Assignment 1 has been released, and can be obtained by clicking on the image or on this link. It will be due on **January 21st at 11:00pm**. Solutions can be submitted either electronically (cc’ing both kmcrane@cs.cmu.edu and nsharp@cs.cmu.edu), or physically either in class or outside 215 Smith Hall. (Note that Smith Hall is typically locked in the evening, so if you want to submit physically you should plan to do so a bit earlier!) Solutions will be graded not only on correctness, but also on *legibility* and *quality of exposition*—part of our goal this term is to help you become better mathematical communicators; solving problems in a vacuum helps nobody. We *will* take points off for terrible handwriting—for those of you who suffer from this affliction, we strongly suggest you use $\LaTeX$. Please make sure to put both your name and your Andrew ID on your submission.

The coding portion of this assignment is forthcoming—basically you will just need to get the code skeleton up and running on your machine, and we will be glad to assist you (potentially in a recitation session).

* If you don’t know where to get started* with some of these proofs, just ask! We are glad to provide further hints, suggestions, and guidance either here on the website, via email, or in person. Office hours are still TBD, but let us know if you’d like to arrange an individual meeting. Also, don’t be afraid to chat with your classmates—you will learn much more together than you will in isolation.

Good luck!

Hi,

I got stuck solving Exercise 8.

According to the hint, I guess the result of Exercise 7 would be useful. However, I don’t know how to get the connection between Exercise 7 and those interior angles at the vertex. Could anyone give me further hint?

Thank you so much!

Best,

Derek

Hi Derek,

The key here is to think about the relationship between angles on the polyhedral surface (left image) and angles in the spherical figure (right image). There are actually a couple interesting relationships here; one of them is the key to this exercise!

Let us know if you need more help (I’m trying not to give too much away the first time through. :-))

-Keenan

Small hints: cross product can be very useful!

As well as the following property:

$$(\mathbf{a} \times \mathbf{b}) \times (\mathbf{a} \times \mathbf{c}) = (\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}))\mathbf{v}$$

There is a typo, $\mathbf{v}$ should be $\mathbf{a}$.

Thank you for all the help :))

Derek

For Exercise 4 the word “minimum” suggests that we not only need to prove the number of irregular valence vertices is at least m(K), but that we also need to construct surfaces of genus g that achieve the minimum. Is this correct? (In particular, this means the problem is asking us to also construct simplical surfaces of genus $g\ge 2$ with exactly 1 irregular valence vertex.)

You should be able to show that the bound holds algebraically. However, if you’re interested in tessellations that actually achieve this bound, then see this post, which has some nice pictures.