Assignment 8B (Due Monday 11/25)

8-5 Dissections and Tiling

You got a handout with a tiling by one big square superimposed on a tiling by two squares. 

·        In one corner of the sheet, indicate with little squares the 90-degree centers of symmetry and with rhombi the 180-degree centers. 

·        Call one of the 90-degree centers A. Some of the other 90-degree centers are translations of A by translations that are symmetries of the pattern.  Color all these centers red (or circle with red or something).  Color the other ones blue.

·        Now in another corner of the page, mark some of the centers again and draw or cut out a square whose vertices are red 90-degree centers.  Also, draw or cut out a separate square (not overlapping) whose vertices are blue centers.

·        If the sides of the squares are a and b, what is the size of the squares you cut out?  How do these squares relate to the Pythagorean theorem?

8-6 Report From Lab 8

Write up and turn in the analysis of Patterns 2 and 3 from the Lab.  Use a printout or graph paper for your pattern.

8-7 Generating Patterns

Draw points A and B on paper.  Let S = A₁₈₀ and T = B₆₀. 

• Construct (with straightedge and compass or Sketchpad) the centers of the rotations ST and TS.
• Construct the other nearby centers of symmetry of a pattern created by repeating S and T over and over as in Lab 8.
• Also, tell what isometry ST³ is and give an exact description.
• Tell what are the translations in this pattern.

8-8 Fibonacci and Golden Ratio

• Learn the definition of the Fibonacci numbers and why the ratios of Fibonacci numbers approximate the Golden Ratio.  References:

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#Rabbits

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#golden

• Read about the puzzle below that proves 64 = 65.  Make a copy of the 8, 13, 21 puzzle on graph paper and cut it out and check it out.

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibpuzzles2.html#jigsaw1

• Challenge:  Give a good explanation of the true dimensions of the rectangle and the reason that Fibonacci numbers are used in the puzzle.