EXAM I, Informal Geometry Spring 2004

1. [20]i. Define what a tessellation is.

ii. Define a regular tessellation.

iii. What is the measure of an interior angle in a regular polygon? Explain how we would derive that formula.

iv. Explain as carefully as you can why there are only three regular tessellations.

2. [15] Sketch a proof of the fact that all quadrilaterals tessellate. Give as much detail as you can. (Drawing pictures might help getting your point across.)

3. [15] i. Explain what rotational and reflectional symmetries are.

ii. What types of rotational symmetry do you see in the Angel and Bats tessellations?

List all, and say where they can be seen (for instance: "at the elbow" if that is where the rotation takes place)

If there is reflectional symmetry, then mark all visible mirror lines in the tessellation.

ii.i What types of rotational symmetry do you see in the Lizard and Bats tessellations?

List all, and say where they can be seen .

If there is reflectional symmetry, then mark all visible mirror lines in the tessellation.

4. [20] i. Explain in your own words how isometries can lead to tessellations.

ii. What is a fundamental domain?

iii. What is a torus?

iv. What is a Klein Bottle?

v. How can we create tessellations on the torus and the Klein Bottle?

5. [20] True or False. If false, explain why.

a. Every square is a rectangle.

b. Every equilateral triangle is an isosceles triangle.

c. Every rhombus is a parallelogram.

d. The sum of the angles in a triangle is 180¡.

e. Every diagonal bisects a quadrilateral into two triangles.

f. There exists a diagonal that bisects a quadrilateral into two triangles.

6. [10] Show, giving a detailed explanation, that if the opposite angles in a quadrilateral have equal measure, then that quadrilateral has to be a parallelogram.

7. [15] Draw a tessellation using this tetromino. Make sure that there is a clear pattern.

Draw a tessellation using this tetromino. Make sure that there is a clear pattern.

Draw a tessellation using this tetromino. Make sure that there is a clear pattern.

8. [10] Given a geometric tessellation (square or parallelogram) create an Escher-like tessellation. (I will leave it up to you how (un)complicated to make this.)

9. [10]

This is a famous print by Salvador Dali ("Cinquenta...tigre Real"). What are the geometric shapes that make up the tessellation? Is the tessellation regular? Is it semi-regular? Explain.

This painting by Mondriaan ("Composition A: Composition with Black, Red, Gray, Yellow, and Blue 1920") is also a tessellation. What are the geometric shapes that make up the tessellation? Is the tessellation regular? Is it semi-regular? Explain