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: 
The reasons trees grow upwards (for the most part) is because of their need to photosynthesize to create food for themselves. Different trees have different adaptations to acquire the maximum about of sunlight, while also protecting themselves from other threats. 

One reason to have a dodecahedron shaped plant is if there is something on the faces or edges or corners of the dodecahedron that prevents other animals from eating it, thereby protecting it. Another thing could be to minimize the amount of itself an animal can eat. If the plant's main feeder is a larger animal that needs to bend over to consume it, it would help to be a small, stocky plant without any external protrusions and smooth faces to make it hard for an animal to eat. 

Another reason could be that the plant's in-built 'factories' to absorb and process its food requires it to be that way. Much like why solar panels are flat, it helps to have an even and flat surface to acquire sunlight (or whatever it needs). Being many-sided allows it to have a direct interface to sunlight from any and all angles at all times. Sunflowers, in order to maximize how much sunlight they get, actually turn their 'face' towards the sun. Maybe this is similar?
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.