MATH 427, Fall 2026
FOUNDATIONS OF GEOMETRY
(Euclidean and various non-Euclidean geometries
and their development from postulate systems.)

Syllabus - Extra credit problems - Tickets

• HWA-03, due 11:15, Mon, Sep 14.
  • Read and understand: Chapters 3 and 4.
  • Read and think: Chapter 5.
  • Exercises: 3.14, 3.15, 4.3.
  • Constructions:
    • Copy of angle,
    • Side-median-side,
    • Angle-side-side,
    • Bisected line.
• HWA-02, due 11:15, Fri, Sep 4.
  • Read and understand: Chapter 2.
  • Read and think: Chapter 3.
  • Exercises: 1.15(b), 1.18, 2.5, 2.10.
  • Register at geogebra.org and do the following constructions.
    • Side-angle-side,
    • Motion.
    (Once you see "Well done", make a screenshot and upload it to CANVAS.)
• HWA-01, due 11:15, Fri, Aug 28.
  • Read and understand: Chapter 1.
  • Read and think: Chapter 2.
  • Exercises: 1.2, 1.4, 1.11, 1.13, 1.15(a).
    (Scan to pdf and upload to CANVAS.)

Extra credit:

Exercises: 1.8, 3.17, TBC.

0. Finding a mistake or misprint in the book earns a tiny bit of extra credit, depending on the nature of the mistake.

1. Show that every 4-point metric space is isometric to a subset of the Manhattan plane \((\mathbb{R}^2,d_1)\).

2. Let \(f\colon S\to\mathbb R\) be a distance-nonexpanding function defined on a subset of a metric space \(\mathcal X\); that is, suppose that \(|f(A)-f(B)|\leqslant AB\) for all \(A,B\in S\). Show that \(f\) can be extended to a distance-nonexpanding function defined on the whole \(\mathcal X\).

3. Describe all lines in the Manhattan plane \((\mathbb{R}^2,d_1)\). Namely, show that a subset \(\ell\) of the Manhattan plane is a line if and only if it has a unique point of intersection with every graph \(y=m\cdot x+b\) with \(m>0\), or with every such graph with \(m<0\).

4. Let \(H_4\) be the 4-dimensional Hamming cube; that is, the set of all sequences of four zeros and ones, with the distance defined as the number of positions in which two sequences differ. (For example, \(\mathrm{dist}(0101,0011)=2\) since the sequences differ in exactly the second and third positions.) Show that any two isometric 3-point subsets \(S\) and \(S'\) of \(H_4\) are congruent; that is, there is a motion of \(H_4\) that maps \(S\) to \(S'\). Show that the analogous statement fails for 4-point subsets.

5. On several occasions, hordes of zombies entered a house. Each time, the zombies scattered thruout the house and went dormant. The next morning, a student chooses a place in the house and sits still; all the zombies start moving toward him to eat his brain. Prove that on each occasion the student could choose his place so that the average distance the zombies would have to travel was the same on every occasion, regardless of how many zombies there were or where they were. Assume that the house did not change. Distance is measured by the length of a shortest path, and such a path exists between any two points of the house.