Tright Treeing

Here is a very simple tree with two branches:

Two-branch tree


These are the steps that a simple computer program follows to draw the tree, with a red arrow indicating where the computer’s focus is at each stage:

Two-branch tree stage 1


2-Tree stage 2


2-Tree stage 3


2-Tree stage 4


2-Tree (animated)


If you had to give the computer an explicit instruction at each stage, the instructions might look something like this:

1. Start at node 1, draw a left branch to node 2 and colour the node green.
2. Return to node 1.
3. Draw a right branch to node 3 and colour the node green.
4. Finish.

Now try a slightly less simple tree with branches that fork twice:

Four-branch tree (static)


These are the steps that a simple computer program follows to draw the tree, with a red arrow indicating where the computer’s focus is at each stage:

4-Tree #1


4-Tree #2


4-Tree #3


4-Tree #4


4-Tree #5


4-Tree #6


4-Tree #7


4-Tree #8


4-Tree #9


4-Tree #10


4-Tree #11


4-Tree (animated)


If you had to give the computer an explicit instruction at each stage, the instructions might look something like this:

1. Start at node 1 and draw a left branch to node 2.
2. Draw a left branch to node 3 and colour it green.
3. Return to node 2.
4. Draw a right branch to node 4 and colour it green.
5. Return to node 2.
6. Return to node 1.
7. Draw a right branch to node 5.
8. Draw a left branch to node 6.
9. Draw a left branch to node 7 and colour it green.
10. Return to node 6.
11. Draw a left branch to node 8 and colour it green.
12. Finish.

It’s easy to see that the list of instructions would be much bigger for a tree with branches that fork three times, let alone four times or you. But you don’t need to give a full set of explicit instructions: you can use a program, or a list of instructions using variables. Suppose the tree has branches that fork f times. If f = 4, you will need an array variable level() with four values, level(1), level(2), level(3) and level(4). Now follow these instructions:

1. li = 1, level(1) = 0, level(2) = 0, ... level(f+1) = 0
2. level(li) = level(li) + 1
3. If level(li) = 1, draw a branch to the left and jump to step 7
4. If level(li) = 2, draw a branch to the right and jump to step 7
5. li = li - 1 (note that this line is reached if the tests fail in lines 3 and 4)
6. If li > 0, jump to step 2, otherwise jump to step 11
7. If li = f, draw a green node and jump to step 5
9. li = li + 1
10. Jump to step 2
11. Finish.

By changing the value of f, a computer can use those eleven basic instructions to draw any size of tree (I’ve left out details like changes in the length of branches and so on). When f = 4, the tree will look like this:

16-Tree (static)


16-Tree (animated)


With simple adjustments, the program can be used for other shapes whose underlying structure can be represented symbolically as a tree. The program is in fact a fractalizer, that is, it draws a fractal. So if you use a version of the program to draw fractals based on right-triangles, you can say you are “tright treeing” (tright = triangle-that-is-right).

Here is some tright treeing. Start with a simple isoceles right-triangle. It can be divided into smaller isoceles right-triangles by finding the midpoint of the hypotenuse, then repeating:

Right-triangle rep-2 stage 1


Right-triangle #2


Tright #3


Tright #4


Tright #5


Tright #6


Tright #7


Tright #7 (no internal lines)


You can distort the isoceles right-triangle in interesting ways by finding the midpoint of a side other than the hypotenuse, like this:

Right-triangle (distorted) #1


Distorted tright #2


Distorted tright #3


Distorted tright #4


Distorted tright #5


Distorted tright #6


Distorted tright #7


Distorted tright #8


Distorted tright #9


Distorted tright #10


Distorted tright #11


Distorted tright #12


Distorted tright #13


Distorted tright (animated)


Here’s a different right-triangle. When you divide it regularly, it looks like this:

Right-triangle rep-3 stage 1


Rep-3 Tright #2


3-Tright #3


3-Tright #4


3-Tright #5


3-Tright #6


3-Tright #7


3-Tright #8


3-Tright #9


3-Tright (one colour)


When you distort the divisions, you can create interesting fractals (click on images for larger versions):

Distorted 3-Tright


Distorted 3-Tright


Distorted 3-Tright


Distorted 3-Tright


Distorted 3-Tright


Distorted 3-Tright


Distorted 3-Tright (animated)


And when four of the distorted right-triangles (rep-2 or rep-3) are joined in a diamond, you can create shapes like these:

Creating a diamond #1


Creating a diamond #2


Creating a diamond #3


Creating a diamond #4


Creating a diamond (animated)


Rep-3 right-triangle diamond (divided)


Rep-3 right-triangle diamond (single colour)


Distorted rep-3 right-triangle diamond


Distorted 3-tright diamond


Distorted 3-tright diamond


Distorted 3-tright diamond


Distorted 3-tright diamond


Distorted 3-tright diamond


Distorted 3-tright diamond


Distorted 3-tright diamond


Distorted 3-tright diamond


Distorted 3-tright diamond


Distorted 3-tright diamond


Distorted 3-tright diamond (animated)


Distorted rep-2 right-triangle


Distorted 2-tright diamond


Distorted 2-tright diamond


Distorted 2-tright diamond


Distorted 2-tright diamond


Distorted 2-tright diamond (animated)


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