MAT 360 Geometric Structures

Spring 2015



The main goal of the course will be to study the foundations of the Euclidean geometry. We will learn how to work with geometric images and how to construct careful mathematic arguments. As this is an upper-level class, familiarity with proofs is expected. Indeed, you will have to write rigorous proofs in your homework.

Week 1 (01/26 – 02/30) Read sections (not pages!) 1-8, 13-16, 21-27, 34-38. (We have discussed most of this in class.) The reading assignments are important because they teach you how to follow the proof carefully, and how to build your own proofs and to use correct notation.

Homework 1, due Feb 10
Read pages 1-18 of the textbook (contained in the the sample pages). Find all the instances where properties of isometries of the plane are implicitly used.
One of the properties is stated explicitly on page 2, lines 2-5: (i) One can superimpose a plane on itself or any other plane in a way that takes one given point to any other given point...
Other properties were given in the first lecture:
(ii) One can superimpose a plane on itself or any other plane in a way that takes one given ray to any other given ray.
(iii) A plane can be superimposed on itself keeping all the points of a given straight line fixed. This "flip" can be done in a unique way.
In other words,
(ii) There exists an isometry which maps a plane onto itself or any other plane in a way that takes one given ray to any other given ray.
(iii) There exists a unique non-identity isometry of a plane onto itself keeping all the points of a given straight line fixed.

Your task is to list all the places in the first 18 pages of the textbook where these properties are used implicitly, ie without explicit mentioning. Present your solution as a table with rows:
page number, line number, the property, how the property is used. (Example: p. 27, line 2, property (iii) is used to "fold" the diagram along the line BD.)

Please also do the following questions from the textbook: 40, 61, 67, 69.

Week 2 (02/02 – 02/06) Read sections 31-33, 39-50.
Homework 2, due Feb 17
From the textbook: 76, 77, 80, 81, 90, 93. Please give detailed proofs/explanations as appropriate.HW2 Solutions

.

Week 3 (02/09 – 02/13) Read sections 51-60.
Homework 3, due Feb 24
From the textbook: 86, 96, 101, 102, and the following extra question.
Suppose two convex broken lines ABCD.. and A'B'C'D'... share the same endpoints and lie on the same side of the line connecting the endpoints, so that A'B'C'D'... is contained inside ABCD.... Show that the sum of the segments in ABCD... is greater than the sum of the segments in A'B'C'D'..., ie AB+BC+CD+..>A'B'+B'C'+C'D'+... The broken lines can have different number of segments. See picture. Start with the case when both broken lines consist of two segments; we considered this in class. What happens if the broken lines are not convex?
For every question, please give detailed proofs/explanations as appropriate. HW3 Solutions

Week 4 (02/16 – 02/20) Read sections 61-69.
Homework 4, due Mar 3
From the textbook: 63, 109, 122b, 124, 138.
In construction problems (122b, 124, 138), you can refer to sections 62-67 as known, without repeating all steps. Justify your construction (prove that the thing you've drawn has required properties). In 63, try to consider all possible cases, and explain why there can't be more answers.

Week 5 (02/23 – 03/27) Read sections 70-82, 84, 85.
Homework 5, due Mar 10
From the textbook: 143, 150, 154, 163, 165, 167.

Week 6 (03/2 – 03/6)
Read sections 86-97, 98-100.
Homework 6, due Mar 24
From the textbook: 179, 180, 190, 191, 192, 197, 210.

Week 7 (03/9 – 03/13) Read sections 9, 10, 103-114, 122-124, 126, 135-136(1), 138-139.
Homework 7, due Mar 31
From the textbook: 237, 245, 265, 275.

Week 8 (03/16 – 03/22) Spring Break There is no homework due Tuesday, 03/17.

Week 9 (03/23 – 03/27)
Read Viro's Lecture notes on isometries (updated 04/06)
Homework 8, due April 7: pdf

Week 10 (03/30 – 04/3)
Finish reading lecture notes on isometries. Read sections 143-160 in the textbook.
Homework 9, due April 14:
From the textbook: 332, 336, 338a, 214a (hint for 214: use lemma in section 159), and one additional question :
We know that isometries of the plane come in four types (rotations, translations, reflections, glide reflections). Determine which of these types can be represented by:
a) Composition of a rotation and a translation; b) Composition of a rotation and a reflection.
Prove your answer. A complete solution must include examples showing that particular types (say reflections and translations) can be represented by a required composition, and a proof that the remaining types (say rotations and glide reflections) cannot be represented.

Week 11 (04/6 – 04/10)
Read Lecture notes on similarity, along with sections 143-154, 156-162, 181-183 in the book.
Homework 10, due April 21: pdf

Read Lecture notes on inversions and sections 194-197 of the textbook. (The latter are about the degree of a point with respect to a circle).
Homework 11, due April 28: pdf



Students with Disabilities: If you have a physical, psychological, medical, or learning disability that may impact on your ability to carry out assigned course work, you are strongly urged to contact the staff in the Disabled Student Services (DSS) office: Room 133 in the Humanities Building; 632-6748v/TDD. The DSS office will review your concerns and determine, with you, what accommodations are necessary and appropriate. A written DSS recommendation should be brought to your lecturer who will make a decision on what special arrangements will be made. All information and documentation of disability is confidential. Arrangements should be made early in the semester so that your needs can be accommodated.