International Mountain Day: Seeing Beauty in Roughness

The geological processes that shape mountain ranges are staggeringly complex, but understanding them can add a whole new dimension to our appreciation of the mountain environment.

In this article, Chair of the Alpine Club Library Council Philip Meredith and Librarian Beth Hodgett explore how a fresh perspective on geometry can help us think about mountains in a whole new way.

^{Sgurr Alasdair from Sgurr Dearg by Charles Pilkington}

People have been drawn to mountains for centuries, and a large part of their appeal lies in the breathtaking aesthetic qualities of mountain ranges. The Alpine Club holds a globally important collection of paintings and drawings dating back to the earliest days of mountaineering which document this obsession. Many prominent mountaineers have also been notable artists, and this is certainly the case for one of the most famous climbers of the ‘Golden Age’ of Alpinism, Edward Whymper (1840-1911).

While Whymper is most well known for his controversial first ascent of the Matterhorn in 1865, he also had a promising career as a wood engraver. Whymper came from a family of artists, his father Josiah (1813-1903) was a watercolour painter, and Whymper himself began his artistic apprenticeship at the age of fourteen. Numerous examples of Whymper’s wood engravings can be found in early issues of the Alpine Journal, as well as in his famous publication 'Scrambles Amongst the Alps'.

 

Further Reading: Whymper's London Diary, January-June 1858 | British History Online ----- Scrambles Amongst the Alps in the Years 1860-69 | Google Books

 

As climbers and alpinists we are used to examining rock faces and mountain ridges in detail, inspecting them to assess potential lines and the likelihood of protection. As Whymper himself put it, “None but blunderers fail to do so”. The process of preparing a wood engraving requires many of the same techniques of close observation, in order to understand and accurately represent the form and proportions of a mountain. It was this attention to detail that led Whymper to make an astute geological observation. On 25th June 1864 Whymper was part of a party that made the first ascent of the Barre des Écrins. In his account of the climb Whymper wrote:

"According to my custom I bagged a piece from off the highest rock (chlorite slate), and I found afterwards that it had a striking similarity to the final peak of the Ecrins. I have noticed the same thing on other occasions, and it is worthy of remark that not only do fragments of such rock as limestone often present the characteristic forms of the cliffs from which they have been broken, but that morsels of mica slate will represent, in a wonderful manner, the identical shape of the peaks of which they have formed a part. Why should it not be so, if the mountain’s mass is more or less homogeneous? The same causes which produce the small forms fashion the large ones; the same influences are at work; the same frost and rain give shape to the mass as well as to its parts."

 

Whymper’s interest in this more unusual kind of summit bagging is also evident in his account of his infamous first ascent of the Matterhorn, which is illustrated in Scrambles… by an engraving of a rock taken from the summit of the Matterhorn. Once again, the similarity between the fragment and the overall form of the peak is striking. We can see another example of this in the Alpine Club’s collection.

 

Compare this picture of the Matterhorn with the fragment of rock taken from near the Matterhorn’s summit and gifted to the Alpine Club as part of the celebrations commemorating the club’s 150th anniversary, and you can very clearly see Whymper’s point. In fact, recognising that tiny rock fragments and far larger rock structures can look identical in form has led to the universal practice of including a scale-bar in geological photographs. Without the scale it is essentially not possible to tell the size of the object, as demonstrated in the pair of photographs below.

 

But is it possible to prove that the piece of rock from the Matterhorn summit doesn’t just look qualitatively similar to the whole mountain but is actually quantitatively identical in structure?

The mathematical theory to describe such structures was developed by the Polish-French-American mathematician, Benoit Mandelbrot. In 1975 Mandelbrot coined the term 'fractal geometry'; drawing on the latin root of the word for ‘fractional’ to describe shapes that maintain their ‘roughness’ or complexity regardless of the level of detail they are examined at. A classic example of this is the Romanesco Broccoli.

If you look closely at the photograph, you can see that each segment of the broccoli is made up of a number of smaller segments whose shape mimics that of the larger structure. No matter how closely you zoom in, the structure of each segment appears the same. This similarity of structure across different scales is called self-similarity.

One way to prove that a small rock fragment and a mountain are mathematically self-similar rather than merely looking alike is to compare how they both take up space. However this is easier said than done. While some shapes are relatively straightforward to measure, others are much more challenging. We are all used to thinking in one, two and three dimensions; that is dimensions of whole integers. For example we know that a cube fills a three dimensional volume, but how do we measure the volume of something more complex like a tree or an alpine ridge, which only partially fills its surrounding volume?

^{The tree will not perfectly fill the surrounding space, but we can measure what fraction of the space it fills.} 

Mandelbrot’s great insight was the theory of fractal geometry. Within this concept, the tree is less than three-dimensional but more than two-dimensional; it has a fractal (or fractional) dimension between 2 and 3. This occurs because natural forms like rock fragments or mountains are not made up of smooth planes, but of complex, rough surfaces that are much harder to measure.

Mandelbrot demonstrated this in a 1967 article in which he posed the question: ‘How Long is the Coastline of Britain?’ The problem with solving such questions, Mandelbrot argued, is that when trying to measure a ‘rough’ shape like a coastline, you get a different answer depending on the unit of measurement that you use. Much like the example of the Romanesco Broccoli, the more you zoom in to look at the coastline, the more complex the shape becomes, with ever-decreasing wrinkles in the rock continuously adding to the overall length to the extent that the problem becomes intractable and an accurate measurement simply cannot be made.

One of the main insights of Mandelbrot’s theories of fractal geometry was his proposal of mathematical ways of measuring this roughness. Using these methods, we can determine the fractal geometry of a mountain ridge and of a piece of rock that comes from it and see that commonly they are quantitatively the same.

Over 100 years after Whymper first observed the relationship between the fragments of rock he collected and the peaks he climbed, Mandelbrot’s theories enable us to move from Whymper’s qualitative observation about the aesthetic similarity between rock samples and peaks, to being able to describe and quantify the relationship between rock samples. In doing so, it is possible to show that Whymper was right, small rock fragments really do fracture in ways that are self similar to the shape of the peaks that they come from! 

Being able to measure variations in roughness opens up a whole new realm of possibilities for understanding the formation of mountains. For example, being able to measure and quantify the microstructural geometry of a rock at the crystallographic scale can help us to understand the mechanics of where, when and how it might fracture. Understanding details like this can help us work backwards from the form that mountains take in the present to reconstruct the processes that formed the mountain many millions of years ago. The mathematics of fractal geometry can also explain the distribution of crevasses in a glacier, the arrangement of mountain ridges, or even the distribution of boulders on a scree slope.

The beauty of Mandlebrot’s work is that it helps us understand and describe the patterns that underlie the seeming chaos of the natural world, giving us new ways to appreciate not only the complexity of the natural world, but the skilled perception of artists like Whymper who study it.

So next time you're on the hill and you notice a loose piece of rock, take a longer look. There's a whole mountain hidden in its geometry.