## 311 – HW 8

April 23, 2015

HW 8 is due Tuesday, April 28, at the beginning of lecture.

Work in hyperbolic geometry. Given a triangle $\triangle ABC$, recall that its Saccheri quadrilateral $\Box ABB'A'$ based at $\overleftrightarrow{AB}$ is defined as follows: Let $M$ be the midpoint of $\overline{AC}$ and $N$ be the midpoint of $\overline{CB}$. Let $A',B'$ be the feet of the perpendiculars from $A$ and $B$ to $MN,$ respectively.

Continuing with the same notation, suppose now that $G$ is an arbitrary point on $\overleftrightarrow{MN}$, and let $H$ be a point on the ray $\overrightarrow{AG}$ with $GH=AG$. Show that $\Box ABB'A'$ is also the Saccheri quadrilateral of $\triangle ABH$ based at $\overleftrightarrow{AB}$.

## 311 – HWs 6 and 7

April 15, 2015

HW 6 is due Thursday, April 16 and HW 7 is due Tuesday, April 21, both at the beginning of lecture.

HW6

Work in hyperbolic geometry.

1. Let $\ell$ and $m$ be two parallel lines admitting a common perpendicular: There are points $P\in\ell$ and $Q\in m$ with $\overleftrightarrow{PQ}$ perpendicular to both $\ell$ and $m$. Suppose that $A,B$ are other points in $\ell$ with $P*A*B$, that is, $A$ is between $P$ and $B$. Let $C$ be the foot of the perpendicular from $A$ to $m$, and let $D$ be the foot of the perpendicular from $B$ to $m$.

Show that $PQ. That is, $\ell$ and $m$ drift apart away from their common perpendicular.

(Note that $\Box PACQ$ and $\Box PBDQ$ are Lambert quadrilaterals, and therefore $PQ and $PQ. The problem is to show that $AC.)

As an extra credit problem, show that for any number $r>0$ we can find $B$ (on either side of $P$) such that $BD>r$, that is, $\ell$ and $m$ not just drift apart but they do so unboundedly.

2. Now suppose instead that $\ell$ and $m$ are critical (or limiting) parallel lines, that is, they are parallel, and if $Q\in m$ and $P\in\ell$ is the foot of the perpendicular from $Q$ to $\ell$, then on one of the two sides determined by the line $\overleftrightarrow{PQ}$, any line through $Q$ that forms with $\overleftrightarrow{PQ}$ a smaller angle than $m$ does, cuts $\ell$ at some point.

On the same side as just described, suppose that $C,D$ are points on $m$ with $Q*C*D$, that is, $C$ is between $Q$ and $D$. Let $A$ be the foot of the perpendicular from $C$ to $\ell$, and let $B$ be the foot of the perpendicular from $D$ to $\ell$.

Show that $PQ>AC>BD$. That is, $\ell$ and $m$ approach each other in the appropriate direction.

As an extra credit problem, show that for any $r>0$ we can choose $D$ so that $BD. That is, $\ell$ and $m$ are asymptotically close to one another. Do they drift away unboundedly in the other direction?

HW 7

Show that the critical function $\kappa$ is continuous. Recall that $\kappa:(0,\infty)\to(0,\pi/2)$ measures the critical angle, that is, $\kappa(x)=\theta$ iff there are critical parallel lines $\ell$ and $m$ and a point $Q\in m$ such that if $P$ is the foot of the perpendicular from $Q$ to $\ell$, and $PQ=x$, then $m$ and $\overleftrightarrow{PQ}$ make an angle of measure $\theta$ in the appropriate direction.

(In lecture we verified that $\kappa$ is strictly decreasing. This means that the only possible discontinuities of $\kappa$ are jump discontinuities. We also verified that $\kappa(x)$ approaches $0$ as $x\to\infty$, and approaches $\pi/2$ as $x\to0$. It follows that to show that $\kappa$ has no jump discontinuities, it suffices to verify that it takes all values between $0$ and $\pi/2$, that is, one needs to prove that for any $\theta\in(0,\pi/2)$ there is an $x>0$ such that $\kappa(x)=\theta$.)

## 311 – HW5

March 31, 2015

(I am counting as HW3 the homework exercises supervised by Sam Coskey during the two weeks following Isabel’s birth, and as HW4 the two written exercises assigned by Sam that I collected on March 17.)

This exercise is due April 7 at the beginning of lecture.

Provide a proof verifying that the function $d:\mathbb R^2\times\mathbb R^2\to\mathbb R$ given by

$d((x,y),(z,w))=\sqrt{2(x-z)^2+2(y-w)^2-3(x-z)(y-w)}$

is a distance function.

## 311, 403/503 – Update

March 1, 2015

Our daughter decided to rush things up a bit, and is being born right now. You probably will have someone else covering class for a couple of weeks, sorry for the inherent inconvenience. I’ll be updating as I know more.

[Edit (11:00am): Isabel. Office hours are cancelled March 3 and 10.]

## 311 – Another extra credit exercise

February 11, 2015

This optional set is due Tuesday, February 24, at the beginning of lecture. Some of these problems are harder than others, you do not need to address all of them. Contact me if you would like additional time to keep working on some of the more difficult ones.

Recall the theories $T_n$ introduced in homework 2:

• For any two distinct points passes a unique line.
• Any line has at least $n$ points.
• There are three non-collinear points.

Recall also the axioms of projective geometry:

1. For any two distinct points passes a unique line.
2. Any two distinct lines meet at a point.
3. There are at least four points, no three of them collinear.

Theorem ($T_{n+1}$). There are at least $n^2+n+1$ points.

Proof. There is a line $\ell$ with at least $n+1$ points $a_1,\dots,a_{n+1}$, and there is at least one point $a$ not in $\ell$. The lines $\overleftrightarrow{aa_1},\dots,\overleftrightarrow{aa_{n+1}}$ are all different, and each of them has at least $n-1$ points other that $a$ and the $a_i$. It is not hard to see that all these points are different, giving us at least: $n+1$ points in $\ell$, $n-1$ points in each of the lines $\overleftrightarrow{aa_i},$ and $a$, that is, $(n+1)+(n-1)(n+1)+1=n^2+n+1$ points. $\Box$

Recall that if $P$ is a finite model of the projective axioms, then there is a natural number $n$ (the order of $P$) such that $P$ has exactly $n^2+n+1$ points and $n^2+n+1$ lines, each point belongs to exactly $n+1$ lines, and each line goes through exactly $n+1$ points. (See here for a reference with proofs of these facts, from Perspectives on projective geometry, by Jürgen Richter-Gebert.)

• Show that the only model of $T_3$ with $7$ points is the Fano plane.
• Show that all finite projective planes of order three are isomorphic (an example is described here).
• Is any model of $T_4$ with $13$ points the finite projective plane of order three?
• Is any model of $T_5$ with $21$ points the finite projective plane of order four? More generally, if $n>1$, is any model of $T_{n+1}$ with $n^2+n+1$ points a finite projective plane of order $n$?

## 311 – Extra credit exercise

February 10, 2015

This optional exercise is due Thursday, February 12, at the beginning of lecture.

Show that the projective plane $\mathbb Z_2P^3$ obtained through the vector space construction applied to the field $\mathbb Z_2$ of two elements is (isomorphic to) the Fano plane.

I review the construction: To form $KP^2$ we start with a field $K$ and consider $K^3$.

A point in the projective plane is just a line in $K^3$ through the origin. Note that any such line is obtained by fixing a non-zero element $(x,y,z)$ of $K^3$ and considering all its scalar multiples, all the vectors $(\alpha x,\alpha y,\alpha z)$ with $\alpha\in K$.

(When $K=\mathbb Z_2$, the only values of $\alpha$ are $0$ and $1$, so a line consists of only two points, one of which is the origin.)

A line in the projective plane is actually the collection of lines contained in a plane in $K^3$ through the origin. In other words, a set $L$ of points (that is, of lines through the origin) is a line iff there is a plane in $K^3$ that contains the lines that are the elements of $L$ and no others. Note that any such plane can be described by fixing coefficients $\alpha,\beta,\gamma$ in $K$, not all equal to zero, and considering the set of vectors $(x,y,z)\in K^3$ such that $\alpha x+\beta y+\gamma z=0$.

To show that when $K=\mathbb Z_2$ this construction produces the Fano plane it is not enough to indicate that the resulting space consists of $7$ points and $7$ lines, you also need to verify that these can be identified with the points and lines of the Fano plane in such a way that all incidence relations in the Fano plane are verified in this space and no, others.

## 311 – HW2

January 29, 2015

Following Venema’s book, we examine a “toy” collection of incidence axioms: Our primitive (undefined) terms are point, line, and the relation “to lie on” (between a point and a line)

1. For every pair of distinct points $P$ and $Q$ there exists exactly one line $\ell$ such that both $P$ and $Q$ lie on $\ell$.
2. For every line $\ell$ there exist at least two distinct points $P$ and $Q$ such that both $P$ and $Q$ lie on $\ell$.
3. There exist three points that do not all lie on any one line.

Axiom 3 gives us in particular that there are at least three points. However, as shown in lecture (or see example 2.2.2 in Venema’s book), we cannot prove from these axioms that there are more than three, because there is a model of these axioms with precisely three points, and three lines, each line containing exactly two of the points.

For a fixed positive integer $n\ge 2$, replace axiom 2 with axiom $2^n$:

For every line $\ell$ there exist at least $n$ distinct points, all of which lie on $\ell$.

(In particular, axiom $2^n$ is the same as axiom 2.)

Now consider the theory $T_n$ consisting of axioms 1, $2^n$, and 3. The comment above indicates that the smallest possible number of points that a model of $T_2$ can have is three.

1. Try to find the smallest possible number of points that a model of $T_3$ can have. That is, describe a model of $T_3$ with as few points as you can manage. (To show that the number, say $k$, you find is optimal, that is, it cannot be reduced, one would need to prove from the axioms of $T_3$ the theorem that says that there exist at least $k$ points. That would be great, but I am not requiring that. The number $k$ you identify does not need to be optimal, but try to make it as small as possible. Of course, for all we know at this point, it could well be that any model of $T_3$ is infinite.)
2. Do the same for $T_4$ and $T_5$.
3. If possible, can you say something in general about the number of points of the smallest model of $T_n$ (as a function of $n$)?

The implicit suggestion here is to play with these theories, trying to understand what can and cannot be deduced from them. Ryan suggested to look at a variant $3^n$ of axiom 3, namely: Given any $n$ points, there is another point such that the $n+1$ resulting points do not all lie on the same line. Is this a theorem of $T_n$? Are there other interesting variants we can consider?

Feel free to include in your homework any results you find about these theories or their models, even if not directly related to the three questions above.

(In principle, this set is due February 2 at the beginning of lecture, but if needed, I am fine with extending the deadline so you have time to further explore the axioms. Let me know.)