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 

 

1/19
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		 Due:1/24
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
1.2
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 
Isometries 
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)
M&I:1.6:1-12,17,18 
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
8
 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


10
 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
11
 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.
M&I:4.1:7,15,16; 
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. 
12
 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.
13
 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.
M&I: 
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.
14
 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 
4.11

15
 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*. 
16
 4/30Conics revisited.
Inversion and the final exam.
Pascal's and Brianchon's Theorem.
Quiz #3
5/2
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
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

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}.