- Title: Modern Geometry
- Instructor: Trevor Wilson (twilson@math.uci.edu)
- Lectures: 12 noon – 12:50 p.m. MWF in MSTB 124
- Office hours: 1:30–3:30 p.m. on Mondays in 510V Rowland Hall
- TA: Andrew Thomas (andrewjt@math.uci.edu)
- Text:
*"Modern Geometry"*(UCI custom reprint of*"Geometry with Geometry Explorer"*by Michael Hvidsten.)

Your course grade will be calculated as follows:

- 30% homework
- 30% midterm exam
- 40% final exam

- Sample midterm exam and Solutions.
- White midterm exam and Solutions.
- Yellow midterm exam and Solutions.

- Mar. 31: 1.4–1.5 (Axiomatic method and systems). Exercises 1.5.4, 1.5.7.
- Apr. 2: 1.6 (Euclid's axiomatic geometry). Exercises 1.6.8, 1.6.11. For these exercises, "true or false in spherical geometry" means "true or false in the spherical model described in Exercise 1.6.7." You don't need to give a formal proof, but the following definitions may be helpful. (1) The measure of a straight angle (that is, an angle $\angle ABC$ where $A$, $B$, and $C$ are collinear) is defined as $180$ degrees. (2) A right angle is defined as half of a straight angle; that is, an angle is a right angle if it has a supplementary angle to which it is congruent.
- Apr. 4: 2.1 (Angles, lines, and parallels.) Exercises 2.1.2, 2.1.8. For Exercise 2.1.8, you may use Theorem 2.9.

- Apr. 7: 2.2 (Congruent triangles.) Exercises 2.2.7, 2.2.12.
- Apr. 9: 2.2, 2.4 (Pasch's axiom; Measurement and area.) Exercises 2.2.6, 2.2.11. For Exercise 2.2.6, you will probably want to use the notion of "same side" in some way. Note: Exercise 2.2.11 does not require any of the new material covered today; also you can disregard the hint (there is a direct proof as well as a proof by contradiction.)
- Apr. 11: 2.4 (Measurement and area.) Exercises 2.4.4, 2.4.9. For Exercise 2.4.4, you may assume the second configuration because we did the first configuration in class. For Exercise 2.4.9, see Definition 2.25 for "median". You may assume that the three medians intersect at a common point (see Exercise 2.4.7). You may also assume these facts about area which we did not have time to cover properly on Friday.

- Apr. 14: 2.5 (Pythagoras's Theorem; Similar triangles.) Exercise 2.5.3. You may use Theorem 2.29 (SAS Similarity) even though we we did not cover it yet today. (By the way, some axiomatic systems for geometry take SAS similarity as an axiom.)
- Apr. 16: 2.5 (Similar triangles.) Exercises 2.5.4, 2.5.5, 2.5.6. For Exercise 2.5.5, just prove that $\sin$ is well-defined. For Exercise 2.5.6, just prove the special case where the angles are acute.
- Apr. 18: 2.6 (Circles.) Exercises 2.6.5, 2.6.8. For Exercise 2.6.8, see Definition 2.33 for "mutually tangent".

- Apr. 21: 3.1 (Cartesian coordinates.) Exercises 3.2.1, 3.2.2.
- Apr. 23: 3.2 (Vectors.) Exercises 3.2.5, 3.2.6. In Exercise 3.2.5, $\vec{A}$ and $\vec{B}$ denote the vectors represented by directed line segments from $O$ to $A$ and from $O$ to $B$ respectively, where $O$ is a point chosen as the origin. This problem is not quite as trivial as it looks. There are at least two ways to do it. Here is a hint for a geometric way: Recall that the vector $\vec{A} + \vec{B}$ is defined in terms of a certain parallelogram. The problem amounts to proving an interesting property of the diagonals of this parallelogram.
- Apr. 25: 3.4 (Angles in coordinate geometry.) Exercises 3.4.1, 3.4.2

- Apr. 28: 4.1 (Euclidean constructions.) Optional exercises: 4.1.1, 4.1.8. These will not be collected. No homework will be due on the week of the midterm exam.
- Apr. 30: 4.1 (Euclidean constructions)
- May 2: Review

- May 5: Midterm exam
- May 7: 5.1 (Isometries.) Exercises 5.1.2, 5.1.4, 5.1.6
- May 9: 5.1, 5.2 (Reflections.) Exercises 5.1.8, 5.2.12. (Only five exercises will be assigned for this week.)
For Exercise 5.2.12, a line $\ell$ is called
*invariant*under a transformation $f$ if $f[\ell] = \ell$. Note that this does*not*say that every element of $\ell$ is a fixed point of $f$. The exercise can be rephrased to say: "Given a reflection $r_m$ with line of reflection $m$, and given a line $\ell$, show that $r_m[\ell] = \ell$ if and only if $\ell = m$ or $\ell \mathbin{\bot} m$."

- May 12: 5.3 (Translations.) Exercises 5.2.13, 5.3.6. In Exercise 5.2.13, "the reflection of $\ell$ across $m$" means the pointwise image $r_m[\ell]$ of $\ell$ under the reflection $r_m$. In Exercise 5.3.6, for a translation $T$ to be in the same direction as a line $\ell$ means that the displacement vector of $T$ is parallel to $\ell$. It is possible to do Exercise 5.3.6 with or without Cartesian coordinates.
- May 14: 5.4 (Rotations.) Exercises 5.4.4, 5.4.5. In Exercise 5.4.4, the notion $R(\ell)$ means $R[\ell]$, the pointwise image of $\ell$ under the rotation $R$.
- May 16: 5.4 (Rotations.) Exercises 5.4.11, 5.4.12. For Exercise 5.4.11, please show the result directly, rather than by quoting the theorem that says that the product of two reflections across intersecting lines is a rotation. The exercise asks you to prove a particular case of the theorem that is simple yet instructive to verify. For Exercise 5.4.12, a hint is to use Cartesian coordinates.

- May 19: 5.6 (Glide reflections.) 5.2.14, 5.6.13. For Exercise 5.2.14, only prove the following cases: if (1) the lines $\ell$ and $m$ are parallel and the lines $n$ and $q$ intersect, or (2) the lines $n$ and $q$ are parallel and the lines $\ell$ and $m$ intersect, then prove that the composition of four reflections $r_q \circ r_n \circ r_m \circ r_\ell$ is equal to a composition of two reflections. In other words, you will be proving that the composition of a translation and a rotation (in either order) is equal to a translation or a rotation. Hint: for (1) first consider the subcase (1a) where $\ell$, $m$, and $n$ are parallel; for (2) first consider the subcase (2a) where $q$, $n$, and $m$ are parallel. Then explain why we can assume without loss of generality that (1a) or (2a) holds. For Exercise 5.6.13, you may use Exercise 5.2.14 even though you only proved cases (1) and (2).
- May 21: 7.1, 7.2 subsection 1 (Non-Euclidean geometry, Poincaré model.) Exercise: Show that the Poincaré model has triangles with angle sum as small as desired. In other words, for every real number $\epsilon \gt 0$ there are points $A$, $B$, and $C$ in the unit disk such that the sum of the three angles between the hyperbolic line segments $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$ is less than $\epsilon$. Hint: let the vertices $A$, $B$, and $C$ get very close (in the sense of Euclidean distance) to the disk boundary but remain far away from one another. Argue that as $A$ approaches the disk boundary the angle at $A$ approaches zero (and similarly for $B$ and $C$.) You may argue informally, using a picture, but you will need to refer to the definitions of "hyperbolic line" and "angle" in the Poincaré model.
- May 23: 7.3 (Limiting parallels.) Exercise: Let $\ell$ be a line
and let $P$ be a point not on $\ell$. Let $Q$ be the point on $\ell$
such that $\overline{PQ} \perp \ell$. Show (in hyperbolic geometry) that there is a least angle
$\theta$ with the property that the line $m$ through $P$ making an angle
of $\theta$ with $\overline{PQ}$ (on either side) is parallel to $m$.
Hint: is there a greatest angle $\theta$ with the property that the line
$m$ through $P$ making an angle of $\theta$ with $\overline{PQ}$ (on either side) is
*not*parallel to $m$? Why or why not, and how is this relevant? Also, angles are measured by real numbers, so the completeness property of the real numbers is relevant.

- May 26: Holiday (Memorial Day)
- May 28: 7.3 (Limiting parallels.) Exercises:
(1) Complete the following proof that if $m$ is a limiting parallel to $\ell$, then $\ell$ is a limiting parallel to $m$: Take a point $P_1$ on $\ell$. Let $Q$ be the point on $m$ such that $\overline{P_1Q} \perp m$. Let $P_2$ be the point on $\ell$ such that $\overline{QP_2} \perp \ell$. Use a continuity argument to show that there is a point $P$ between $P_1$ and $P_2$ such that the segment $\overline{PQ}$ makes equal angles with $\ell$ and $m$. Then consider the perpendicular bisector of $\overline{PQ}$ and use a symmetry argument. (I think that a proof along these lines is simpler than the book's proof.)

(2) Show that there is an isometry $f$ of the Poincaré model (in other words a function from the open unit disk to itself that preserves the hyperbolic distance) and a line $\ell$ in the model such that $\ell$ is (represented by) a Euclidean line segment and $f[\ell]$ is (represented by) a Euclidean arc. The point is that the property of looking "straight" or "curved" is not an intrinsic property of hyperbolic lines; it depends on how we model them in Euclidean space. Hint: our work with reflections doesn't depend on the parallel postulate, so it is still valid in hyperbolic geoemtry.

(3) Exercise 7.3.3. I don't know what the hint "use the parallelism properties of reflections" means, but anyway it should not be hard to show that if $m$ is a limiting parallel to $\ell$, then $r_\ell[m]$ is a limiting parallel to $\ell$ on the same side. The part about omega points turns out to be trivial once you untangle the definitions (including what it means for an isometry to fix an omega point) so I'll say you can skip this part, but you should think for a minute about what it means.

Remark: To visualize exercises (3) and (4) it may be helpful to consider the special case where the line $\ell$ is represented in the Poincaré model by a diameter of the unit disk. Then we can extend the reflection $r_\ell$ to act on the boundary of the unit disk in a natural way. Two and only two of the boundary points will be fixed by this extension of $r_\ell$.

- May 30: 7.3 (Omega-triangles.) Exercises:
(4) Exercise 7.3.4. Hint: Let $\ell'$ be a line that is

*not*right-limiting parallel to $\ell$, which means that it has a different omega point on the right side (the same argument will work for the left side.) We want to show that this right omega point of $\ell'$ is*not*fixed by the reflection $r_\ell$. In other words, we want to show that the line $\ell'$ is*not*right-limiting parallel to its own reflection $r_\ell[\ell']$. Consider two cases: (a) $\ell'$ intersects $\ell$; (b) $\ell'$ is parallel to $\ell$ but is not right-limiting parallel to $\ell$.(5) Exercise 7.3.10. You may use Exercise 7.3.6 even though we did not prove it (note that it's easy to prove the special case of Exercise 7.3.6 where one of the angles is a right angle, because the other angle will be an instance of the notion of "angle of parallelism", which is always less than a right angle.) Hint: you can think of the statement in Exercise 7.3.10 as an AAS congruence theorem for omega triangles, because you can think of the sides $\overline{P\Omega}$ and $\overline{P'\Omega'}$ has both having infinite length. Then think about how you can prove AAS congruence for ordinary triangles from SAS congruence for ordinary triangles.

- June 2: 7.4 (Saccheri quadrilaterals)
- June 4: 7.5 (Lambert quadrilaterals)
- June 6: Review

- June 11 (Wed.): Final exam 4:00-6:00pm