastolic flow velocity of muscles in the heart of a growing fetus increases by 1.6 mm/s
per week. It’s remarkable how these and other metrics of cardiovascular health hover close to
this special value. But even in these examples, there’s still a bias in choice of units. For
example, why is this particular metric in mm/s and not inches/s? Such choices could alter these
quantities and take away the divinity in them. But why? If there isn’t so much evidence
behind the divinity of the golden ratio, why do people have this obsession over it? Well,
it’s the same reason why we once believed the earth was the center of the universe. We
want to believe these things. Of course, people have moved on from those particular
beliefs to more modern and sophisticated ones, like how the earth is flat and all. But
the core philosophy remains the same. We love patterns, and we love them
even more if they make us seem special. Why did the golden ratio only have to do with
our faces and not those of other animals? What about the body proportions of a snake?
It’s only human to only care about ourselves. This also highlights the age-old question
about whether math is invented or discovered. You see, the phenomenon of the golden ratio
existed long before we had any idea of it. The flowers arranged themselves without us
having a cognition of numbers or mathematics. Even when we tried forcing our hand on
certain elements of the golden ratio to suit our narratives of existence, the
golden ratio itself did not change. If the golden ratio were from a different
time and it were named differently, we would still be talking about the
same thing, only with a different name. It was what it was, and it will always be,
whether we have any cognition of it or not. There are things, properties of math and
science out there that exist currently, right in front of us perhaps, and
we have zero idea what they are or what they entail. We may define things along
the way, just as we have given the ratio a name, but the underlying mathematical
properties remain unchanged. In truth, the divine proportion owes its divinity
not to some all-pervading transcendental property, but rather to the idea of beauty and
perfection that we’ve imposed on it. This beauty we seek is quite possibly
not concentrated in one golden number, and even if it is, the golden
ratio is probably not the one. In trying to find perfection, we could nonetheless
keep defining Phi. But remember, it’s irrational. Phi does not end, and neither does our
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