The next step is to put a vector (type double[15], in these examples) of outputs at every vertex.

At every timestep at every vertex you get the calculation inputs (double[45] here) with three times as many inputs as outputs. This is so the inputs can be the output of (1) this neuron, (2) the average of its neighbours and (3) the difference between its neighbours. All inputs are from the outputs in the previous step.

Some inputs are overwritten with the vertex's position and growth direction.

These outputs are fed into a neural network with 15 outputs, 30 hidden neurons and 45 inputs. Every vertex has an identical neural network.

A simplified diagram looks like this:

The neural network itself is probably best described with code, since it's fairly simple but a bit fiddly to describe:

double[] Evaluate(double[] input) { for (int j = 0; j < input.Length; j++) { neuron[0][j].value = input[j]; } for (int i = 1; i < NLayers; i++) // The zero-th layer is skipped. { for (int j = 0; j < number of neurons in layer i; j++) { double a = -neuron[i][j].threshold; for (int k = 0; k < number of neurons in layer (i-1); k++) { a += (neuron[i - 1][k].value - 0.5) * neuron[i][j].weights[k]; } neuron[i][j].value = 1.0 / (1.0 + Exp(-a)); } } for (int j = 0; j < number of neurons in final layer; j++) { output[j] = neuron[last][j].value; } return output; }

The next step is to grow the mesh. In each timestep, after the vertex outputs have been calculated, you move each vertex according to the outputs.

Each vertex has a growth direction. The distance it moves in each timestep is an increasing function of one of the outputs. If any face grows too large, then a new vertex is placed in its centre, and given a growth direction that is a combination of the normal of the face and also the outputs of the three parent vertices (since all faces have 3 vertices) - so the growth direction of new vertices can be controlled by the neural network.

The final step is to occasionally retriangulate the mesh so that you don't have too many long thin faces. This needs to be done with some care to preserve the overall structure of the object.

This shows what the growing plant looks like:

If you randomise the neural network and grow a plant from it many times, then you get a group of plants like this:

This provides quite a lot of different meshes, which is good, but they are all quite blob-like. The next step is to search for neural networks that are more interesting. A genetic algorithm can be used to select shapes that optimise a heuristic: Heavily selected meshes tend to be more interesting, and they should lend a sense of correctness to the game: The simulated game world would presumably have competition between plants, so a mesh that competes well is more correct for the game. The algorithm starts with a population and generates child neural networks from high-fitness parents using mutation and sometimes crossover. The worse neural networks (according to whether the plant that they grow has a high score) are replaced with the child. In this way the space of possible neural networks is gradually searched for interesting meshes.

If I draw the highest scoring plant in each generation according to maximising horizontal area above a certain height, I get this:

## Saturn's ringsA simulation of Saturn's rings --- a few thousand particles are simulated, in a repeating tiled region. You use the mouse and keys to fly in it. |

## Physics Algorithms BookThis is a work in progress to write a book on physics algorithms. At the moment, it is about 1/3 finished though, but the latest version can be downloaded for free. |

## Three Crystal StructuresA ThreeJS model of three crystal structures |

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