After the post on Mount Fuji, there was a discussion on which was the most beautiful volcano. Beauty can be a rather vague term, and people can disagree with each other as to the best volcanic beauty. Who is the judge in the Volcano World beauty contest? Or can we solve the question here, and define beauty in a way that leaves one clear winner? Our definitive i-cone-oclast?
The volcanoes that were proposed in the previous discussion have things in common. Like Mount Fuji, they tend to be fairly steep cones, isolated with a clear contrast to the rest of the landscape, and symmetric. Does this give a way to define what a perfect volcano should look like?
Beauty is in the eye of the beholder, as the expression says. This of course is very much true. A new-born baby is the most beautiful thing on earth – if you are the parent. But a new baby does not live up to any standard of beauty. In fact, dare I say it, they can at first be quite ugly. Patience: they quickly beautify over the first months of life, and keep improving until puberty hits and life begins for real. Old paintings show how our perception of beauty has changed. What was a beautiful face for a renaissance painter would not pass muster in the age of Hollywood. Both use standards, but the standards have changed. Some things remain: symmetry and skin suppleness survive while the ideal shape changes. Leonardo da Vinci described beauty as a combination of symmetry and proportion. Nowadays, plastic surgery uses a set of linear and angular parameters to beautify a face. But a universal description of the shape of a perfect face does not exist: it varies with time, race and culture.
So it is with a landscape. A perfect Japanese landscape differs from a German one (the former has space, the latter probably has a forest in it somewhere). But these are variations in a theme. Both themes strive for proportion, a degree of symmetry, and for harmony. There should also be an element of surprise. Like a perfect and unblemished face looks artificial and made up, a perfect landscape looks boring. There is nothing happening. A landscape can be shown from many different points of view. In some of the images of Hokusai’s 36 Views of Fuji, the mountain itself is the focus, always showing a different but domineering aspect. In others the focus is elsewhere, with Fuji in the background to give meaning to a human land. The Big Wave falls in that category. Just a landscape is not enough. It needs context.
Beauty rubs off. Studies have shown that people have more favourable impressions of people who have attractive partners – even if those persons themselves do not satisfy typical definitions of beauty. A beautiful partner is more than a trophy: it is an asset which adds value. (Of course, for the partner it can be a curse to be seen as a trophy and asset, rather than being admired for talent!) This transfer of beauty applies to both people and volcanoes – any landscape looks better with a stratovolcano in it, just like that volcano looks better with its surroundings.
There are other aspects of beauty. Colour is one: lava is ugly in the light of day (really!) but it becomes a wonder at night when the lava turns into a light show worthy of Christmas. Music is another: it can leave a lasting impression of beauty and harmony. But beauty can be bland and appreciation does not need beauty. Much of the music each of us enjoys cannot be called ‘beautiful’. Paintings can have great visual impact without beauty (Picasso’s Guernica comes to mind). But those pieces are much more affected by personal preferences. Different people enjoy very different music, but we can agree that the waving melody of Greensleeves and the lightfootedness of Mozart’s Eine Kleine Nachtmusik have beauty.
The previous post mentioned that at close view, Fuji is far from perfect. But in fact, that can add to the beauty. It adds character to symmetry – as long as the deviations do not dominate. Pure beauty can seem lifeless. An average human face has beauty but has lost all character, perhaps because it lacks those minor, surprising asymmetries.
The golden ratio
Skin tone and texture is relevant in people’s perception of beauty. So are proportions. The volcanoes we like are all fairly steep mountains. Does that mean anything?
Originally this was called the divine ratio. It is a weird number that was discovered already by the ancient Greeks. Take a line, and divide it into two unequal lengths. If the ratio of the two lengths is the same as the ratio of the larger length to the total, that is the golden ratio. It is commonly denoted with the greek letter φ.
The definition above in mathematics can be written as follows:
If we now make the choice to take b=1, then automatically a=φ. The equation now becomes
Rearranging this gives the equation
The ancient Greeks were fascinated by this equation. It has a funny solution which they couldn’t quite figure out. It could not be written as a ratio between two whole numbers, say n/m. They were even able to proof this. No such number had been found before. Nowadays we call this an irrational number. To the Greeks this was as strange as imaginary numbers are to us. We can phrase the solution as an equation, but not as a precise number. The solution is in fact
If you had written the original ratio the wrong way around, with +φ rather than -φ, you would have calculated 1/φ rather than φ. It turns out, that gives exactly the same number with the first ‘1’ replaced by ‘0’: it gives 0.61803.. That was funny. Later it was found that the golden ratio is related to the Fibonacci numbers. If you don’t quite remember what this is, don’t worry – most people just pretend that they know. The Fibonacci numbers start with 0 and 1 (now you already know more than most), and then make more numbers by adding the previous two together. So the next number in the sequence is 1 (from 0+1), the following one is 2 (1+1) and now it gets harder: 0,1,1,2,3,5,8,13,21, .. The numbers quickly become larger.
(Actually, you could argue that starting with 0 is wrong. 0 is not really a number in mathematics. It doesn’t exist. For instance, you would instantly see the difference between 10 elephants and 10 apples. But how about 0 elephants and 0 apples? You know you can add apples together, and you could add elephants together if you had enough of them. But adding apples and elephants makes no sense: 2 apples plus 4 apples makes 6 apples, but 2 apples plus 4 elephants just makes a mess. Could you add 2 apples plus 0 elephants? Mathematics has no real answer to this, except to say that zero doesn’t really exist. So there is an other series which is like the Fibonacci numbers but begins with 1,2 rather than 0,1. It is called the Lucas sequence, designed by a mathematician who did not believe in zero.)
Back to the original topic. If you take the ratio of the last two numbers in the Fibonacci sequence above, 21 and 13, you can calculate the ratio. (Just pop it into the google search bar. You might even get some refreshing adverts later, when google figures out your new-found love of mathematics.) That ratio is 1.615, very close to the golden ratio. In fact when we continue the sequence, the next two numbers (34 and 55) have a ratio even closer to 1.618. The same is true for the Lucas series. If you continue them long enough, the ratio between your last two numbers becomes equal to the golden ratio. The 16th Fibonacci number is 2207: we can now immediately say that the next one must be 3571, just by multiplying 2207 by the golden ratio. But we can never get to the exact value of the golden ratio in this way. That is because it is an irrational number which cannot be written as a fraction of two whole numbers. But you can get very close.
(If you want to impress your friends: the higher Lucas numbers are almost exactly the golden ratio to that power. The golden ratio to the power 20, for instance, will give you the 20th Lucas number.) (It is 1.6180339920, or 15127, by the way.)
So far this is of interest to mathematicians, not volcanoholics. You eyes may be glazing over and you are already hoping that the next post will be a bit more explosive, or at least involve more latin (lava, from the latin lavare) and less greek. I would have agreed with you. Except, this number began popping up in strange and unexpected places. It kind of erupted everywhere.
It describes the pentagon (the shape, not the US military) and the dodecahedron (the shape, not the dead bird). It came up in nature: the distribution of leaves on a plant follow the golden ratio.
Where mathematics and nature lead, people follow. There has been a lot of discussion on whether the golden ratio appears in our creations. People have found it in old buildings, in paintings and in writing. But it is hard to prove these things. We don’t have written evidence from the architect of the pyramids that he (presumably) used the golden ratio. (This is claimed by some but is more than a little unlikely as the ratio was not known at that time, and wasn’t considered as important outside of mathematics until the late middle ages.) Some of the lines in cathedrals have the golden ratio. Was that by design? Or just because it is easy to scale things up by adding lines together, a bit like making a Fibonacci sequence out of planks? If a building design includes lines drawn as a pentagon, the golden ratio automatically appears even though it was never included in the design.
The first claims that the golden ratio appears in art date to the time of Leonardo da Vinci. In fact, these claims were applied to his work while the master was still working. We don’t have a statement from the artist himself, so this has remained controversial. It has appeared in more recent, famously in the geometrical designs of Mondrian and in the Sacrament of the Last Supper of Salvador Dali.
Technology may use the golden ratio. The computer screen I am working on has a wide screen with dimensions 16:10. This is, within the accuracy, the golden ratio. But it is not deliberate design. It is just a size that is comfortable to work on.
And that is the bottom line about the golden ratio. Mathematically, it is a marvel. In nature, it is sometimes the optimum design, eventually favoured by natural selection. In the human world, it is a comfortable, pleasing ratio. The golden ratio became a measure of beauty. Is that because it happens to be present in nature? Or because of our own faces? The ratio of the length to the width of a face is approximately the golden ratio. It is the first shape a baby sees – and that is worth a smile. You can find calculators on-line which will take a photo of a face, calculate various parameters such as the facial index defined above, distances between eyes, etc, all compared to the golden ratio, and look at the symmetry of the face. It will now give you a score which indicates how beautiful the face is. It is of course artificial: these calculators try to make believe that one size fits all. Faces vary and people (and races) have different shapes. Face recognition software measures many of the same parameters, but without assigning a beauty index. (I think!)
So the golden ratio (or at least numbers not too far from it) are seen as an aesthetic indicator. It is something pleasing to look at. But how does this apply to volcanoes? Volcanoes don’t look like faces – at least being told you have a face like a volcano is not seen as as compliment.
Fuji has the reputation of being a beautiful, perhaps the most beautiful volcano on Earth. Why would that be?
Here is a classic view of Fuji. In Japanese tradition, it is framed by cherry blossom with one corner left open, creating space on one side. The lake and snow create a layered composition with the white bringing the summit into sharp focus. Sadly the reflection is disturbed by the waves on the lake, otherwise this would be the perfect Fuji composition.
Let’s look at this white summit. I have drawn a triangle which fits the upper slopes and show where the peak would have been without the central crater. Each half of the triangle (left and right) have horizontal and vertical lengths which have a ratio of about 1.6 – the golden ratio. In fact, if you were to combine the two halves to make one rectangle, that rectangle would be a so-called golden rectangle – with about the same aspect ratio as my computer screen.
(The triangles are not golden triangles. A golden triangle is defined differently, and in my opinion wrongly.)
So the summit of Mount Fuji can be approximated by the golden ratio. Fuji is a golden volcano! But only the summit peak: the lower, shallower slopes become part of the landscape instead.
In case you wondered, the angle of the slope in the triangle is about 31 degrees. Steep, but not mountaineering level. 30 degrees is typical for the upper slopes of a stratovolcano. Cinder cones are steeper, and shield volcanoes are shallower. So now it is official: stratovolcanoes are the most beautiful. They will score highest on the beauty algorithms. Hollywood, look no further.
But there are many stratovolcanoes in the world. Is there anything that makes Fuji stand out, and grades it as a 10 rather than a 9? The image shows an obvious one: the white summit forms a sharp contrast with the dark lower slopes, and delineates the ‘golden area’ very nicely. Of course, this is only during the right season, but picking the right time is one sign of a brilliant photographer.
The other thing about Fuji is that the foreground perfectly frames the mountain. The next picture shows the same triangles, but now I have made them larger – in fact the dashed lines are 1.6 times longer (where have I seen this number before!) than the solid lines. Amazingly, this traces the edge of the lake, and the cherry blossom approximately borders this new triangle. This is again a sign of brilliant photography: the cherry blossoms mirror the shape of the golden mountain.
So the beauty of Mount Fuji is a combination of the shape, the contrast and the environment. Maybe it is the perfect volcano, at least if you are at the right place at the right time with the right camera. The 36 Views of Mount Fuji of Hokusai show how this mountain can dominate, in a brooding way, even when it is just a distant background as it is in the famous Big Wave where the mountain and the people seem swamped by the storm at sea. But the mountain, small and distant, still dominates the scene. Older versions of this work, without the mountain, lack that focus.
So what makes the beauty of Mount Fuji? It it a combination of the shape, the symmetry, the youthful regularity and the surroundings. It is not perfect: the shape in particular has distortions and the snow is only seasonal. But that just adds character to the beauty. And beauty plus character makes impact.
Is it the most beautiful volcano in the world? Below are some other candidates for this which have been suggested in the comments on the previous post. I will invite people to make their case in support of their favourite Volcano World beauty contest entry! Write your nomination in the comments and I will add the text to the post. Glory awaits the winner.
The following have been added from the comments
“I guess my aesthetics run in a different direction, and there’s the question of what counts as a volcano for this purpose, but I’d take a gothic plug like Shiprock or a dramatically eroded summit like Mt. Thielsen over a symmetrical cone any day.” (Jackson Frishman)
“Kronotsky definitely deserves a shoutout here. Highly symmetrical, quite pointy, yet it doesn’t have historical eruptions that people are aware of. It’s a bit more eroded than others here as a result, yet still maintains the perfect conical profile in a beautiful region.” (cbus20122)
“I’ll always have soft spot for Rainier as well. Anybody who has been to the Seattle / Tacoma region on a clear day knows how impressive and almost otherworldly Rainier is in the skyline. It’s just as beautiful up close too despite not being a perfect conical volcano.” (cbus20122)
“For the whole package, the view from Panajachel to Lake Atitlán with its 3 volcanoes is my personal favorite” (Virtual)
“Since others have put forth ancient volcanic edifices, I nominate Diamond Head on O’ahu, HI, USA as a beautiful & welcome sight for sailors returning to Pearl Harbor from the Western Pacific.” (Mike Steussy)
Over to you!
Albert, September 2022, at the end of the second Elizabethan age. The first Elizabethan age ended soon after the devastating Huaynaputina. The second one lasted until just after Hunga Tonga, the loudest volcano since Krakatau. They were worthy endings.