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.
| Week | Tuesday | ?? | Thursday | Reading/Videos for the week. | Problems Due on Wednesday of the next week |
|---|---|---|---|---|---|
| 1 | 1/18 1.1
Beginnings What is Geometry? |
|
1/20 |
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:1/26 M&I p5:1-8,11 |
| 2 | 1/25
Euclid. Prop 1-3 The Pythagorean Theorem |
. The Pythagorean Theorem 1.2 |
1/27 More Euclid Constructions. |
M&I
1.2,
1.3 E: I Prop. 16, 27-32, 35-45 Watch online: Here’s looking at Euclid. |
Due: 2/2 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
Equidecomposable Polygons Constructions Isometries 1.1 Def'ns- Objects 1.2 Constructions 1.3 Geometry: Constructions and numbers |
1.4 Continuity | 2/3
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: 2/9 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 continuity and rational points. Similar triangles | 2/10 More on similar triangles and rational
points. Start Inversion. Orthogonal Circles |
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 |
||
| 5 Reading Report due: 2/15 | 2/15
Coordinates and Proofs Inversions and orthogonal circles. Odds and ends on continuity axiom More on Cantor |
Isometries / | 2/17
Transformations - Isometries. coordinates/ |
M&I:
2.1,
2.2 OPTIONAL [E: V def'ns 1-7;VI: prop 1&2 ] E:IV Prop. 3-5 Isometries (Video # 2576 in Library) . |
Due : 2/23 [Previously due 2/16] M&I: p23: 9,10 (analytic proofs) Problem Set 2 |
| 6. |
2/22 More Isometries: classification |
More
on
Isometries |
2/24 .... Finish Classification of Planar Isometries. | M&I: 2.1,2.2 | |
| 7 Reading Report due: 3/1 |
3/1
Isometries
and
symmetries |
More
on
Similarity |
3/Begin
Affine
Geometry Proportion and Similarity |
M&I: 2.2 again;3.1,3.2, 3.5 | |
| 8Quiz #1 on Thursday 3/10 in class | 3/8
3
Inversion
and
Affine
Geometry
(planar
coordinates) The Affine Line. |
Affine geometry- Homogeneous coordinates and visualizing the affine plane. | 3/10
Visualizing
the
affine
plane.
Seeing
the infinite. 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) |
Due 3/23 M&I:1.6:1-12,17,18 Problem Set 3 (Isos Tri) [3 Points for every distinct correct proof of any of these problems.] |
| 9 Spring break | 3/15 No Class | No Class | 3/17 No Class | ||
| 10
Reading Report due: 3/24 Project progress report Thursday, March 25th |
3/22
More
on
The
affine
Plane
- A first look at
a "Projective plane." Axioms, consistency, completeness and models. A non-euclidean universe. |
Begin
Synthetic
Geometry
[Finite]
Algebraic-projective
geometry:
Points
and lines. Spatial and Planar |
3/24Axioms
for
7
point
geometry. Begin Synthetic Projective Geometry |
M&I
:3.1,
3.2,
3.5,
3.6,
3.7;
4.1 |
Due
:
3/30 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 |
| 11 |
3/29
Homogeneous
Coordinates
with
Z2 and Z3 More on Finite Synthetic Geometry and models. Proof of Desargues' Theorem -Projective Geometry -Visual/algebraic and Synthetic.. |
|
3/31
No
Class
CC Day |
M&I:4.1, 4.2, 4.3, 2.4 | |
| 12 Reading Report due: 4/7 (changed 4-3) | 4/5
Axioms
1-6 Projective Planes. More on the axioms of Projective Geometry.RP(2) as a model for synthetic geometry. Proofs of some basic projective geometric facts. |
Applications
of
Projective
Geometry
Postulates.1-6 |
4/7
Triangle
Coincidences
(Perpendicular Bisectors- the circumcenter) Desargue's Theorem and Duality |
M&I:4.1, 4.2, 4.3, 2.4 | Due: 4/13 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 Problem Set 5 |
| 13 quiz #2 4/14 |
4/12Conic
Sections.
Pascal and More Duality |
Projective transformations. Perspectivities and Projectivities. |
4/14Sections |
M&I:
4.5,4.6(p94-97).4.7, p105-108 (Desargues' Thrm) |
|
| 14 Reading Report due: 4/19 | 4/19
Perspective
More duality. Graphs and duality. Complete quadrangles Postulate 9. |
4/21Projectivities.
Transformations
of
lines
with
homogeneous
coordinates. Conics Pascal's Theorem ? More on Projective Line Transformations with Coordinates. Begin Harmonic Sets and Construction of Coordinates. |
4.10,
5.4,
2.4
4.11 |
Due
:4/27 M&I:4.5:2; 4.6:7,8,9; 4.7:4,7 4.10: 4,5,7,9,10 [Prove P9 for RP(2),optional] |
|
| 15 Projects are due 5pm Thursday, 4/29 |
4/26More
on
coordinates and transformations. Projectivities
in
3
space: Harmonic
sets:
uniqueness
and
construction
of coordinates for a Projective
Line, Plane, Space. Projective generation of conics |
More on Transformations, Coordinates and Harmonic sets. . | 4/28Matrices
for
familiar
Planar
Projective
Transformations.
|
5.1,5.4;,5.2, 5.3,5.5, 5.7, 6.1, 6.2 | Due: OPTIONAL! M&I: 4.10:1,3,6; 5.1:5; 5.4:1-8,10; 5.5: 2,3,7 |
| 16 Reading Report due: 5/3 | 5/3Conics
revisited.
Inversion and the final exam. Quiz #3 |
Inversion angles, circles and lines. | 5/5
Pascal's and Brianchon's Theorem. The Big picture in Summary. . |
|
|
| 17 Final Exam Week. |
Office Hours for Exam Week MTWRF 8:15-10:00 and by appointment or chance |
Student Presentations will be made Tuesday 5-10 3:00-4:50 | Final Exam DUE Friday, May 13, before 5 P.M. |
||
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:
| 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 |
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.]
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 |
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.
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.
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.
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 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.
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}.