## MATH 371 Assignments and Project Spring, 2016

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TEXTS: Fundamentals of Geometry by B. Meserve and J. Izzo, A.W. (1969) - ON LINE at Moodle.
The Elements by Euclid, 3 volumes, edited by T.L. Heath, Dover (1926)
Proof in Geometry by A.I Fetisov, Mir (1978)
Flatland By E. Abbott, Dover.

Tentative assignments and topics for classes.1/18/16 Blue cells Subject to Revision
Week Monday Wednesday Friday Reading/Videos for the "week"-through Monday. Problems
Due on Wednesday
of the next week
1

1/20 1.1 Beginnings
What is Geometry?

1/22

Start on Euclid- Definitions, Postulates, and Prop 1
Convexity defined

Watch online: Here’s looking at Euclid
M&I:1.1, 1.2
E:I Def'ns, etc. p153-5;
Prop. 1-12,22,23,47
A:.Complete in three weeks
Due:1/27
M&I p5:1-8,11
2 1/25 Euclid- Definitions, Postulates, and Prop 1. cont'd

.1/27
Pythagorean plus.
1/29 Euclid Postulates/ Pythagoras
M&I 1.2, 1.3
E: I Prop. 16, 27-32, 35-45

Due: 2/3
M&I: p10:1,2,5,10,11-13
Prove:The line segment connecting the midpoints of  two sides of a triangle is parallel to the third side and is congruent to one half of the third side.
[ HELP! Proof outline for the midpoint proposition.]
3 Reading Report due: 2/1 2/1 Euclid early Props/ Pythagoras

2-3 Pythagoras and Dissections- equidecomposeable polygon .
2-5 More on Equidecomposeable polygons
Begin Constructions and the real number line
M&I's Euclidean Geometry
Isometries
1.1 Def'ns- Objects
1.2 Constructions 1.3 Geometry: Constructions and numbers
M&I 1.3,1.4
E: III Prop. 1-3, 14-18, 20, 21, 10
F. Sect. 11, 25, 30, 31
[Optional Sections:1-10; 12,13,18,20]
Watch Equidecomposable  Polygons
Due 2/10
M&I: p17:5, 8-11
p11: 16-19, 24, *27  Problem  Set 1
4Preliminary Project Proposals  first review 5 p.m. Wednesday, February 9th. 2/8 More on Details for triangulation of planar polygonal regions
2-10 Begin Constructions and the real number line
M&I's Euclidean Geometry
2-12 Constructions from M&I. M&I 1.3,1.4
E: III Prop. 1-3, 14-18, 20, 21, 10
F. Sect. 11, 25, 30, 31
[Optional Sections:1-10; 12,13,18,20]

5 Reading Report due: 2/15 2/15 1.4 Continuity. Continuity and rational points. Similar triangles 2/17 More on similar triangles and rational points. Isometries.
Start Inversion.
Orthogonal Circles
2/19 Coordinates and Proofs
Inversions and orthogonal circles. Odds and ends on continuity axiom
More on Cantor
M&I:1.5, 1.6, 2.1
OPTIONAL[E: V def'ns 1-7;VI: prop 1&2 ]
F. [Sect. 32] pp 55-61on Axioms of Continuity
Video:
Isometries (Video # 2576 in Library) .

6
2/22 Inversions and orthogonal circles.  2/24Odds and ends on continuity axiom
More on Cantor More Isometries.
2/26 Begin Transformations - Isometries.
coordinates.....
M&I: 2.1,2.2
E:IV Prop. 3-5
Due 2/26 M&I: p23: 9,10 (analytic proofs)
Problem Set 2
7 Reading Report due: 2/29
2/29 Finish Classification of Planar Isometries and coordinates
3/2 Isometries and symmetries
More on Similarity
3/4/ M&I: 1.6, 2.1, 2.2 again
Video: "Central similarities" (in Library #4376 )  (10 minutes)
"On Size and Shape"
(How big is too big? "scale and form")(in Library #209 cass.2)  (about 30 minutes)
Due 3/7
M&I:1.6:1-12,17,18

Flatland Essay #1
8 Quiz #1 on Wed. 3/9
3/7 Begin Affine Geometry
Proportion and Similarity
3/9 Quiz #1
Proportion and Similarity in Euclid.
3/11More on Proportion a la Euclid.

M&I: 2.1, 2.2 again; Start 3.1,3.2

.
Due 3/23
Problem Set 3 (Isos Tri)   [3 Points for every distinct correct proof of any of these problems.]
9 Spring break 3/14 No Class  3/16No Class 3/18 No Class

10 Reading Report due: 3/21
A progress report on the project is due March 23rd
3/21 Similarity, proportion with  numbers vs euclid.
The Affine Line. Seeing the infinite.
Affine geometry-
3/23 -
Homogeneous coordinates and visualizing the affine plane. Inversion and

3/25 Affine Geometry (planar coordinates)
A first look at a "Projective plane."
M&I :3.1, 3.2, 3.4, 3.5, 3.7; 4.1 Due : 4/4!
M&I: 3.5: 1,3,4,5,10,11
3.6: 3,7-15
Problem Set 4
M&I: 3.5: 1,3,4,5,10,11
3.6: 3,7-15
Submit Outlines of Content for Videos 346 and 628

11
3/28 A non-euclidean universe. 3/30Axioms for 7 point geometry.
Begin Synthetic Projective Geometry
4/1 The Conics: From cones to equations.
Non-Euclidean Geometry (24 minutes) Open University (History of Math)
A Non-Euclidean Universe. (24 minutes)(Open University VIDEO346)
The Conics (24 minutes) (Open University HSU Libray VIDEO628)
12Reading Report due: 4/4 4/4 The affine plane and homogeneous coordinates for points. Lines and homogeneous linear equations.
4/6 -Homogeneous Coordinates with Z2 and Z3
More on Finite Synthetic Geometry and models.
Proof of Desargues' Theorem
4/8Projective Geometry -Visual/algebraic and Synthetic..
Synthetic Projective Geometry Algebraic-projective geometry: Axioms, consistency, completeness and models.
Points and lines.
Spatial and  Planar
Orthogonal Projection [HSU Library videorecording]
VIDEO 4223 (11 minutes)
Central Perspectivities [HSU Library videorecording]
VIDEO 4206 (14 minutes)
Central Similarities - [HSU Video 4376]
YouTube Video for central similarities10:36
Submit Outlines of Content for Videos 4223, 4206, and 4376.

M&I:3.7: 1,4,7,10,13
4.1:7,15,16;
Prove P6 for RP(2);
4.2: 2,3, Supp:1
4.3: 1-6, Supp:1,5,6

13
4/11
4/13 Quiz #2?
4/15

14
4/18 Proofs Using  PlaneProjective Geometry Postulates
4/20 Duality. Intro to Transformations of RP(1) and matrices.
4/22 The complete quadrangle.
Perspectivities and projectivities.
M&I:
4.5,4.6(p94-97).4.7,
4.10, p105-108 (Desargues' Thrm)
5.4

Projective Generation of Conics Video 2574

Math History 8

(about 1 hour) A Summary and extension
of our work on the projective plane.
Submit Outline of Content for Video 2574

4.1:7,15,16;
Prove P6 for RP(2);
4.2: 2,3, Supp:1
4.3: 1-6, Supp:1,5,6

Problem Set 5
Reading Report on Monday 4-25.
15 Last Reading report due 4/25
Final Exam Distributed 4/29
4/25
4/27
4/29

16
5/2
5/4 Quiz # 3
5/6

17 Final Exam Week.
Office Hours for Exam Week
MTWRF 8:15-10:00 and by appointment or chance

Student Presentations will be made Wednesday 5-11 10:20-12:10 Final Exam
DUE Friday, May 13, before 5 P.M.

Problem Set  5

1. D is a circle with center N tangent to a line l at the point O and C is a circle that passes through the N and is tangent to l at O as well.
Suppose P is on l and PN cap  C = {Q}; Q' is on C so that Q'Q is parallel to ON; and {P'} = NQ' cap l.
Prove: a) P and Q are inverses with respect to the circle D.
b) P' and Q' are inverses with respect to the circle D.
c) P and P' are inverses with respect to the circle with center at O and radius ON.

2. Suppose C is a circle with center O and D is a circle with O  an element of D.
Let I be the inversion transformation with respect to C.
Prove: There is a line l, where I(P) is an element of  l for all P  that are elements of  D -{O}.

Problem Set  4

1. Use an affine line with P0 , P1 , and P_{oo} given. Show a construction for P1/2 and P2/3.

2. Use an affine line with P0 , P1 , and P_{oo}  given. Suppose x > 1.
Show a construction for  Px2 and Px3 when Px is known.

Problem Set 3

1.  Prove: Two of the medians of an isosceles triangle are congruent.

2.  Prove: If two of the medians of a triangle are congruent then the triangle is isosceles.

3.  Prove: The angle bisectors of congruent angles of an isosceles triangle are congruent.

4.  Prove:  If two of the angle bisectors of a triangle are congruent then the triangle is isosceles.

Flatland Essay #1

The Sphere has brought the Tetrahedron and the Cube to visit his new friend the Square in Flatland. Describe two possible successions (for each) of different planar shapes the Tetrahedron and the Cube might  appear as while passing through Flatland. Compare how would these experiences would differ from what the square saw when the sphere passed through Flatland. You may draw the sequences as they would be seen in Flatland but you need to describe these drawings verbally in your response.

Problem Set 2

1. Suppose n is a natural number. Given P_0 and P_1 , prove by induction that you can construct with straight edge and compass (SEC) a point  P_{sqrt(n)}   which will correspond to the number  sqrt(n) [square root of n] on a Euclidean line.

2. Suppose we are given P_0, P_1, and P_a where P_a corresponds to the real number a>0. Give a construction with SEC of a point P_{sqrt(a)}which will correspond to the number  sqrt(a) [square root of a]  on a Euclidean line.

3. Given points P_0, P_1, P_x, and P_y on a Euclidean line corresponding to the real numbers x>0 and y>0, give constructions with SEC for the following points.

 a) P_{x + y} b) P_{x - y} c) P_{ x *y} d) P_{1/x}
4. Construct with SEC on a Euclidean line:   sqrt(5)/sqrt(2)  + sqrt(sqrt(3)) .

5. Suppose that d(A,B) = d(A',B') and that l is the perpendicular bisector of the line segment  A A'. Let B'' be the reflection of B across l, i.e., B''= T_l(B). Prove that if  B' is not equal to B''  then A'  lies on the perpendicular bisector of the line segment.

Problem Set 1

DEFINITIONS: A figure C is called convex if for any two points in the figure, the line segment determined by those two points is also contained in the figure.
That is, if A is a point of C and B is a point of C then the line segment AB is a subset of C.

If F and G are figures then F ∩ G  is { X : X ε F and X ε G }.
F ∩ G is called the intersection of F and G.
If A is a family of figures (possibly infinite), then  ∩A = { X : for every figure F in the family A, X ε F }.
A is called the intersection of the family A.
-----------------------------------------------------------------

1. Prove: If F and G are convex figures , then F ∩ G is a convex figure.

2. Give a counterexample for the converse of problem 1.

3. Prove: If A is a family of convex figures, then ∩ A is a convex figure.

4. Prove: The line segment RS is convex. [ Refer to M & I pg.2.]

Project Proposal Guidelines and Suggestions
The Project. Each student will participate in a course project either  as an individual as a part of a team. Each team will have at most three. These projects will be designed with assistance from myself . The quality of the project will be used for determining letter grades above the C level. Ideas for projects will be discussed during the third week.
Preliminary Project Proposals should be submitted for first review by 5 p.m. Monday , February 8th.
A progress report on the project is due March 23rd.
Final projects are due for review Friday, April 29th. (These will be graded Honors/Cr/NCr.)

A Project Fair will be organized for displays and presentations during the final exam period. Details will be discussed later.

Guidelines for Preliminary Proposals:

• Proposal Format: The proposal should be typed (neatly hand written proposals are acceptable).
• Contents: The content of your proposal should describe, explain or otherwise demonstrate what your project is as you currently envision it. It should also indicate how you will go about completing the project.

• Below are some specific suggestions on features your proposal description might include:
• Title: Include a name( or list of possible names ) for your project.
• Introduction: (Your topic's core idea.) You should explain the idea of your project. Remember that the Introduction is the first place where the reader hears about your idea. You should also explain how the Proposal is organized in the introduction.
• Form and result: Indicate your vision of the final project's form(s), that is, the appearance of the FINAL PRODUCT. What will your project look like in its final ideal form? Note that all forms must include some written explanatory  component.
• Variations: (optional) Since this is a preliminary proposal, indicate some of the possible variations of both substance and form. It might be useful to distinguish the ideal from what may be a minimal project in both substance and form, and perhaps to see the project in stages from minimal to ideal, just in case you run into practical or time problems.
• References and Tools: List references and tools (books, journals, software, people, etc.) that are relevant to your project and that you might use. If you don't have any specific references yet, then indicate the kind of references you might use and where you will find them.
• Methods- Timeline and Task Delegation (for partnerships): Who will do what? When will they do it? If your project has definite parts or subdivisions, then indicate target dates for the completion of each stage.

• For partnerships:This project is a collective effort and should reflect the work and effort of all. Indicate when and where you will meet outside of class and how often. When possible, estimate the number of hours you are allocating to each task.

• Record keeping: Indicate how you will keep track of the progress of your project and the time spent by each individual participant on the project's work.

Results of Brainstorming and other suggestions from previous courses :)

 Tiling patterns - tesselation 3d tiling MC Escher perspective Curves: conics, etc. optical illusions knots fractals Origami Kaleidescope Symmetry The coloring problem Patterns in dance and other performance arts Flatland sequel (4d) Maps Juggling structural Rigidity dimension Polyhedra bridgemaking (architecture) Models (3d puzzles) paper mache or clay mobiles sculpture A play - movie build three dimensional shapes power point performance website