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Anthropo-Fractals

type | RESEARCH // BIOPHILIC DESIGN    status | ONGOING     year | 2020

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Cities and Planning

While fractals are generally found in geo- or bio-morphologies, anthropogenic design often mimics or creates fractal patterns. Most commonly, fractal patterns can be found in the way we plan and design our cities - fractally branching service and transportation networks are often the most efficient ways to provide utilities to individual houses in a vast city. However, branching service systems are not the only way fractal forms emerge in city planning. 

In cases where cities grow organically over centuries, there is often no discernible pattern when the growth of the city is looked at as a whole entity. However, gradual 

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The coastline of Maharashtra, where the fractal is at a lower resolution.

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The coastline of Brazil, where the fractals are more apparent.

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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. 

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Norway's Coastline fractal, at level 1: The main fjord, flowing into the sea.

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Norway's Coastline fractal, at level 2: The cirque-formed valleys, smaller water-bodies feeding into the main fjord.

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Norway's Coastline fractal, at level 3: snow- and ice-melt streams and rivers flood the smaller glacier lakes, which in turn over-flow into the main glacier.

Farmland and Deforestation Patterns

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

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The Chenab River Fractal, at level 1.

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The Chenab River Fractal, at level 2.

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The Chenab River Fractal, at level 3.

Fractal Technology

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. 

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Fractal lava flows on Isla Fernandina, Galapagos; fractal at level 1.

Architectural Fractals

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.

Fractally branching Timber columns in a

Sand dunes in the Sahara Desert, Libya; fractal at level 1.

Tropical islands as geological form are shaped almost entirely by fluid effects, the wind, rain and the sea sculpting away at the land over several millennia. Islands, therefore, are awash with fractal forms. 

Here, we look at the estuary formation an island of The Bahamas: Andros. Water channels have eroded entire tracts of the island away, leaving a series of straits splitting the land form into about 4 major parts. At level 1 of the fractal, each of these straits appear to be extremely chaotic, rendering the island into shapes with a "random" form. Closer to the mouth of the straits, we see a familiar fractal structure emerge, one that is common in river systems and other coastlines (level 2 of the estuarine fractal). This is where we see freshwater rivers merge into the brackish straits on the coast of the island that is at a lower elevation. The rivers often flow parallel to the straits, in the sand flats. Freshwater streams diverge from these rivers and flood brackish pools; brackish water from the pools may form rivulets that flow into the freshwater river. These rivulets and streams interchanging freshwater and saltwater in the estuary are level 3 of the fractal. 

When compared to rivers and other coastal fractals, there is no novelty to this fractal. The fractal-water-body/land ratio on this island, however, is a great example of how nearly every geological form is subject to erosion into a fractal at some level - and how our whole planet is shaped by these fractals. 

Computational Columns by Michael Hansmey

Sand dunes in the Sahara Desert, Libya; fractal at level 2.

You can see the previous page in this series, "Space Fractals", or look at an example of how fractals were applied to a cluster-building housing project under "Generative Microhousing".

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