Kepler's Conjecture - Polydisperse and Spheroid packing (Spring 2027)

In 1611, Johannes Kepler conjectured that the familiar pyramid stacking of oranges in a grocery store fills space as tightly as possible (≈ 74.05%). Proving that no arrangement of identical spheres can ever beat this limit took nearly 400 years. 

Our project asks: What happens when the shapes aren't identical round spheres, and we would explore two possible directions

1. 1. The M&M Mystery (Spheroids):
      If you squish or stretch spheres into M&M-like shapes (spheroids) and keep them aligned on a grid, their packing density doesn't change at all (≈ 74.05%). If you allow neighboring spheroids to tilt in different directions, they can pack up to 77% - strictly denser than round spheres!
      Your task: Model and simulate how changing the shape's stretch factor (aspect ratio) affects the maximum possible packing density.

   2. The Big & Small Marble Puzzle (Polydisperse Packings):
      If you have large spheres and small spheres, you can tuck the small ones into the gaps between the big ones.
      Your task: For any size ratio \gamma = r/R (r <R), what is the absolute, mathematically proven ceiling on packing density?

For more information contact Sagar Shrivastava (sagars@iastate.edu)

People:

• Sagar Shrivastava (Postdoc)

• TBD

Pre-requisites:

• Required: Multivariable Calculus (Calc III) and Linear Algebra.

• Helpful: Basic programming experience in Python or SageMath.