Assignment 5

November 11

Assignment 5

Transformation Exercises | Finite Geometries Exercises

Exercises

Transformational Geometry

1) The product of two transformations is defined to be one transformation followed by another. What is the product of a rotation of a figure A about an angle with a rotation of the figure A about a second angle?
2) Give an example in which a segment and its image are parallel under reflection.

3) Describe the inverse of a glide reflection.

4) What specific transformation could be used in a proof of each of the following:

a) For an isosceles triangle, the medians to the two congruent sides are congruent.
b) If two parallel lines are cut by a transversal, then the corresponding angles are congruent.

5) Create a tessellation and explain which transformation you used to create it. Your tessellation could be an Escher or a Penrose tessellation or a semi-regular not covered in class or a nonuniform or non-periodic or a tessellation using a non-regular polygon of your choice.

6) Add your name to your tessellation and do a Save As under the File menu. Save your tessellation as a HTML/JavaSketchpad Document. Then e-mail it to me so that I can put them all up on the web. Here is an example of how this will work. (NOTE: This will only work using IE in the Math lab! It works fine with Netscape or IE outside of that lab.)

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Finite Geometries

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1. For the three point geometry:

a) Exactly how many points are on each line?
b) Do any squares exist?
3) Prove that a line cannot contain three distinct points.

2) For the four point geometry:

a) Draw another model for this geometry different from those presented.
b) Rewrite the set of axioms for this geometry using tree for point and row for line.
c) If the points are A, B, C, and D, name all sets of parallel lines.

3) For the four line geometry:

a) Draw a different representation for this geometry from those presented.
b) Do each two points of the geometry lie on a common line?
c) How many triangles exist in the geometry in which all three sides are lines of the geometry?

4) For the 5 point geometry:

a) Which axioms are also true statements in Euclidean geometry?
b) How many triangles exist in the geometry in which all three sides are lines of the geometry?

5) For Fano's geometry:

a) Rewrite the set of axioms for Fano's geometry using book for point and library for line. Does this representation of the geometry give your more understanding of the geometry?
b) Prove that each point lies on exactly three lines.

6) Choose one of the models for a finite geometry given below and develop your own set of axioms and state at least one theorem for your geometry.

7) Answer the questions about Desargues' Finite Geometry.