Cutting and Shuffling: A New Dynamical Systems Paradigm for Mixing

Project: Research project

Project Details


Page A
"Cutting and shuffling," much like what takes place in shuffling a deck of cards or mixing
the colors on a Rubik’s cube, is a means of mixing. However, the fundamental properties of
mixing by cutting and shuffling, which can be described in the formalism of a new mathematics
called Piecewise Isometries, remains unexplored. The key problem is to connect the formalism
of Piecewise Isometries with a dynamical systems approach and thereby to develop a framework
that extends from one- and two-dimensional "toy" systems to fully three-dimensional systems
relevant to practical industrial problems.
The focus of the proposed work is a dynamical systems approach to cutting and shuffling
using the mixing of granular material in a three-dimensional tumbler as a model system. Not
only does the mixing of granular materials offer a practical industrially-relevant application
of the approach, but it also provides a three-dimensional physical system in which the concepts
of cutting and shuffling can be experimentally tested. The 125th anniversary issue of Science
identified the flow of granular materials as one of the 125 big questions in science. Although
granular flows are important in many disciplines ranging from geophysics to industrial processing
(powders, pharmaceuticals, grains, ores, foodstuffs, etc.), the emphasis here is on granular
systems as a physically relevant model system in which to explore dynamical systems approaches
for 2D and 3D cutting and shuffling as a mixing mechanism.
for 2D and 3D cutting and shuffling as a mixing mechanism.
The proposed research is an integrated effort to develop new theory inspired by abstract
mathematical concepts that is confirmed and validated by computational and experimental results.
The theoretical work will focus on the application of Piecewise Isometries in three dimensions
and its connection to dynamical systems approaches such as Poincare sections as well as its
practical implications with respect to cutting and shuffling as a paradigm for mixing. The
approach is based on the geometry and symmetries of 3D mixing by cutting and shuffling, but
is aimed at integrating the abstract mathematics of Piecewise Isometries into the dynamical
systems toolkit while accounting for non-ideal complications such as diffusion. Complementary
experiments and simulations will be used to confirm the applicability of the theoretical approaches
to a model physical system for mixing granular materials.
The ultimate objective of this research is to develop and utilize new mathematical tools
in a dynamical systems framework to predict mixing by cutting and shuffling in 2D and 3D geometries
with eventual applications to practical mixing systems.
Intellectual Merit :
The merging of abstract new mathematics and physical applications could result in an entirely
new paradigm for mixing. This research is at the intersection of three fields: 1) the new
mathematics of Piecewise Isometries (cutting and shuffling itself), 2) dynamical systems approaches
and tools as applied to cutting and shuffling, and 3) the implications of cutting and shuffling
as a mixing process. It is at intersections like this that transformational research occurs.
Broader Impacts :
First, there is a link between fundamental research and practical applications. Results in
pure mathematics often find surprising applications but require the marriage of two components:
a physical picture and the mathematics to support the picture. The new mathematical approaches
considered here have potential for broad
Effective start/end date9/1/148/31/18


  • National Science Foundation (CMMI-1435065)

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