Animating Tessellations of the Plane with Statistic Models Ben McCabe Advisor: Rolfe Petschek
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Animating Tessellations of the Plane with Statistic Animating Tessellations of the Plane with Statistic Models Models Ben McCabe Advisor: Rolfe Petschek Department of Physics, Case Western Reserve University 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. Abstract Abstract Introduction 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 Where Z is the partition function and is the inverse temperature (defined as the reciprocal of the product of the Boltzmann constant and the absolute temperature) Methods 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. 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 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 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 then we can define a new variable K=-J. When h=0 there is a phase transition for the triangular lattice at At which the model exhibits fractal looking patterns that can be considered aesthetically pleasing. Acknowledgements Acknowledgements 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 If K 6 <0 then sites try to be as close to their neighbors as possible on the color wheel, and if K 6 >0 they try to be as far from their neighbors as possible on the color wheel. If K 2 <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 K 2 >0 primary and secondary colors will each try to neighbor with the other. If K 3 <0 sites will try to neighbor with sites of the same or opposite color, and if K 3 >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 I would like to thank Professor Rolfe Petschek for his continued guidance and support, as well as Professors Gary Chottiner and Kenneth Singer. j E E s j s i s i e Z e P A tessellation by M.C. Escher i i i j i j i ij S h S S J H 275 . 0 6 tan ln 2 1 c K Six-State Clock Model Six-State Clock Model Output of the program for the Ising model with no external magnetic field at the phase transition j i j i S S K H 3 cos 6 j i j i S S K cos 2 j i j i S S K 3 2 cos 3 Output of the program for the six-state clock model with K 6 =-0.35, K 2 =-0.3, and K 3 =0.0
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Animating Tessellations of the Plane with Statistic Models Ben McCabe Advisor: Rolfe Petschek Department of Physics, Case Western Reserve University. Abstract. - PowerPoint PPT Presentation
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Animating Tessellations of the Plane with Statistic Animating Tessellations of the Plane with Statistic ModelsModelsBen McCabeAdvisor: Rolfe PetschekDepartment of Physics, Case Western Reserve University
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.
AbstractAbstract
IntroductionIntroductionWe 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
Where Z is the partition function and is the inverse temperature (defined as the reciprocal of the product of the Boltzmann constant and the absolute temperature)
MethodsMethodsThe 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.
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 ModelIsing ModelThe 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
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 then we can define a new variable K=-J. When h=0 there is a phase transition for the triangular lattice at
At which the model exhibits fractal looking patterns that can be considered aesthetically pleasing.
AcknowledgementsAcknowledgements
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
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
I would like to thank Professor Rolfe Petschek for his continued guidance and support, as well as Professors Gary Chottiner and Kenneth Singer.
j
E
E
sjs
is
i
eZ
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A tessellation by M.C. Escher
i
iiji
jiij ShSSJH
275.06tanln21
cK
Six-State Clock ModelSix-State Clock Model
Output of the program for the Ising model with no external magnetic field at the phase transition
jiji SSKH
3cos6
ji
ji SSK cos2
jiji SSK
3
2cos3
Output of the program for the six-state clock model with K6=-0.35, K2=-0.3, and K3=0.0
