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What are...?

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by

Kenneth Tynan

on 28 January 2015

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Transcript of What are...?

What are Eclipse Groups?
. Defining features:

- Formed of n circles of the same size all overlapping a common point equally where
n
is always a whole number

- Each individual group has a distinct number of pieces, small sections formed by the overlapping of the circles’ lines.

- As more circles are added, the number of pieces increases and individual pieces become smaller.
Eclipse Groups
Obstacles
We faced few but difficult obstacles

- What we started with was not what we ended with

- Only one of us could reproduce the figures

- The larger the circle got the harder it was to count the pieces
Discoveries
Though we experienced trouble we had many breakthroughs
Eclipse Groups
by
Andrew Mitchell & Kenneth Tynan

2 circles- 3 pieces

3 circles- 7 pieces

4 circles- 13 pieces

5 circles- 21 pieces

6 pieces- 31 pieces

7 pieces- 43 pieces

8 pieces- 57 pieces

9 pieces- 73 pieces

10 pieces- 91 pieces
(n) Circles = (P) Pieces
1 circles - 1 increase

2 circles - 2 increase

3 circles - 4 increase

4 circles - 6 increase

5 circles - 8 increase

6 circles - 10 increase

7 circles - 12 increase

8 circles - 14 increase

9 circles - 16 increase

10 circles - 18 increase
There is an increase of 2 pieces when a new circle is added
*excluding one
P=(n)(n-1)+1
Equation of P for value of n
Examples
91 = 10 × 9 + 1
p=(n)(n-1)+1
- Eclipse groups have 2n symmetrical actions; rotation and reflection
- The Eclipse groups will develop a n-sided polygon in the center with that has curved sides.
(proved up to n=10)
Areas of Further Study
- Will our proofs hold true the conjectures for all values of n?

For example:

- Will n continuously increase by multiples of 2?

- How is the pattern in the increasing number of pieces related to the equation we found?

- How do the pieces change size with the addition of more circles? Is there a pattern or ratio?

- Can we produce similar results with different overlapping patterns?
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