Nature contains a multitude of beautifully complex patterns, which often leave us wondering how they are produced. Fascinatingly, many of nature’s most iconic patterns are created by Reaction-Diffusion systems.

The order that creates nature’s complexity is often shrouded in mystery and has puzzled thinkers for millennia. Yet, from animal camouflage patterning, to their territorial areas or the self-organisation of ecosystems, Reaction-Diffusion models elegantly recreate some of nature’s most distinctive forms.

Reaction-Diffusion therefore offers a window to explore how simple processes, unfolding from clearly defined rules, can create dynamic and intricate patterns, like those found in biology and the natural world.

Nature contains a multitude of beautifully complex patterns, which often leave us wondering how they are produced. Fascinatingly, many of nature’s most iconic patterns are created by Reaction-Diffusion systems.

The order that creates nature’s complexity is often shrouded in mystery and has puzzled thinkers for millennia. Yet, from animal camouflage patterning, to their territorial areas or the self-organisation of ecosystems, Reaction-Diffusion models elegantly recreate some of nature’s most distinctive forms.

Reaction-Diffusion therefore offers a window to explore how simple processes, unfolding from clearly defined rules, can create dynamic and intricate patterns, like those found in biology and the natural world.

Reaction-Diffusion, put simply, simulates how ‘chemicals’ or ‘agents’ react and diffuse out in space. The colours within the animations illustrate the varying concentrations of each agent at any given point.

One prominent Reaction-Diffusion system is the Gray-Scott Model, which works by digitally simulating how just 2 chemicals diffuse and interact.

It is made from a few simple rules that govern how the digital agents spread out and interact. The basic rules of the system are:

· There are 2 chemicals: A and B

· When you get 2xA and 1xB at a certain point, they fuse to become 3xB

· Chemical A and B diffuse outwards at specific rates

· Chemical A is continually generated within the system at a specific rate

· Chemical B decays and is thus removed from the system at a given rate

These rules are then calculated across time to create the evolution of the system.

Reaction-Diffusion creates a wide diversity of patterns and dynamically moving structures, as is illustrated by the videos within this post. The importance of initial conditions and the calibration of the settings mean only minor tweaks can drastically alter the system. This is seen in the diversity of forms within the videos where only the parameters and starting conditions have been slightly adjusted.

Reaction-Diffusion patterns, often referred to as ‘Turing patterns’ and are found all across the natural world. Turing patterns takes their name from Alan Turing, one of the founding thinkers of mathematical biology, whose work examined the rules governing morphogenesis in nature.

The underlying processes which generate the patterns we find in nature will of course be more complex than the perfect abstraction of any Reaction-Diffusion computer model. Yet formal models offer valuable insights into the dynamics that drive pattern formation.

Beyond simply replicating the visual patterns, Reaction-Diffusion models help to make clear the underlying dynamic forces that create pattern. The models help show how tensions between ‘activating’ and ‘inhibiting’ forces conspire to produce form. So spots can get larger to become stripes, which can then join to become mazes, all depending upon the relative pressures of activation and inhibition.

Equally, yet in a vastly different domain, predator-prey relationships can also resemble Reaction-diffusion systems. Animal population sizes and their territories also create Turing patterns. This is due to the activating and inhibiting tensions between species, with predation, reproduction and territorial expansion dynamically interacting. Hence Reaction-Diffusion offers a mathematical lens to better understand how symbiotic relationships function within nature.

All of this points to the ways in which complexity can arise from simplicity. From a relatively simple system, with just a few interacting elements, we can get intricate, dynamic and complex structures.

Working backwards to decipher the processes underpinning the complexity of nature is exceptionally difficult. However, Reaction-Diffusion reveals one of the ways complexity can be produced inside of a simplified closed system. As without the messy realities of nature, we can explore how the underlying dynamics of pattern formation operate inside of a system that operates in well-defined ways.

Reaction-Diffusion also shows the ways evolution can play out in a concrete manner. It allows us to consider how small changes to systems could mutate and shift across generations, allowing major shifts in form to occur gradually. Where the patterning that survives would be best suited to the environment that it exists within.

**In addition, the forms of Reaction-Diffusion have beauty, depth, and intrigue that speak to the complexity of the natural world we inhabit.**

The process of Reaction-Diffusion as outlined above featured as a recent t-shirt design. This design was made in collaboration with Simon Alexander-Adams also known as @polyhop on Instagram

The video below shows the exact graphic used on our stonewashed burgundy t-shirt

The Penrose tiling is an iconic geometric form, with not only an intriguing visual elegance but also a fascinating conceptual underpinning.

From just two simple quadrilateral tiles we are able to endlessly tile a 2-dimensional plane without the pattern ever repeating. Penrose tiles are built upon a captivating aperiodic structure, seamlessly blending perfect order with endless irregularity.

The Penrose tiling is an iconic geometric form, with not only an intriguing visual elegance but also a fascinating conceptual underpinning.

From just two simple quadrilateral tiles we are able to endlessly tile a 2-dimensional plane without the pattern ever repeating. Penrose tiles are built upon a captivating aperiodic structure, seamlessly blending perfect order with endless irregularity.

The Penrose tiling gets its name from its creator, Roger Penrose. Penrose discovered the tiling structure during the 1970s, working through various iterations, clarifying, and simplifying the concept down along the way.

Penrose initially worked by subdividing a regular pentagon into 6 smaller ones, with 5 triangular gaps. From here he set about configuring tiling patterns to ‘solve the jigsaw puzzle’ and recombine this basic shape into tile configuration.

Initially, he created the ‘P1’ tile set composed of 4 tiles of different shapes (shown below). Whilst it can tile the plane, much like the later versions, the tiles required to do so are much more complicated and lack the mathematical elegance that is so often strived for. While the ‘P1’ set showed that the tiling patterns were possible, the ‘P2’ version (known as the kite and dart), elegantly simplified the pattern down to just two quadrilaterals.

Perhaps the most iconic form is the ‘P3’ Tiling; just two unremarkable 4-sided tiles that within them hold remarkable properties. The tiles endlessly combine to perfectly tile the plane without any awkward overlays or gaps, coming close to repeating but never doing so.

The Penrose tilling’s beauty arguably comes from the irregularity and complexity of the 5-fold symmetry upon which it is built. While triangles (3-fold), squares (4-fold), and hexagons (6-fold) each tesselate in straightforward, periodic ways (think of square grids and isometric triangular lattices), no clear way appeared possible for 5-fold pentagonal symmetries. It always appeared to leave a remainder.

5-fold symmetries have long been held with a degree mysticism and wonder, from Plato and the Dodecahedron. In many ways this reverence makes sense, with the Dodecahedron the last frontier of geometric rationalism, perfectly balancing 12 regular pentagons into 1 whole object. With the irrational beyond the limits of human comprehension at the time.

There are many fascinating characteristics of the Penrose tiles, one being how it relates to the Golden ratio, phi Φ. The relative side lengths of the Penrose tiles conform to the Golden ratio, balanced to these proportions. The Golden ratio is fascinating ratio found across the natural world.

Additionally, the Penrose tiles have the fractal property of self-similarity with inflation and deflation, meaning that each tile can be split into smaller versions of itself. This is yet another facinating way in which the tiles embody big mathematical concepts.

The Penrose tiles are a beautiful example of a pure mathematical discovery that is only later observed within nature, alluding to geometric relationships that underpin the natural world. While there is not a perfect example of the Penrose tiles in nature, quasicrystals share the same 5-fold geometric structure.

Designers and architects often reach for simple repeating structures to build with and nature often seems to follow a similar logic. Crystals also typically grow with repeatable triangular, square, and hexagonal lattices.

Quasicrystals are materials found in nature that are ordered but not periodic, much like Penrose tiles. Their atoms are arranged in lattices that follow 5-fold symmetries with aperiodic structures, rather than more regular structures found in their crystal cousins.

Penrose tiles are not only theoretically intriguing but also possess fascinating visual elegance. One can see clear structure yet at the same time there is a curiously pleasing irregularity to them. This has led to the tiles being the inspiration for numerous design projects.

One beautiful example is the floor outside the Mathematical Institute in Oxford. Made from stone and clay tiles in the shape of the iconic P3 tiles, it also shows with steel lines the joining pattern for how the tiles are arranged. Whilst there are countless variations for how the tiles can be configured, once certain tiles are laid down this can dictate how other tiles must be placed. The steel lines illustrate this relationship.

The Penrose tiling is an iconic geometric form, with not only an intriguing visual elegance but also a fascinating conceptual underpinning.

From just two simple quadrilateral tiles we are able to endlessly tile a 2-dimensional plane without the pattern ever repeating. Penrose tiles are built upon a captivating aperiodic structure, seamlessly blending perfect order with endless irregularity.

Max Cooper & Jessica In (@shedrawswithcode) collaborated on the mesmerizing track and video named ‘Penrose Tiling’, with Max creating the track and Jessica the visuals. The video plays with Penrose tiles with them unfolding and dancing around the screen to the song, morphing and shifting along the way.

From here we set about creating a final graphic that perfectly captured the story of the Penrose tile. We settled on a final design that plays with negative space, subtly alluding to the missing tiles to create a visually intriguing silhouette. The finished t-shirt is bold and striking yet also has a minimalist elegance.

We created a limited run of the t-shirts made from custom fabric that is available while stocks last. Available in a classic unisex cut as well as in a womens fit

For more design that fuse fascinating concepts with modern style explore our wider collection