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author: niplav, created: 2019-06-21, modified: 2019-12-02, language: english, status: notes, importance: 4, confidence: highly likely

It is currently unknown how many ways exist to arrange n circles in the affine plane. However, for up to 5 circles, the number is known. This page attempts to classify the current known combinations.

# Circle Combinations

## Characteristic Description of a Circle Combination

Any unintersected plane is contained by n circles. How many planes exist that are contained in n circles?

The characteristic description of a circle combination lists the number of planes contained by n circles, in ascending order. For example, the following circle has the description 1,2,1,1:

There is only one plane not contained by any circles (the outer plane), 2 planes only contained by one circle, 1 plane contained by 2 circles, and 1 plane (the innermost circle) contained by all 3 circles.

As one can see, the outer plane is counted as well.

Conjecture: The characteristic description of a circle is a unique identifier of the circle combination described (no 2 circle combinations have the same characteristic description).

## 1 Circle

 Image Unintersected lines Line intersections Unintersected planes Characteristic description 1 0 1 1,1

## 2 Circles

 Image Unintersected lines Line intersections Unintersected planes Characteristic description 2 0 3 1,2,0 4 2 4 1,2,1 2 0 3 1,1,1

## 3 Circles

 Image Unintersected lines Line intersections Unintersected planes Characteristic description 3 0 4 1,3,0,0 5 2 5 1,3,1,0 8 4 6 1,3,3,0 5 2 5 1,2,2,0 8 4 6 1,2,2,1 5 2 5 1,2,1,1 12 6 8 1,3,3,1 3 0 4 1,2,1,0 3 0 4 1,1,2,0 5 2 4 1,1,2,1 3 0 4 1,1,1,1 12 6 8 1,4,2,1 12 6 8 1,2,4,1 12 6 8 2,3,3,0