Branching patterns are a common and diverse phenomena in nature – occurring in trees, leaf venation, streams, lightening bolts, lungs, arteries and many other places. The geometric nature of these branching patterns have broad similarities. These similarities arise due to the common reason behind the branching phenomena – the distribution of energy to or from a large area in the most efficient way. In the first activity we will demonstrate how the requirement for efficiency dictates the nature of the branching. In the second activity we will explore a way to quantify the branching patterns so that similarities can be objectively investigated.

1. On the top half of one piece of triangular grid paper pencil in a hexagon of dots which is five dots on each side. This hexagon represents an area the requires a source of energy or nutrients. Mark the central dots with a pen. The central dot is the source of nutrients for the rest of the dots. The first way we will feed the rest of the dots is to create a single path starting at the center dot which does not cross itself and passes once through every dot on the grid. After you have created your path compare with your group members. There will likely be a few different paths which satisfy this simple rule. This kind of path is called a meander

1. The total path length
2. the average distance of each point from the center along the path connecting it to the center

1. Next we will try an different strategy for feeding the points. Create a second hexagon of dots on the bottom half of your paper. This time connect each dot directly with the center. You do not need to follow the lines on the grid. This kind of pattern is called an explosion.

Once again find the total path length and the average distance of each point from the center. Compare your results with those in part 1. Try to come to a conclusion about explosion and meander patterns.

2. Now we will try to find a compromise between the meander and star burst patterns. On a new sheet of paper draw out 2 new hexagons of dots as before. On each hexagon create a system of paths from the center which include some kind of branching. Make sure there are not loops – that is make for any point there is only one path back to the center. Do a different type of branching on each hexagon. You may want to try an approximation of some types of branching you see in leaves. (note: a path does not need to follow along the lines of the grid.)

Variations: Try locating the feeding point on a corner of the hexagon rather than in the center. You may also be concerned that the triangular grid will unduly bias your results. Try using a square grid instead.

1. Mark off a rectangular array of dots on your grid which is 10 dots wide by 20 dots high. For the sake of this activity think of this area as a drainage field for a river. Water at each point will drain down left or right to the next point. For each dot in your grid toss a coin. For heads draw a line in pencil going down to the left to the next point. For tails draw a line to the right. After you have completed this for all points in your grid you will have a collection of lines representing the flow water down a gradient. (The observant among you may recognize this pattern as similar to one formed by the receding wave at the beach). Some of these lines will have naturally joined into branched networks. Find the longest such network and colour it in with a pen.
2. For this random branching network complete a Horton analysis. To do a Horton analysis first count the number of first order branches (those that originate from a point that has no other lines. Then count the number of second order branches (those that originate when two first order branches join). A third order branch is one which originates when two second order branches and so on. Note when a first order branch joins a second order branch the second order branch does not changes its order. It may be helpful (and even esthetically pleasing) to shade second order branches thicker than first order branches and so on. To complete the Horton analysis find the ratio of first order to second order branches. Then find the ratio of second order to third order branches and so on. Typically these ratios will range between 3 and 5 for streams. In fact this ratio is remarkably constant for a wide range of branching phenomena. Can you explain why that might be?
3. Complete a Horton analysis of your branching structures in Activity 1. Due to the small size of hexagons in part 1 the number of 3^rd order of branches may be small. Try constructing a larger hexagon and creating a branching pattern which has 3^rd or even 4^th order branches. What are the quantitative similarities and differences you observe between the random stream branch network in Activity 2 and the constructed branching patterns in Activity 1.