MATH 371 Assignments and Project Spring, 2007

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TEXTS: Fundamentals of Geometry by B. Meserve and J. Izzo, A.W. (1969) - ON LINE with  HSU ONCORE through Moodle.
The Elements by Euclid, 3 volumes, edited by T.L. Heath, Dover (1926)
Proof in Geometry by A.I Fetisov, Mir (1978)
Here's Looking at Euclid..., by J.Petit, Kaufmann (1985).
Flatland By E. Abbott, Dover.

Due: 4/25
M&I: 4.10:1,3,6,7;
 5.4:1-8,10;  5.5: 2,3,7
Tentative assignmentsand topics for classes. 2/9/07 Blue cells Subject to Revisions
Week Monday Wednesday Friday Reading/Videos for the week. Problems 
Due on Wednesday 
of the next week
Math 480 Lab Assignments
1 1/15 No Class
1/17 1.1 Beginnings 


What is Geometry? 
Starting to look at Euclid. Prop 1.

M&I:1.1, 1.2
E:I Def'ns, etc. p153-5; 
Prop. 1-12,22,23,47 
A:.Complete in three weeks
M&I p5:1-8,11

2 1/22 The Pythagorean Theorem 
Lab: Intro to Geometer's Sketchpad/ Wingeom
1/24 Convexity defined.
The Pythagorean Theorem
1/26 Guest Lecture
 on Euclid
Here’s looking at Euclid
M&I 1.2, 1.3 
E: I Prop. 16, 27-32, 35-45.
Due: 1/31
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.]
Lab Exercises 1:
Due: 1/29

Construct a sketch with technology of 
1. Euclid's Proposition 1 in Book I. 
2. Euclid's Proposition 2 in Book I. 
3. One "proof" of the Pythagorean Theorem.
31/29 Equidecomposable Polygons Constructions 
1.1 Def'ns- Objects 
1.2 Constructions 1.3 Geometry: Constructions and numbers 
1/31 1.4 Continuity2/2   Breather:
Start: Transformations - Isometries
M&I 1.3,1.4 
E: III Prop. 1-3, 14-18, 20, 21, 10 
F. Sect. 11, 25, 31
Watch Equidecomposable  Polygons
Due: 2/7 [extended to2/14]
M&I: p17:5, 8-11 
p11: 16-19, 24, *27  Problem  Set 1

Lab Exercises 2:
Due by : 2/5
Do Construction 3, 4, 6, 7, and 8 from Meserve and Izzo Section 1.2.
BONUS:Show how to "add" two arbitrary triangles to create a single square.
4 2/5 More on continuity and rational points. Similar triangles 2/7 More on Cantor
Start Inversion.
2/9Orthogonal Circles
Odds and ends.
Transformations - Isometries. Coordinates.
M&I:1.5, 1.6, 2.1 
E: V def'ns 1-7;VI: prop 1&2 
F. Sect. 32
Due 2/19 [Changed  as of  2-9]
M&I: p23: 9,10 (analytic proofs)
Problem Set 2
Lab Exercises 3: Due 2/12
1. Construct a scalene triangle using Wingeometry. Illustrate how to do i) a translation by a given "vector", ii) a rotation by a given angle measure, and iii) reflections across a given line..
2. Create a sketch that shows that the product of two reflections is either a translation or a rotation
5 2/12 More Isometries: Coordinates and Transformations   2/14 Isometries /
2/16 coordinates/
M&I: 2.1, 2.2
E: V def'ns 1-7;VI: prop 1&2 
E:IV Prop. 3-5
Isometries (Video # 2576 in Library) .

Lab Exercises 4: Due 2/19
1. Draw a figure showing the product of three planar reflections as a glide reflection.
2. Draw a figure illustrating the effects of a central similarity on a triangle using magnification or dilation that is a) positive number >1, b) a positive number <1, and c) a negative number.
6 2/19 classification 2/21  More on Isometries
2/23  .... Finish Classification of Planar Isometries. M&I: 2.1,2.2 Due 2/28 Extended to 3/7!
Problem Set 3
(Isos Tri)   [1 Point for every distinct correct proof of any of these problems.]
Lab Exercises 5: Due  2/26.
1. Construct the inverse of a point with respect to a circle a) when the point is inside the circle; b) when the point is outside the circle.
2.  Given a circle O and two interior points A and B, construct an orthogonal circle O' through A and B. 
3. Draw two intersecting circles O and O' and measure the angle between them.
7 Quiz #1 on Monday in class.
2/26Isometries and symmetries
Begin Affine Geometry
Proportion and Similarity
2/28 More on Similarity
3/2 Inversion and Affine Geometry (planar coordinates) the Affine Line. M&I: 2.2 again;3.1,3.2, 3.5
Lab Exercise 6:  Due 3/5
1.Draw sketches for each of the following triangle coincidences:
1. Medians. 2. Angle Bisectors. 3. Altitudes. 4. Perpendicular Bisectors
3/5 Visualizing the affine plane. Seeing the infinite. 3/7 Affine geometry- Homogeneous coordinates and visualizing the affine plane. 3/9 More on Homogeneous coordinates for the plane. M&I:3.6, 3.4,3.7
View video (in Library #4376 ) on "Central similarities" from the Geometry Film Series. (10 minutes)
View video (in Library #209 cass.2) on similarity (How big is too big? "scale and form")  "On Size and Shape"  from the For All Practical Purposes Series. (about 30 minutes)

Lab Exercise 7: Due  3/19. 
1. Inversion: Investigate and sketch the result of inversion on lines and circles in the plane with a given circle for inversion. 
When does a line invert to a line? When does a line invert to a circle? When does a circle invert to a line? when does a circle invert to a circle?  Show sketches where each case occurs. [ Remember the inverse of the inverse is the original figure.] 
2. Use inversion with respect to the circle OP to invert <BAC to <B'A'C'. Discuss briefly the effects of inversion on angles. 
3. Draw a sketch of the affine plane showing the horizon line and label the lines X=1,2,-1, Y= 1,2,-1 and points (1,2) and (2,-1).
9 Spring break 3/12 No Class 3/14 No Class 3/16 No Class

3/19 More on The affine Plane - A first look at a "Projective plane."
Axioms, consistency, completeness and models.
A non-euclidean universe.
3/21 Begin Synthetic Geometry [Finite] Algebraic-projective geometry: Points and lines.
Spatial and  Planar

3/23Axioms for 7 point geometry.
Begin Synthetic Projective Geometry
M&I :3.1, 3.2, 3.5, 3.6, 3.7; 4.1
Due : 3/28
 M&I: 3.5: 1,3,4,5,10,11
3.6: 3,7-15 
3.7: 1,4,7,10,13
Problem Set 4
Lab Exercise 8: Due 3/26
Draw a sketch for Desargues' theorem in the plane.
Optional: Draw a spatial skectch for Desargues' Theorem
3/26 Homogeneous Coordinates with Z2 and Z3
More on Finite Synthetic Geometry and models.
Proof of Desargues' Theorem
-Projective Geometry -Visual/algebraic and Synthetic..Axioms 1-6 Projective Planes. .
3/28 More on the axioms of Projective Geometry.RP(2) as a model for synthetic geometry. 
3/30 No Class.
CC Day.
M&I:4.1, 4.2, 4.3, 2.4 Due 4/11! - watch for additional problems.
Prove P6 for RP(2); 
4.2: 2,3, Supp:1 
4.3: 1-6, Supp:1,5,6
Lab Exercise 9: Due 4/2
A.1. Construct a sketch showing ABC on a line perspectively related to A'B'C' on a second line with center O.
2. Construct a sketch of  ABC on a line projectively (but not perspectively) related to A'B'C' on a second line. Show two centers and an intermediate line that gives the projectivity.
B.1'. Draw a dual sketch for the figure in problem 1.  2'. Draw a dual sketch for the figure in problem 2. 
4/2Proofs of some basic projective geometric facts.
Triangle Coincidences (Perpendicular Bisectors- the circumcenter)
4/4 Applications of Projective  Geometry Postulates.1-6

4/6 Desargue's Theorem and Duality

No Lab this week.
4/9Conic Sections.
Pascal and More Duality
4/11 Complete quadrangles Postulate 9. 
Projective transformations. Perspectivities and Projectivities. 
4/13 Conics
Pascal's Theorem ?
More on coordinates and transformations.
4.5,4.6(p94-97).4.7, p105-108 (Desargues' Thrm)
Due : 4/18
M&I:4.5:2; 4.6:7,8,9; 4.7:4,7 
4.10: 4,5,9,10
[Prove P9 for RP(2),optional]
Lab Exercise 10: Due 4/16
Pascal's configuration: Hexagons inscribed in conics. Points of intersections of opposite sides lie on a single line. 
Construct a figure for Pascal's configuration  with  a) an ellipse , b)a parabola,  and c) an hyperbola.
4/16Projectivities. Perspective
4/18 Transformations of lines with homogeneous coordinates. 4/20 Projectivities in 3 space: More on Projective Line Transformations with Coordinates. Begin Harmonic Sets and Construction of Coordinates.
4.10, 5.4, 2.4 

4/23Harmonic sets: uniqueness and construction of coordinates for a Projective Line, Plane, Space.
Projective generation of conics 
4/25 More on Transformations, Coordinates and Harmonic sets. . 4/27Matrices for familiar Planar Projective Transformations.
5.1,5.4;,5.2, 5.3,5.5, 5.7, 6.1, 6.2 Due: 5/2 M&I: 4.10:1,3,6,7; 
5.1:5; 5.4:1-8,10;  5.5: 2,3,7

Lab Exercise 11: Due 4/30
1.Draw a sketch showing H(AB,CD) and H(CD, AB).
2. Draw a sketch that shows that if H(AB,CD) and H(AB,CD*) then D= D*. 
4/30Conics revisited.
Inversion and the final exam.
Pascal's and Brianchon's Theorem.
Quiz #3
5/5 Inversion angles, circles and lines.
5/4The Big picture in Summary. . Student Presentations

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., February 13th.
A progress report on the project is due March 26th.
Final projects are due for review Tuesday, May 1st. (These will be graded Honors/Cr/NCr.)

A Project Fair will be organized for displays and presentations during the last day of class. Details will be discussed later.

Guidelines for Preliminary Proposals:

Results of Brainstorming and other suggestions from previous courses :)

Tiling patterns - tesselation
3d tiling
MC Escher
Curves: conics, etc.
optical illusions

The coloring problem
Patterns in dance and other performance arts
Flatland sequel (4d)

structural Rigidity

bridgemaking (architecture)

Models (3d puzzles) paper mache or clay

A play - movie
build three dimensional shapes
power point

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

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

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

Problem Set 2 (revised 2-7-07)

1. Suppose n is a natural number. Given P0 and P1 , 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) on a Euclidean line.

2. Suppose we are given P0, P1, and P a where P a corresponds to the real number a>0. Give a construction with SEC of a point Psqrt(a) which will correspond to the number  sqrt(a) on a Euclidean line.

3. Given points P0, P1, Px, and Py 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(3)  + sqrt(sqrt(6)) .

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

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. 

Problem Set  4

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

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

3. 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 intersect C = {Q}; Q' is on C so that Q'Q is parallel to ON; and {P'} = NQ'  intersect 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.

4. 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}.