Question: Premise
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I was inspired by [Karl Niklas](https://en.wikipedia.org/wiki/Karl_J._Niklas) who used computer modeling to simulate the ideal structure for a tree when different traits were favored. Here is a chart summarizing his work:

[![enter image description here](https://i.stack.imgur.com/llkgX.jpg)](https://i.stack.imgur.com/llkgX.jpg)

**Explanation**

* L (light): horizontal branches
* M (mechanical stability): fewer horizontal branches
* R (reproduction): taller to spread seeds further

When he combined them all, his model yielded trees that closely resemble trees in the real world, with enough of each trait to survive and reproduce.

**A new tree shape**

From this point I wanted to take things to a more extreme angle. I want to have trees that are shaped like a dodecahedron:

[![enter image description here](https://i.stack.imgur.com/xg8ug.png)](https://i.stack.imgur.com/xg8ug.png)

Question
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If we take the point from Niklas' research that trees take a shape that helps them best survive given what the environment deems favorable traits, then what kind of a world do I need for a dodecahedron tree to be deemed favorable?

**Further Clarifications:**

* Roots: as pictured
* Foliage: configurable
* World: configurable (Earth-like or non-Earth like), we just assume no matter what the world, evolutionary theory, as we understand it, holds to a reasonable degree

Answer: 
Since the structure defines an inside and an outside, it might be useful for situation where the nutrients are not absorbed via the roots but via the branches/leaves, while being carried by the medium. More or less what sponges do in water.

Such a tree would therefore have the outer leaves being the photosynthetic ones, while the inner ones would be the absorbing ones.

The roots would be used only as supporting mechanism, to anchor the tree to the substrate.

The environment could, for example, be:

* Highly windy, with nutrients carried by the wind. The hollow shape allows for the carried particles to precipitate.
* Liquid, like for the sponges.
A permanent tumbling plant may evolve a shape that as closely resembles spherical as possible, while using minimal mass. Natural objects also tends to respect symmetry as this reduces the necessary genome size. Like the icosahedral viruses which only have to code for a protein that forms a pentameter of trimers for its capsid.

I imagine a world with constantly blowing high winds, which will uproot anything that is not adapted against it's force, but will also give any plants that can harness it's force plenty of opportunity to disperse and thrive. For example, plants that can tumble in the wind and put down roots where ever there is plenty of water and nutrient supplies, and have a shape that can always photosynthesize no matter which side it lands on.

For a given shape, the fewer the faces it has, the less amount of edge length it has for a give surface area. Of course, there will also be tetrahedral, cubic and octagonal plants, but they are unable to roll due to having acute to right angles on their edges and vertices.

The plant needs to roll or tumble, so therefore it have to have leaves on all sides, and needs to be rounded for rolling. It also have to be the lightest for its given size. A tree's mass is largely concentrated in its trunk and branches, therefore: any higher fullerenes or spherically symmetric structures will be less mass-efficient for it's cross section, and any lower symmetric structures won't roll because of the angles, therefore a rolling plant optimizes to adopt a dodecahedral shape, to both maximize it's surface area and minimize it's mass, while still being able to roll with the wind.

As for growth, I imagine the tree grows as a trimer of pentamers, I.e. having individual seedlings being a simple bent shape with a bend angle of 108 degrees, every five seedlings locks together to forme a Pentagonal face, then the outside of every three seedlings adheres to each other, forming a vertex. This naturally self assembles into a dodecahedron when high winds jitters the field of newly sprouted seedlings around, causing them to be stired up and bump into each other. As each seedling sprouts roots at the bent point, the dodecahedral structure can always take root nomatter which side it lands on. Also, The leaves will be on the inside of the seedlings, forming the face of the dodecahedron. This ensures the plant always have photosynthetic tissue facing the sun/light source nomatter which face it lands on.