Chladni Figure

Photograph by Steve Mould. Full video here.

Chladni figures are the strange patterns created when a violin bow is used to vibrate a metal plate with sand on it. Based on where the plate is held down, different patterns are created. This is caused by standing wave on the plate. An image of a standing wave is shown below, which can be thought of as an exaggerated cross section of the plate.

The wave has nodes which are stationary and antinodes which move up and down. The sand settles towards the antinodes, creating the patterns. The pattern on the plate can be modeled by a partial differential equation. This equation has boundary conditions that are set by where the metal plate is held down. Depending on these boundary conditions, different patterns are created.

 Design a puzzle such as “Crystal Ball”

The crystal ball puzzle is one example of a polyhedra puzzle. There is a variety of ways to build these types of puzzles but typically the simpler puzzles are based on orthogonal grids and more complex puzzles are built on non-orthogonal grids like the shape on the right.  These puzzles often use principles of combinatorics to decide how the pieces should be arranged, color combinations, and the set of different connection possibilities given these puzzles can often be solved in more than one way. They also rely on group theory particularly symmetry and permutation groups to establish rotational or reflective symmetry within the puzzles, reduce problem complexity, and develop cycles of different moves.

Illustrate the four-color theorem in different topological settings

The four-color theorem employs graph theory to prove that any 2D surface can be colored using just four colors in a way where no two adjacent regions share the same color. The Szilassi polyhedron is a non-orientable polyhedron with 7 vertices, 14 edges, and 10 faces. This polyhedron is an interesting example because its surfaces are non-orientable. This project would include using topology to apply the four-color theorem to different surfaces such as a sphere, torus, or 2D plane. Euler’s formula would be topical here as it relates the number of vertices, edges, and faces of a polyhedron.