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Time:      MWF period 8

Place:     Little 221

Phone:   352-294-2339

Office:    438 Little Hall

Email:    avince@ufl.edu

 

Textbook: 

Office Hours:        Monday, Wednesday, Friday  –  period 6

                                            (or by appointment)

 

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Links

 

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Homework

    Show that the sum of the angles of an n-gon is (n-2)pi.
    Show (by pictures) that any convex quadrilateral admits a tiling of the plane.   (due Monday 27 Aug)
    List the symmetries (translations, rotations, reflections glides) of the square tiling of the plane.    (due Fri 31 Aug) 

    Finish the list (above) that we started in class.

    Express the following transformations in matrix notation:  
(a)  30 degree counterclockwise rotation about the origin, 
(b)  60 degree clockwise rotation about the origin,
(c)  reflection in the diagonal line y=x,
(d)  the composition of a 90 counterclockwise rotation about the origin 
and a reflection in the y-axis
(two problems: first the rotation, then the reflection; 
first the reflection, then the rotation).

    Show that AFF(2) is closed under composition.  

    Chess board domino tiling puzzle.    (due Fri. 7 Sept)

Find an example of:
   (a)  a tiling with only one symmetry (the identity).
   (b)  A tiling with a translational symmetry in only one direction.
   (c)  A tiling whose translation subgroup equals the symmetry group.   (due Mon 10 Sept)


Find the orbits of the points given in class under the action of the symmetry groups.

What are the possible periods of rotational symmetries of the square tiling? 
the hexagonal tiling?    (due Mon 24 Sept)

Group project for the week of Sept 17-23:

    Classify the edge-to-edge tilings of the plane by regular polygons 
such that the cyclic pattern at each vertex is the same.  Sketch as many
as you can.  This is due on Fri 5 Oct. 

Although this is a group project, each student should write up the results 
her/his self - to be turned in on Mon 24 Sept. A presentation 
in class by all six students should also take place on Mon 24 Sept. 

Orbit problem for square tiling continued  (due Wed 26 Sept)

Find the length of a diagonal of a regular pentagon of side length 1.  (due Mon 8 Oct)

    Can a 7x10 board be tiled by 2x3 tiles?
    Can a 17x28 board be tiled by 4x7 tiles?
        (due Mon 15 Oct)

Prove: For integers a,b,n prove: n|(a-b) if and only if a and b have the
same remainder upon division by n.   (due Wed 17 Oct)

Find the lengths of the two tiles in cut and project tiling of the line from class. 
What are the coordinates of the endpoints of the tiles in 1-dim tiling discussed in class? 
(due Fri 19 Oct)

Prove that conditions (1) and (2) from class imply that the axb rectangle can be tiled
by copies of the cxd rectangle.   (due Mon 22 Oct)
 
Explain, to the class, one thing about the Cantor set, Koch curve, or Sierpinski triangle. 
    (starting Wed 31 Oct)

L-system example  (due Fri 9 Oct)
Experiment with online software (note: using weitz.de/lindenmayer/ F goes forward; any 
   other symbol does nothing. Try:  
      F -> FF 
      X -> --FXF++FXF++FXF-- 

     Axiom: FXF--FF--FF
     Angle: 60  
     StartAngle: 0

Find the fixed point of the affine function given in class.  (due Fri 16 Nov)

Find an affine iterated function system whose attractor is the Koch curve.  the Sierpinski triangle.  (due Mon 19 Nov)
   
Due by Mon 26 Nov (L-system homework)

Iterated function system and chaos game attractors (experiment with the IFS Construction Kit software)

Using the definition, find the dimensions of the five examples from class.  (due Wed 28 Nov).

Find the box counting dimension of {0,1,1/2,1/4,1/8,...}.  (due Fri 30 Nov)

L-systems and IFS assignments due by noon, Fri 7 Dec.  

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Topics

• Tilings
  • A little history
  • What is a tiling?
  • Tilings by polygons
  • Symmetry and periodicity of tilings
  • Non-periodic tilings and aperiodic prototile sets
  • Penrose tilings
  • Quasicrystals
  • Matching rules, substitution, cut and project
  • Tilings by rectangles
• Fractals
  • Examples of fractals
  • L-systems
  • Iterated function systems
  • Fractal dimension
  • The Mandelbrot set
• Tiling with Fractals
  • Construction of fractal tilings
  • Properties of fractal tilings

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Messages

 

Welcome to Tilings and Fractals 

Free tutoring at the Teaching Center, SW Broward Hall. Check Teaching Center for the time schedule.

Students with disabilities requesting accommodations should first register with the Disability Resource Center (352-392-8565) 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 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: attendance policies.

Information on current UF grading policies for assigning grade points may be found at: grades.

Students are expected to provide feedback on the quality of instruction in this course by completing online evaluations. Evaluations are typically open during the last two or three weeks of the semester, but students will be given specific times when they are open. Summary results of these assessments are available to students at https://evaluations.ufl.edu/results/.

Grades

Grades are based on weekly assignments, class presentations, and attendance.

 

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