### Geom-e-Tree for the Math-Minded

These exercises are suitable for high schoolers and college students. Let's hear it from Math Professors on the Open Questions!

It may help to refer to the Explore page for sample patterns.

### Exercises

1. How can the tree make the hexgonal, isometic, and square grids!? Once you have one, you can just vary the angle to get the others. Why? Hint: What do you suppose the common ratio needs to be so these plane tiling patterns can happen?

2. How do the Hex tiling patterns vary depending on N? (N is the branching factor.) You'll need to generate a hex pattern for various n's, if possible, and compare them. Can you explain why they are different? Does every value of n have a hex tree? Can you predict what the higher order trees will produce? Does the number of 'levels' of branches affect the pattern produced?

3. How do the Isometric tiling patterns (grids) vary depending on N? (N is the branching factor.) You'll need to generate a isometric pattern for various n's, if possible, and compare them. Can you explain why they are different? Does every value of n have an isometric tree? Can you predict what the higher order trees will produce? Does the number of 'levels' of branches affect the pattern produced?

4. How do the Square tiling patterns vary depending on N? (N is the branching factor.) You'll need to generate a square pattern for various n's, if possible, and compare them. Can you explain why they are different? Does every value of n have a square grid tree? Can you predict what the higher order trees will produce? Does the number of 'levels' of branches affect the pattern produced?

5. For what values of n do inverted isometric grids appear?

6. Where do you suppose the tree lines actually run in the (hex, isometric, square) tiling patterns? That is, how many and which lines are drawn over? How many times are those lines drawn over? The colored themes offer a hint. Save one of the a tiling patterns to Photos and then print it if you can. Label each line with a number of times you think it must have been overdrawn... Take this further if you can.

7. Research: Can you find any literature exploring patterns made by trees when the Common Ratio is less than 1.0? (I.e., tree branches get longer as you move away from the root of the tree.)

8. Can you develope a formula for the common ratio that will make a self-contacting tree for given values of n and θ? How about just for a binary tree (n=2) and a given angle?

9. Can you map trees of various degrees onto polyhedra? Which trees onto which polyhedra? Describe the mapping and coverage.

### Open Questions

1. Why does (3,144°,0.5) make a five-pointed star, while (5,144°,0.5) makes an apparent pentagon!? (6,144°,0.5) is also a pentagon, but (4,144°,0.5) is halfway between - a star with a pentagon inside. Does the 3-branched tree-star have a pentagon in it? Our forth-coming poster might be of help here.

2. What about the n=7 pentagon? Is it the same inside as the n=5 pentagon? BTW, it looks like pentagons are possible "all the way" up for 144°.

3. Consider all the basic shapes that can be made by Geom-e-Tree with n from 2-6. Are any other shapes possible with higher branching factors? Say n=12, n=25 ... what happens to the set of patterns as n gets larger?

4. Can you explain how Geom-e-Tree can make polygonal shapes in terms of a generator function for the endpoints of all the branches? Restating: The set of all endpoints for certain trees seem to be bounded by a polygon. What is the relation between the degree of the tree, its branch angle, and the number of sides in the bounding polygon?

### Reminder

We currently have the common ratio (CR) constrained to ≥0.5... The author thinks that allowing it to go any lower just results in more pinwheel-like patterns, with lines shooting off (and drawing times increasing radically). In later versions of Geom-e-Tree, we will consider a way to allow you to disable the CR limiter. We don't want any exploding iPhones out there. ;^)

### Caveat

The questions and/or numbers will change over time, so be careful if you reference them!!

### Talk to Us!

Please tell us about Geom-e-Tree being used in high school or a university class!
support@geom-e-tree.com