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Investigating Octagon Loops

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Introduction

Investigating Octagon Loops

Introduction

I am primarily investigating Octagon Loops. In the investigation I plan to find out the relationship between the number of tiles used in the loop and the total number of free edges. Also, I will find the relationship between the number of tiles and the inside free edges and the number of tiles and the outside free edges.

What I Will Do

I plan to do this by first using octagon shapes and positioning them into loops, I will then put the results into a table so that it’s easier to analyse.

I’ll start by making a loop with the least amount of shapes and then build up the number of shapes to give me a good range of values. Then I will analyse the results and hopefully discover some rules.

What I did

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Middle

rd 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).

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Conclusion

n="1" rowspan="1">

14

20

12

8

16

24

14

10

18

28

Total Edges: 2N                 Inside Edges: N-4                 Outside Edges: N+4

Hexagons

No. Of Hexagons

Inside Free Edges

Outside Free Edges

Total Free Edges

6

6

18

24

8

10

22

32

10

14

26

40

12

18

30

48

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.  

...read more.

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