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Pattern Formation

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(1)

Pattern Formation

Necessary and Sufficient Condition

on the Set of Initial Configurations (with chirality)

(2)

Pattern formation (with chirality)

2

If we know the initial configuration is not symmetric, can we form a pattern that is not a regular n-gon?

(3)

Pattern formation (with chirality)

2

If we know the initial configuration is not symmetric, can we form a pattern that is not a regular n-gon?

Yes!

(4)

Pattern formation (with chirality)

2

If we know the initial configuration is not symmetric, can we form a pattern that is not a regular n-gon?

Yes!

What are the conditions on the initial configuration and on the pattern so that the problem is solvable?

(5)

Examples

3

(6)

Examples

3

(7)

Examples

3

Possible

(8)

Examples

4

(9)

Examples

4

(10)

Examples

4

Possible

(11)

Examples

5

(12)

Examples

5

(13)

Examples

5

Not Possible

(14)

Examples

5

Not Possible

More symmetry here than here

(15)

Symmetricity

6

(16)

Symmetricity

6

(17)

Symmetricity

6

Definition: the symmetricity 𝜌 of a configuration is the number of rotations for which the configuration is invariant.

if a robot is in the center of the SEC, then the symmetricity is one.

(18)

Symmetricity

6

Definition: the symmetricity 𝜌 of a configuration is the number of rotations for which the configuration is invariant.

if a robot is in the center of the SEC, then the symmetricity is one.

𝜌 (C) = 4

(19)

Necessary condition

7

Theorem: The pattern formation problem is solvable in FSYNC only if the symmetricity of the initial configuration divides the symmetricity of the pattern

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