Dr. Cleary's Math 360 S26 Homework Assignments:

• HW1, due Tues, Feb 3rd
  Consider the following axiom set.
  • Axiom 1. Every team plays at least two games.
  • Axiom 2. Every game has at least two teams.
  • Axiom 3. There exists at least one team.
  1. What are the undefined terms in this axiom set?
  2. Prove Theorem 1. There exists at least one game.
  3. What is the minimum number of games? Prove.
  4. Find two models that show that the statement "There are exactly two teams" is independent of the axioms.
  5. Find two models that show that the statement "Every team plays every other team" is independent of the axioms.
• HW2, due Tues, Feb 10th
  • Stahl, 1.2 #2: make precise statements of Euclid's propositions 5 to 9 using modern terminology and notation
  • Stahl, 1.2, #10: prove that the angle bisectors of a triangle are concurrent (meet in a single point.) Does this hold in neutral geometry as well?

• HW3, due Thur, Feb 19th
  • Give an example of a statement using only the undefined terms of incidence geometry which is independant of the incidence axioms which is shown to be independent by the model of the usual Euclidean plane and the integer lattice plane: Points = (a,b) with both a and b integers, lines are of the form (y-y0)=m(x-x0) where x0 and y0 are integers and m is rational, and vertical lines at x=n, for n an integer.
  • Which of Hilbert's axioms hold for the integer lattice model of incidence geometry? You may assume that undefined terms in the integer lattice are interpreted the same way as in the Euclidean plane ("between", "same side", "congruent", etc.) and you may ignore the three-dimensional axioms that refer to planes and points in three-dimensional space.
  • Stahl, 2.1 #1
  • Stahl, 2.2 #1a

• HW4, due Thurs, Feb 26th
  • Stahl, 2.2, #1ceghj, #2, #8, #11, #14
  • Stahl, 2.3: #2, #5abdfgi

• HW5, due Thurs, Mar 5th (day of exam 1)
  • Stahl, 2.4: #1, #2,#4,#8
  • For each of the six wallpaper groups on the handout, find the wallpaper pattern with the same symmetries on the reference sheet.

• HW6, due Tues, Mar 10th
  • 3.1: #1, 2, #3acdfhij, (not #4, postponed to Thursday)

• HW7, due Thurs, Mar 12th
  • 3.1: 4abcdf

• HW8, due Thurs, Mar 19th
  • 3.1: #5, #9, #10
  • 4.1: #1, #2, #3
• HW9, due Thurs, Mar 26th
  • 4.2: #1-4
  • 4.4: #1-4
• HW10, due Thurs, Apr 16th
  • Read E. Abbott Abbott's Flatland from 1884, available online in many places, for example here.
  • 4.5: #1, #2, #4
  • Construct a 45-45-? triangle using the same method we used in class, where the vertices are at A=(0,1), B=(0,b) for b>1, and C, a point with x>0. Find the third angle C in the triangle if the first two are each pi/4.

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