This is module to create bren-stars. bren-stars were a creation while taking notes on dot matrix paper (I may have been a little distracted). That is how the square conformation of the star came to be. Constructing square stars add interesting constraints over traditional circle stars. To start, the traditional way to to create stars is to connect every other point around the circle until all points are connected. This however is not as nice in a square conformation as many point lie on the same line. Now we're getting to the point where I need to introduce a few numbers. The first number is the size of the square. The best way that I found to quantify this is by side length which I call n. n accounts for the number of points on each side minus one. The square will have 4n points in its perimeter.
I formalized the definition as follows (but am open to a better description):
A bren-star of order n is a set of 4n points arranged in a square where each point is connected to the points m points away (where m is the smallest number that satisfies: m > n and m is co-prime with 4n)
One interesting observation is looking at m as a function of n:
| n | m |
|---|---|
| 1 | 3 |
| 2 | 3 |
| 3 | 5 |
| 4 | 5 |
| 5 | 7 |
| 6 | 7 |
| ... | ... |
m is equal to either n+1 or n+2. More specifically it is the next largest odd number greater than n.
As this was originally created on dot paper noticed interesting behaviors in the points (having integer coordinates) contained by the the stars. Originally I was interested in which points lie inside the inner most shape created by the star. Since then I have identified 2 classes of points:
- Points on the star: these are any points on one of the lines they include
- Vertex points - points lying on the perimeter of the square that are used in the construction of the star
- Intersection points - points where multiple lines intersect
- Other points on the star - I hypothesize that due to reflectional and rotational symmetries all points with integer coordinates that lie on the star will be points of intersection.
- Points in the star: These are points that do not lie on any of the lines and include:
- Central - the points that are contained in the central most contiguous area
- Peripheral - the points inside the star but not inside the central most shape
| n | Vertices [4n] | Central Interior Points | Peripheral Interior Points* | Points of Intersection* | Total Points [(n+1)^2] |
|---|---|---|---|---|---|
| 1 | 4 | 0 | 0 | 0 | 4 |
| 2 | 8 | 1 | 0 | 0 | 9 |
| 3 | 12 | 0 | 0 | 4 | 16 |
| 4 | 16 | 5 | 4 | 0 | 25 |
| 5 | 20 | 4 | 0 | 12 | 36 |
| 6 | 24 | 13 | 12 | 0 | 49 |
| 7 | 28 | 12 | 0 | 24 | 64 |
| 8 | 32 | 21 | 20 | 8 | 81 |
| 9 | 36 | 24 | 12 | 28 | 100 |
| 10 | 40 | 37 | 44 | 0 | 121 |
| ... | ... | ... | ... | ... | ... |
* Peripheral interior and intersection totals are not necessarily accurate yet howe ever the sum of vetices, central points, peripheral points, and intersection points. Do total the projected total points so that is something in favor of them being accurate.
| + As mentioned before I was originally intrigued by the central points. Here is the first characterization from my notebook. drawing stars 2 through 7 results in a interesting patter where the odd stars have 1 few points than the even star just smaller than them. |  |