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Morphogenesis on Ice

The transformation that turns a faceted ice prism into an intricately branched stellar dendrite is an example of physical morphogenesis—the spontaneous creation of pattern and form by inanimate materials, the process by which order arises from chaos.

There are plenty of other manifestations of physical morphogenesis you can find around you, such as ripples on ponds, banded patterns on snowdrifts and sand dunes, and even the billowing convection cells you see in a bowl of hot miso soup. Of course, our favorite example is always the snowflake—the quintessential example of physical morphogenesis.

The Origin of Branching

The six corners of a snow crystal grow a bit faster because they stick out farther into the humid air, causing branches to sprout. As the crystal grows larger, the same effect causes sidebranches to sprout from the faceted corners of each branch. This process is responsible for the complex shapes of snow crystals.

Branching is an example of a growth instability. The corners of a hexagonal plate stick out a bit, so they grow a bit faster, causing them to stick out even farther, causing them to grow even faster.

As a general rule, faceting dominates when snow crystals are small, or when they grow slowly. Larger, faster growing crystals become branched.

Chaos and Order

The formation of sidebranches sometimes exhibits a chaotic behavior, as in this fernlike stellar dendrite. When the ice growth is especially hurried, the sidebranches are rather erratically spaced, with little symmetry between the branches, or even between the two sides of a single branch.

Chaos and order are both present during snowflake growth, and this is what makes snowflake patterns so intriguing. By itself, the branching instability brings chaos -- the unbridled creation of structural complexity. Faceting, on the other hand, brings order, as in the simple perfection of a hexagonal prism. Bring those two forces together, however, and beautifully intricate, symmetrical snowflakes result.


Branched constructions like snowflakes often exhibit fractal patterns. The defining characteristic of a fractal snowflake is a self-similar structure, where branches have sidebranches, which have their own smaller sidebranches, and so on. The self-similar construction on the right is called a Koch snowflake.

In fact, real snowflakes are only slightly fractal. The first sidebranches rarely have additional sidebranches, so calling the whole assembly self-similar is a bit tenuous. Moreover, noting the fractal qualities of a branched snowflake does not shine any light on its origin. The branching instability is needed to explain why the branches and sidebranches arise in the first place.