Math 11300, Winter 2007
Studies in Mathematics- Geometry

Office hours (Official): Wednesday 4:30-6:00 PM and Thursday 4:30-6:00 PM Math/Stat 202 (here's a map) Also I'm happy to meet other times, just send me an email to make an appointment.

Tutorial: T/Th 12-1:20 PM GB 502 (Tutorial has begun!)

Office Hours before Final

Friday March 9th 9:30-10:20AM in classroom
Monday March 12 4:30-6 PM in my office
Tuesday March 13th (TBA on Friday given your preferences)

Short Essays

Due Wednesday March 7th

How are graphs generalizations of the polygons and polyhedra we've studied this quarter? Why did we need to define distance on graphs in order to study symmetries of graphs? How did we define distance? How do symmetries and automorphisms of graphs relate? How are the ideas of graph isomorphism and group isomorphism similar?

Finally, what are some applications of graph theory to "real life"? What did we learn about Eulerian and Hamiltonian circuits? Planar graphs? ** In particular, which of these problems are "easy to solve"?

Due Wednesday February 28th

The essay topic for the week is symmetries of polyhedra. Discuss our main observations about these groups, and be sure to address the following topics. 1) What's the difference between an "impossible position" and symmetries that we cannot visualize carrying out in 3D (such as flips or rotoreflections). 2) What are the generators for these groups? 3) Orientation preserving vs reversing symmetries. 4) The two main ways to count these symmetries. 5) Also, compare and contrast the symmetries of the tetrahedron vs. the cube.

No essay for Wednesday February 21st- Homework due instead

Due Wednesday February 14th (Happy Heart Day!)

Please discuss permutations in this weeks essay. What is a permutation, and what is S_n? (Explain what the elements of S_n are and what the operation between elements is.) How many elements are in in S_n? Also, what different ways are there to write permutations, and why are they useful? Finally, why are we studying permutations in this geometry class?

No essay for Wednesday February 7th- See homework instead

Due Wednesday January 31st:

This essay should focus on what we have learned about symmetries viewed as a group. Be sure to mention why symmetries of an object form a group and the definition of generators, subgroup, cyclic group, and isomorphism. Connect these definitions with the symmetry groups we considered. Also discuss how you prove two groups are or are not isomorphic and why isomorphism is an important concept.

Due Wednesday January 24th:

I will write the essay this week, but you will fill in the mathematical details. See homework section.

Due Wednesday January 17th:

Please give a brief (brief!) summary of what has happened in the class so far (up to Friday). In particular, mention any main definitions and statements of theorems. What have our main goals been? Also address the following questions in your essay:

1) Give an intuitive informal definition of symmetry. Reconcile your definition with our formal one. Also notice that every symmetry has an inverse. (For example if you rotate a triangle counterclockwise 120 degrees, you could return it to its original position by rotating it clockwise 120 degrees). Last quarter, before we could talk about inverses, we always had to have an identity (Why?). What then is the identity symmetry? Does it fit our definition of a symmetry?

2) What's the point of using a compass and straightedge (Euclid's dead! as Morgan told us)? Why not use a ruler to measure lengths and a protractor to measure angles?

Homework

Due Wednesday March 7th (either in class or in office hours):

Office hours this week: MONDAY and Wednesday 4:30-6:00 PM
Assigned Monday:
1) Finish reading 14.3
2) Chapter 14, Exercise 14.6
3) Prove that K_5 is not planar following the same argument we did for K(3, 3) in class.

Assigned Friday:
1) Start reading 14.3
(Some of these problems include material on planar graphs, which we will cover Monday)
2) Chapter 14, Exercise 14.3 Look for routes which are Eulerian Circuits (begin and end at same vertex) and separately Eulerian trails (can begin and end at different vertices). See section 14.2 for more information.
3) Chapter 14, 14.4, 14.5, 14.15
4) What is Sym(G) for the graph given by Map 1 in 14.3? What is Sym(G) for the underlying graph (with each edge being counted as 1 distance unit, not the distances in the picture) of the map in 14.5?

Due Friday March 2nd:

Assigned Wednesday:
1) Prove that if f is in Sym(G), then f is in Aut (G) (This is the direction of the proof we did not do in class).
2)14.13, 14.16 (See p. 384 for definition of complement)
3) Let G=(V, E) where V={A, B, C, D} and E={(A, B), (A, C), (B, C), (B, D), (C, D)}. Calculate Aut(G)=Sym(G). What group is this isomorphic too?

* FOR TUTORIAL: Practice calculating the distance between two points and calculating Aut(G) for various graphs.

Assigned Monday:
1) Read 14.1, 14.2
2) Draw the following graphs G=(V, E) where V={A, B, C, D} and E={(A, B), (C, B), (B, D)} and G'=(V', E') where V'={F, H, I} and E'={(F, H)}
3) An automorphism of a graph G is an isomorphism function f from G to itself. First list all the automorphisms of the graphs you drew in part 2. Second, show that the set of automorphisms of any graph G is a group with the operation composition of functions. Third, what groups are isomorphic to the groups of automorphisms of the graphs in part 2? Prove it!
4) Draw all the non-isomorphic graphs with four vertices.
5) Exercise 14.7

Due Monday February 26th:

1) Construct your semi-regular or star polyhedron.
2) Read (at least) the first three pages of Chapter 14.

Due Wednesday February 21st:

1) Read Chapter 13.3 and 13.4 Make sure you can do the practice problem!
2) Explain why any permutation of the cube gives rise to a permutation of the diagonals of the cube.
3) For each kind of rotation we found for the cube, pick a representative axis for this rotation. Given that axis, write all the rotational symmetries around that axis as permutations. Then, given each of these rotational symmetries around that axis, write the corresponding permutation on the diagonals of the cube. (Label the diagonals as on p. 325, 326 in the book )
4) What permutations of the diagonals can you get from rotational symmetries of the cube?
5) Describe some subgroups of the group of rotational symmetries of the cube.

Due Friday February 16th:

Assigned Monday:
1) Read Chapter 13.2
2) 13.1 (if you haven't done it yet), 13.2, 13.3, 13.5, 13.6, 13.9 (Just see if you can figure out the formula, no proof needed.) 13.13

Due Thursday February 8th (in tutorial):

Assigned Monday:
1) Read Chapter 13.1
2) Thursday Tutorial Work (Don't have to turn in): 12.7, 12.8, and if time 12.2, 12.9

Assigned Wednesday:
1) Read/review Chapter 12
2) Prove that the group S_n of permutations is closed under composition.
3) Prove that the group S_n has inverses (Make sure to check the order of composition does not matter.)
4) (Material for these questions will be covered Friday.) 12.1, 12.3, 12.4, 12.5, 12.6

Due Friday February 2nd:

Also get started on next week's homework (See above).
Assigned Monday:
1) Begin reading Chapter 12
2) Prove that Z_2 X Z_2 is isomorphic to the Klein 4 subgroup we considered today. (Make Joe explain Z_2 X Z_2 again)

Assigned Saturday:
1) Review Chapter 11
2) Exercises: 11.2, 11.4, 11.5, 11.6, 11.7, 11.8 (Some of this material we will cover Monday)

Due WEDNESDAY January 24th:

(This is the whole assignment...nothing was added Monday.)
Assigned Friday:
1) Read Chapter 11
2) Exercises: 11.1, 11.3, 11.9 (Before using your triangle to figure this out, think about whether the number of flips alone could answer this question...Try to come up with a conjecture and an explanation of why your conjecture works.)
3) Fill in details of my Tessellation Essay on problem 15.6.

Due Friday January 19th:

Assigned Wednesday:
Note! Problem 15.6 will be due as a major part of the essay for next week Wednesday (ie you don't have to turn it this Friday.) It IS a tough problem, but don't lose hope! I also have office hours Thursday afternoon.
1) Read 11.1
2) Exercises: 15.1, 15.2, Formalize the proof we talked through today that regular pentagons don't tessellate.

Assigned Friday:
1) Finish reading 15.1
2) Exercises: Chapter 15, 15.3, 15.6 (This last one is challenging; at least read it over before tutorial on Tuesday. It is an amazing result though. We saw that all 3-gons and 4-gons tessallate, and on Wednesday we'll talk about 5-gons and 6-gons. This problem will show that NO convex n-gon for n>6 can tessallate, a very powerful result.)


Due Friday January 12th:

Assigned Wednesday:
1) Begin reading 15.1 (We'll talk about quadrilaterals and tessalation Friday).
2) Exercises: Chapter 10, 10.14, 10.15, and Chapter 15, 15.2

Assigned Monday:
1) Read Chapter 10.2, the Appendix (namely SAS, SSS, ASA).
2) Chapter 10 Exercises: 10.2, 10.4, 10.6 (You can explain, not necessarily give a formal proof), 10.7, 10.9, 10.10, 10.11, 10.12, 10.13
Note: A lot of these problems you have to play/experiment for a bit before finding an answer. Just try things out first, and see how it goes.
3) Think about when you want office hours!

Due Friday January 5th:
1) Read Chapters 9 and 10.1, begin reviewing the Appendix (namely SAS, SSS, ASA).
2) Chapter 9 Exercises: 9.1, 9.2 b-f, 9.3, (9.7 is interesting to think about, but you don't have to do it).


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