Short essays require the student to reflect upon the topics we have covered. Their purpose is to help the student synthesize the information from class and then organization this information into a cohesive whole. The essays will then be graded so that the students can see if they have gained an appropriate understanding of the topic.

The due dates for all essays will be listed on the class schedule.

1. Solution to your assigned puzzle
   Write a clear description of the solution to one of the puzzles from section 1.4 that was assigned to your group. Include charts, graphs, drawing, etc. as appropriate to make the solution easier to understand. Use the solution to the puzzles in The Man Who Counted to guide you.
   
   
2. Euclid’s proof that there are an infinite number of primes
   • List the hypothesis (the assumption)
   • Give the mathematics that lead to a contradiction, explaining each statement.
   • State the contradiction that was reached.
   • Summarize why we have proven that there are an infinite number of prime numbers.
   
   
3. Aristotle's proof that the square root of 2 is irrational.
   • List the hypothesis (the assumption)
   • Give the mathematics that led to a contradiction, explaining each statement.
   • State the contradiction that was reached.
   • Summarize why we have proven that the square root of 2 is irrational.
     
     
4. Pythagorean Theorem
   Prove the Pythagorean theorem using these figures. Explain the how to construct the figures as well as giving the proofs. Assume you are given the right triangle.
   
   Bhaskara’s proof.
   
   
   
   
5. Flatland
   You are to focus on two topics to demonstrate the relevance of the book in today’s world. Draw a parallel between the treatment of the lower classes in Flatland and prejudice that exists toward a culture or race in today’s world.
   
   Explain several features of the fourth dimension by using Abbott’s analogies.
   
   
6. Infinity Explain why there are at least two types of infinity in mathematics.
   
   
7. Mathematicians describe three types of space: Euclidean, spherical, and hyperbolic. For each of these three types of space:
   • Describe an example of a surface in this space. Draw an example.
   • Describe triangles in this space. How many degrees do triangles have in this space.
   • Draw a line in your space. Set a point not on this line. How many lines go through this point parallel to your original line? Draw an example of this.