Space Fractals
type | RESEARCH // BIOPHILIC DESIGN status | ONGOING year | 2021
Fractal Craters
The phrase "organic form" is often attributed to cities, architecture or objects whose growth and creation seem to lack any underlying structure. Rivers, mountains and trees are naturally occuring entities which seem to grow and take shape through randomness, chance, and sheer chaos. How do you begin to anticipate the growth of a new branch, or map the intricate details of a complex coastline? All things, even chaotic entities, must have some structure that defines them. Simply stating that mountain ranges and coastlines are a series of random zig-zags is not good enough.
At least, it wasn't for mathematician Benoit Mandelbrot, who spent the latter part of his career focusing on deriving a purely mathematical breakdown of the structures we find in nature. He coined the term "fractal", which is now commonly used both in theoretical mathematics and natural sciences. His theory stated that most natural forms, ones that we automatically deem as "organic" or "random", are governed by certain mathematical 'codes' - self-similarity, repetition, and scale. It states that when you look at a whole, the appearance of regularity is a fault of our perception (for example, a relatively straight coastline). When we look close enough, we see the same form broken down into smaller forms, that also appear regular (the bays and peninsula forms on a coast, for example). And upon looking at each of these forms up close, we see them broken down into smaller regular forms (continuing with the coastline example: coves, bights, outcrops and inlets along each peninsula or bay). And so, each form increases in complexity, and therefore, accuracy the closer you look.
The degree of complexity of a particular fractal form is known as it's fractal dimension, or level. Theoretically, a fractal could have an infinite number of levels - however, as this page a display of fractal forms at a macro, geographical scale, I've limited each fractal to level 3 of a particular fractal.
The coastline of Maharashtra, where the fractal is at a lower resolution.
The coastline of Brazil, where the fractals are more apparent.
Norway's coastline (for consideration, a series of about 1200 fractals at level 1)
The coastline of Norway (as seen in the image above), looks terribly complex - and chaotic. So in an effort to reign in the "chaos", we begin breaking the coast into individual units - a geological feature known as a Fjord. Fjords are valleys carved by glaciers as they flow towards the sea. As the ice melts, the entire valley is inundated by water - freshwater from seasonal ice melt and brackish water from the sea. The main fjord can be considered a "level 1" fractal. Further inland, the fjord is fed by a series of smaller glacier lakes, or cirques (these are also known as tarns). These would then be the "level 2" fractals. Above these lakes, the melting of snow and ice runs streams and rivulets into rivers, which flow through ever-deepening ravines to the masses of water - "level 3" of Norway's coastal fractal. The whole coast contains about 1200 such fjords - each with their own glacier lakes, each glacier lake with its own rivers and streams.
Thus, at a metaphysical level, every fjord is identical: Rivers and Streams (level 3) flow into glacier lakes (level 2) flow into the main fjord (level 1) which flows into the sea. The scale of the water bodies increases, the volume of water stored in each kind of water body increases, the dimensions of the valleys that hold the water bodies increase - however, they are all water bodies flowing through a valley. Through repetition, self-similarity and scale, vast geographies can be predicted and mapped with accuracy.
This is the underlying structure of Norway's chaotic coastline.
Norway's coastline (for consideration, a series of about 1200 fractals at level 1)
Deltas and River Valleys on Mars
Easily spotted in satellite imagery, any large river system is the poster-child for large-scale fractals spread over part of a continent. While most gravitate towards the Nile or the Ganges-Brahmaputra delta, both of which are breathtaking formations, I've chosen the Chenab River system.
The Chenab River, a major tributary of the Indus, flows from the Zanskar Range in the east, across Jammu and Kashmir and into Pakistan. It merges with the Indus which then drains into the Arabian Sea. As it flows down from the Zanskar and Himalaya range, it is fed by tributaries from the Siwalik, Kalidhar and Trikura ranges (the entire river system is far larger, fed by tributaries originating in the Hindu-kush range, along with larger rivers such as Sutlej, Ravi and Tawi from the Himalayas).
The whole length of the Chenab river, with the extended tributaries spanning northern India and Pakistan would be level 1 of the fractal. Towards the north, in the mountain ranges, the mid-sized tributaries merging into the Chenab along valleys would be level 2 of the fractal. And finally, the rain-fed streams and rivulets that flow to the tributaries would be level 3 of the fractal.
Here, taking into consideration the fjords in the previous section, you can see that flowing water bodies are a major geological sculptor that work through a fractal structure. They grow not only in dimension, but in volume - adhering to the fractal canons of self-similarity, repetition, and scale.
The Chenab River Fractal, at level 1.
The Chenab River Fractal, at level 2.
The Chenab River Fractal, at level 3.
Lava Flows
Rivers and fjords form clear fractals: apparent in form from satellite imagery. However, there are other parts of the planet where geology breaks down into fractals.
The Galapagos Islands is a cluster of isolated volcanic islands sitting atop a very active magma vent. The plate on which these islands rests moves eastwards, slipping under the neighbouring plate, creating a unique geology that is found nowhere else in the world. The volcanoes on the western islands are still active, with the volcano island of Fernandino seeing a major eruption as recently as 2005. Satellite imagery of the island shows portions of the island blackened by major lava flows along the slopes towards the sea - peppered with a few smaller lava flows. Since the whole island is an active volcano, I have considered the whole volcano as a fractal system at level 1. Closer to the lip of the volcano, large lava flows are apparent as streams of black along the landscape - these would be level 2 of the fractal. And finally, in a particular flow: the lava branches out along water-carved ravines (level 3), but another feature can be observed. Lava flows often burst through a partially cooled and solidified flow, building up a new layer of molten rock, and create a flow-within-a-flow, another fractal form.
The hypothesis that flowing water bodies are the only sculptors of fractal geographies is therefore false. It seems as though fractal geological forms are largely perpetrated by fluid motion, be it water or otherwise. Ravines and valleys carved by water seem to have a sharper definition to them, whereas the lava flows seem to be "blurred" or smoother. This may be owing to the viscosity and thus velocity of flow of the material: lava is more viscous compared to water, and it slows down as it cools and solidifies, blunting the edges of the fractal formation. This comparison can be done with the structure of water systems and ice flows or landslides as well.
Fractal lava flows on Isla Fernandina, Galapagos; fractal at level 1.
Fractal lava flows on Isla Fernandina, Galapagos; fractal at level 2.
Fractal lava flows on Isla Fernandina, Galapagos; fractal at level 3.
Sand Dunes
Sand dunes do not seem to be a fractal structure. At least, not as apparently as river systems or fjords. Their structure is very different from the pattern a fractal should create. And yet, dunes are fractals.
To understand this, we must look at how sand dunes are formed.
Sand particles are picked up by gusts of wind and blown across vast, flat, featureless land. Eventually, acted upon by gravity and friction, they begin to lose energy and fall towards the ground. Here, they may encounter obstacles - rocks or undulations in the land. Several thousands of particles encounter this undulation, and form a single, small dune (these may be about a meter in length and a few centimeters high). These mini-dunes are level 4 of the fractal. Over time, the singular dune grows in size, till it breaks down into a series of two or more dunes, pushed further into formation by eddies or gusts of wind. These dunes form a progression of vertical obstacles that trap more airborne particles of sand, gradually building up their size. A series of such dunes eventually coalesce into a single large dune - ranging from tens of meters in length to hundreds of meters in length. These are the sand dunes that form level 3 of the fractal: shown in the images below. These larger sand dunes form clusters of dunes - tens, maybe hundreds of dunes - with one massive dune along the edge of the cluster, keeping all the cluster intact. This dune can be up to ten kilometers long, spanning two or more clusters. These clusters of dunes are level 2 of the dune fractal. And finally, viewed on a large-enough scale, the whole landscape of the desert is pockmarked by hundreds of dune clusters - level 1, as seen in the first image below.
Dune form and size may vary depending on the structure of the sand particles that form them. The essence of their fractal nature has more to do with the generation of the largest dunes by the amalgamation of mid-sized and smaller dunes. Although they may not exhibit any notable fractal geometry, they can well be considered fractal forms.
Sand dunes in the Sahara Desert, Libya; fractal at level 1.
Sand dunes in the Sahara Desert, Libya; fractal at level 2.
We are in awe of the scale at which fractals operate. Entire mountain ranges, coastlines and plains sculpted into these organic forms. All of them following the same code: self-similar forms, scaling up (or down) of volumes and sizes, and repetition of forms.
Simply focusing on the beauty of the form and attempting to mimic the creation as a whole in our design processes is an exercise in futility. To use fractals, we need to understand the physical reason that fractal forms are so pervasive in nature and what their advantages are. For one thing, fractals use repetition: the same form or typology of object is repeated multiple times. Why design a new form for conducting water across large portions of land, when you can repeat the form of streams or brooks that conduct water over small portions of land? Furthermore, why change the form of something that is good at conducting water, when you can scale up the streams or brooks into rivulets, and eventually rivers? And lastly, why change the typology of the flow of water from rills to brooks to gullies, when this can repeated with similarity in the flow of water from streams to rivulets to rivers?
Another aspect of the macro-fractals we can understand and inculcate into our design methodologies is the expansion of the fractal over time. The fact that all fractal forms in nature are generated and do not simply pop into existence in an instant is a canon we can use to design our every-expanding need for built space over time.
After all, every river is a very slow, very gradual space-filling curve.
You can see the next page in this series, "Man-Made Fractals", or look at an example of how fractals were applied to a cluster-building housing project under "Generative Microhousing".