Transcript Animating Tessellations of the Plane with Statistical Models

Animating Tessellations of the Plane with Statistic Models
Ben McCabe
Advisor: Rolfe Petschek
Department of Physics, Case Western Reserve University
Abstract
Artists, particularly M.C. Escher, have made many attractive tessellations of the plane, incidentally illustrating many of the two dimensional space groups.
Such pictures may have different space groups or attractiveness based on the patterning and symmetry of various colors in the pictures, and in defects in
such symmetries. Many physical systems have "broken" symmetries in which an unspecified state (e.g. color) can be determined by the state of its
neighbors, as in antiferromagnetism. Much is known about such ordering in various two dimensional cases, including various statistical models thereof. In
this project we combine art and physics by animating an Escher-like tessellation of the plane, using a Monte-Carlo or similar simulation of a statistical model
to change the colors or other attributes of the tessellation. As the formation of long range order in such models, and the resulting potential fractional
dimensionality of the patterns is thought to have aesthetic value, this will hopefully achieve all of providing: an interesting illustration of two dimensional
statistical models, an interesting piece of modifiable art, and a potential screen-saver.
Introduction
We perform a Monte Carlo simulation using the Heat Bath
method as follows. We pick a site at random in the
tessellation and calculate it’s contribution to the Hamiltonian
as defined by the statistical model we are considering for
each possible state it can be in. For this project we consider
only nearest neighbor interactions. We then set the state of
the chosen site with the possibility of a given state j being the
new state being given by
 bE si
Psi 
e
Z  e
 bE s j
j
Where Z is the partition function and b is the inverse
temperature (defined as the reciprocal of the product of the
Boltzmann constant and the absolute temperature)
The boundary is periodic in both directions in order to reduce
boundary effects. This makes the simulation topologically
equivalent to a donut. We allow the user to start the
simulation with all the sites in one state or to start with a
random configuration.
In addition, the speed of the
simulation is controlled by specifying the number of iterations
of the model to perform every frame.
Ising Model
The Ising model is a model with two possible states for each
site. The possible states are +1 and -1, and represent spins.
The hamiltonian is given by the equation
H   J ij Si S j   hi Si
i j
i
And our program only considers the case where both J and h
are constant. h is the external magnetic field, and J is the
variable that defines the interactions between spins. If J>0
the interaction between spins is ferromagnetic and spins try
to align, and if J<0 the interaction between spins is
antiferromagnetic and spins try to antialign. If we pull J out of
the summation and absorb it with b then we can define a new
variable K=-bJ. When h=0 there is a phase transition for the
triangular lattice at
  

1
Kc 
ln tan
 0.275
2
6
At which the model exhibits fractal looking patterns that can
be considered aesthetically pleasing.
Six-State Clock Model
As the name suggests, there are six states in the six state
clock model. We decides to associate colors to the states as
follows: 0=red, 1=yellow, 2=green, 3=cyan, 4=blue,
5=magenta. In this way, each primary color is adjacent to the
two secondary colors that contain it and each secondary
color is next to the two primary colors it consists of. The
hamiltonian is given by the equation


H   K 6  cos Si  S j 
3

i j
 K 2  cos S i  S j 
i j
 2

Si  S j 
 K3  cos
 3

i j
If K6<0 then sites try to be as close to their neighbors as
possible on the color wheel, and if K6>0 they try to be as far
from their neighbors as possible on the color wheel. If K2<0
then primary colors will try to neighbor with other primary
colors and secondary colors will try to neighbor with other
secondary colors, and if and if K2>0 primary and secondary
colors will each try to neighbor with the other. If K3<0 sites
will try to neighbor with sites of the same or opposite color,
and if K3>0 sites will try to neighbor with sites that are neither
the same nor opposite color. Varying the three parameters
allows for a wide variety of interesting patterns
A tessellation by M.C. Escher
Methods
The project was written in C++ and uses DirectX for
graphics. We represented the state of the tessellation as an
array and used simple rules to determine the nearest
neighbors. In order to find the nearest neighbors for a
triangular lattice, we consider every other row to be indented.
Each site has two nearest neighbors above, two below, and
two in the same row. If the index representing the column of
a site is j, then the nearest neighbors in the same row are at
j-1 and j+1. If the site is in an indented row then the nearest
neighbors above and below are at j and j+1. Otherwise they
are at j-1 and j.
Output of the program for the six-state clock model with K6=-0.35,
K2=-0.3, and K3=0.0
Output of the program for the Ising model with no external magnetic
field at the phase transition
Acknowledgements
I would like to thank Professor Rolfe Petschek for his
continued guidance and support, as well as Professors Gary
Chottiner and Kenneth Singer.