#### Presentation Title

Coin Configuration Spaces

#### Presentation Type

Poster

#### Keywords

Topology, Configuration Space, Homology, Mathematics

#### Department

Mathematics

#### Major

Math Education (Madison), Mathematics (MacKenna)

#### Abstract

A coin configuration is a collection of coins (closed disks) in a plane such that the union of all coins is connected while the interiors are disjoint, giving the property that each coin is tangent to at least one other coin. The configuration space *C*(*r*_{1},...,*r _{n}*) includes all coin configurations with

*n*coins having radii

*r*

_{1},...,

*r*

_{n}. Each coin configuration has an associated tangency graph which records the tangent relationships between coins. By studying when the tangency graphs change we get a partition of the configuration space into smaller pieces, which are the flexible spaces. A flexible space then consists of all configurations with the same tangency graph.

Building on previous work, we determined the flexible spaces within the configuration space of coins of varying size. We studied how the sizes of the coins affects which tangency graphs are possible and also the boundary relationships between flexible spaces. We use information about boundaries of flexible spaces together with the tools of homology to piece together this configuration space.

#### Faculty Mentor

Joshua Bowman

#### Funding Source or Research Program

Summer Undergraduate Research Program

#### Location

Waves Cafeteria

#### Start Date

24-3-2017 2:00 PM

#### End Date

24-3-2017 3:00 PM

Coin Configuration Spaces

Waves Cafeteria

A coin configuration is a collection of coins (closed disks) in a plane such that the union of all coins is connected while the interiors are disjoint, giving the property that each coin is tangent to at least one other coin. The configuration space *C*(*r*_{1},...,*r _{n}*) includes all coin configurations with

*n*coins having radii

*r*

_{1},...,

*r*

_{n}. Each coin configuration has an associated tangency graph which records the tangent relationships between coins. By studying when the tangency graphs change we get a partition of the configuration space into smaller pieces, which are the flexible spaces. A flexible space then consists of all configurations with the same tangency graph.

Building on previous work, we determined the flexible spaces within the configuration space of coins of varying size. We studied how the sizes of the coins affects which tangency graphs are possible and also the boundary relationships between flexible spaces. We use information about boundaries of flexible spaces together with the tools of homology to piece together this configuration space.