The diagrams page shows what shapes I made with the octagons to gather my results.
Results
My results for the octagon loops are shown below:
Diagrams
1st loop: 2nd loop: 3rd loop:
Conclusion
When I looked at the results I discovered that there could only be even amounts of tiles starting from 4 (no less). I also found out that from one diagram to another there was always a constant difference between the values. The difference between the amounts of tiles in the loop was 2 and both the inside and outside free edges had a difference of 6. The difference between the total edges was 12.
After analysing the results I discovered a rule for finding out the total amount of free edges from the amount of tiles, the rule is: T=6N. This rule works when T is the total number of free tiles and N is the number of tiles is the loop. I discovered the rule when I first found out the constant difference between the total free edges (12). So I was going to put 12N in the formula, but since the amount of octagons went up in twos, I divided 12 by 2 to give me 6N.
I also found out the formula for the amount of inside and outside free edges, they are as follows; inside: I=3N-8 and the outside: O=3N+8. The formulae work when I is inside free edges and O is the outside free edges. I found the formulae when I discovered the constant difference between the inside and outside edges (6), but because the loops go up in twos, I divided it by 2 which gave me 3. The outside and inside formulae are about the same apart from you take away 8 (number of sides of octagon) for the inside and add 8 for the outside.
Extension
Now that I had discovered the rule for octagons, I decided to look at other regular polygons to see if there was any relationship between them all.
First I investigated squares because they only had four sides and were easy to work with and analyse. After finding the formulae for squares I investigated hexagons with began to build up a general picture of the relationships between the polygons.
My results are shown on the next page.
Results for extension work
Squares
Total Edges: 2N Inside Edges: N-4 Outside Edges: N+4
Hexagons
Total Edges: 2N Inside Edges: N-4 Outside Edges: N+4
Conclusion
After gathering all this information I discovered an overall trend in the total edge formula, it seemed to be that what you multiplied by N was the amount of sides of the regular polygon take away 2. The formula is below.
(S-2)N
This formula works when N is the amount of polygons in the loop and S is the amount of sides of the polygon.