In addition to the homework exercises for each week, you should know the names, very rough dates of activity, and relevant notable facts about the mathematicians we've mentioned in class that week. ## Homework 12, due Apr 26 (Wed) at 1pm

Stillwell p99: 6.5.2, 6.5.3

Stillwell p103: 6.7.1, 6.7.2, 6.7.3## Homework 11, due Apr 19 (Wed) at 1pm

Read the exercises on p94-95, but do not do them. Then do the following problems:## Homework 10, due Apr 7 at 1pm

Stillwell p80: 5.5.2, 5.5.3, 5.5.4, 5.5.5

Stillwell p91: 6.2.1## Homework 9, due Mar 31 at 1pm

Stillwell p73: 5.2.1

Stillwell p75: 5.3.1, 5.3.2

Stillwell p77: 5.4.1

Stillwell p80: 5.5.1## Homework 8, due Mar 22 at 1pm

Stillwell p58: 4.2.1, 4.2.2, 4.2.3, 4.2.4, 4.2.5

Stillwell p61: 4.3.2## Homework 7, due Mar 3 at 1pm

Stillwell p47: 3.4.1, 3.4.2, 3.4.3, 3.4.4

Stillwell p49: 3.5.1## Homework 6, due Feb 24 at 1pm

Stillwell p43: 3.3.1, 3.3.2, 3.3.3, 3.3.5 ## Homework 5, due Feb 17 at 1pm

Stillwell p41: 3.2.1, 3.2.2

Stillwell p43: 3.3.4## Homework 4, not to be collected

Stillwell p27: 2.3.3

Stillwell p30: 2.4.1, 2.4.2

Stillwell p41: 3.2.3## Homework 3, due Feb 3 at 1pm

Stillwell p24: 2.2.2, 2.2.3

Stillwell p27: 2.3.1, 2.3.2## Homework 2, due Jan 27 at 1pm

Stillwell p8: 1.3.3

Stillwell p11: 1.4.2

Stillwell p20: 2.1.1, 2.1.2## Homework 1, due Jan 20 at 1pm

Stillwell p5: 1.2.1, 1.2.2, 1.2.4 (see also 1.5.2)

Stillwell p8: 1.3.1, 1.3.2

Stillwell p13: 1.5.2 (see also 1.2.4)

Stillwell p103: 6.7.1, 6.7.2, 6.7.3

**A.** i) Give an example of a shellable 2D simplicial complex with *V* - *E* + *F* = 3.

ii) Explain why *V* - *E* + *F* > 0 for any shellable 2D simplicial complex.

Hint: Problem A from last week might be helpful!

Stillwell p96: 6.4.1, 6.4.2, 6.4.3, 6.4.4, 6.4.5

As well as the following additional problems:

**A.** Let *S* be a shellable 2D simplicial complex, and let *T*_{1}, *T*_{2}, ..., *T*_{m} be a shelling order of its faces (triangles).

Describe how to use the shelling order to compute the number *V* - *E* + *F*, as studied by Euler for polyhedra.

(See also part ii of problem B.)

**B.** i) Find a shelling of the surface of the octahedron.

ii) Euler proved that *V* - *E* + *F* = 2 for any polyhedron. Verify that your method from Problem A gives 2 for your shelling of the octahedron!

Stillwell p91: 6.2.1

Stillwell p75: 5.3.1, 5.3.2

Stillwell p77: 5.4.1

Stillwell p80: 5.5.1

Stillwell p61: 4.3.2

Stillwell p49: 3.5.1

And the following additional problem:

**A.** What is golden about the golden ratio?

i) Show that if you apply the anthyphairesis process to the golden ratio τ = (1 + sqrt(5)) / 2, then it repeats after a single iteration.

ii) Show that τ is the only number greater than 1 so that anthyphairesis repeats after a single iteration. (Hint: set up a quadratic equation!)

Hint on 3.4.2 and related problems: it may be helpful to keep track of whether you are "breaking off a square" horizontally or vertically. That is, it may be helpful to keep track of whether the square of anthyphairesis is to the left or to the top.

It may also be helpful to model 3.4.1 with squares and rectangles.

Stillwell p43: 3.3.4

And the following additional problem:

**A.** Hexagonal numbers

i) Following the pattern for triangular, square, and pentagonal number, give the definition of the *n*th hexagonal number. (Your definition should involve a big Σ, or at least a sum.)

ii) Find an explicit formula for the *n*th hexagonal number.

Stillwell p30: 2.4.1, 2.4.2

Stillwell p41: 3.2.3

Although this homework won't be collected, of course it is excellent practice for the exam!

Hint: There are several ways to approach 3.2.3. If you've seen induction before, that is one method. A more direct method is to write each pentagonal number as a sum of a square number and a triangular number.

Stillwell p27: 2.3.1, 2.3.2

**A.** Calculate the angle defect at each vertex of the polyhedron obtained by "gluing" two regular tetrahedra together along a face. Verify that Descartes' formula holds.

**B.** Calculate the angle defect at each vertex of a (square) pyramid with base sides 1 and height *h*. Verify that Descartes' formula holds.

Stillwell p11: 1.4.2

Stillwell p20: 2.1.1, 2.1.2

**A.** Describe the smallest convex 2-dimensional region that contains the points (0, 0), (1,2), (-1, 2), (2, 2), (-2, 2), (1, 3), (-1, 3), and (0, 4). Explain carefully why your answer is correct.

Stillwell p8: 1.3.1, 1.3.2

Stillwell p13: 1.5.2 (see also 1.2.4)