MAT 4930-2H22
Spring 2017
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Time: MWF period 7
Place: 219 Little
Phone: 352-294-2339
Office: 438 Little Hall
Email: avince@ufl.edu
Suggested Text:
Tilings and Patterns by Grunbaum and Shephard
Office Hours: Monday, Wednesday, Friday – period 8
(or by appointment)
Links
Homework
The class finds the Archimedean tilings
What isometry is the composition of two translation?
What isometry is the compostion of two reflections? (due Fri Jan 13)
List the symmetries of the square tiling.
Every convex quadrilateral admits a monohedral tiling of the plane. (due Wed Jan 18)
Show: A is closed if and only if, for each a in the complement of A there exists a disk centered at a such that D is contained in the complement of A.
O is open if and only if O^c is closed.
The union of any number of open sets is open; the intersection of any number of closed sets is closed.
The interior of a set is open.
The closure of a set X is the intersection of all closed sets containing X. (due Wed Jan 25)
Let G be a group of symmetries of a tiling T of the plane. Call points x and y equivalent if they are in the same orbit of G. Prove that this is an equivalence relation. (due Wed Feb 1)
The set of affine functions on the plane is a group.
Describe the following transformations geometrically: (a 0; 0 a) and (1 1; 0 1). (due Fri Feb 3)The point “group” of a group of isometries is a group. (Due Mon Feb 6)
Describe the orbifold and give the Conway symbol for the patterns given in class. (due Fri Feb 24)
Continue to find all possible Conway symbols with total cost 2. (due Wed Mar 1)
Checkerboard domino tiling problem.
Two domino tiling problems. (due Mon after spring break)
Can a 7×10 rectangle be tiled by copies of a 2×3 rectangle? A 17×28 rectangle by copies of a 4×7 rectangle? A 10×15 rectangle by copies of a 1×6 rectangle? (due Fri Mar 17)
Can a rectangular (checker) board with exactly one black and one white square removed be tiled by dominoes? (You can use Robert’s method.) (due Mon Mar 20)
The following are equivalent (1) n|a-b and (2) a and b have the same remainder on division by n.
Congruence modulo n is an equivalence relation, i,.e, the transitivity property holds. (due Mon Mar 20)
Domino switching problem. (due Fri Mar 25)
Postage stamp problem.
Two irreptile problems. (due Fri Mar 31)
How many domino tilings of a 2xn rectangle?
Prove the Fibonacci identity: [f(n)]^2 – f(n-1)f(n_1) = (-1)^n. (due Mon April 3)
Topics
A little history
What is a tiling?
A little point set topology
Normal tilings
Tiling by regular polygons
Regular and uniform (Archimedean) tilings
Polyominoes
Monohedral tilings by convex polygons
The group of isometries of the plane
The linear and affine groups
The symmetry group of a tiling
Translations subgroup and the lattice
The crystallographic restriction
Periodic tiling
Fundamental domain
Wallpaper (2-dimensional crystallographic) groups
Conway notation
Surfaces, Euler characteristic, and orbifolds
Classification of wallpaper groups
Tiling by a set P of prototiles
Does P admit a tiling? (dominoes)
How many tilings does P admit?
Relationship between tilings (domino flipping)
Tiling a rectangle by smaller rectangles
Perfect tilings, rep-tilings, and irreptilings?
Non-periodic tiling and aperiodic prototile sets
History: Wang tiles, aperiodic prototile sets
The golden ratio
Penrose tiling
Grades
problem
class presentations
attendance
Messages
Welcome to the Mathematics of Tiling
Students with disabilities requesting accommodations should first register with the Disability Resource Center (352-392-8565, www.dso.ufl.edu/drc/) by providing appropriate documentation. Once registered, students will receive an accommodation letter which must be presented to the instructor when requesting accommodation. Students with disabilities should follow this procedure as early as possible in the semester.
The course will be conducted in accordance with the University honor code, academic honesty policy, and policy regarding the use of copyrighted material.
Requirements for class attendance and make-up exams, assignments, and other work in this course are consistent with university policies that can be found at: https://catalog.ufl.edu/ugrad/current/regulations/info/attendance.aspx.
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