Syllabus and and other important documents
Syllabus
This syllabus contains important information about the course, including its goals,
assignments, policies, resources, and expectations.
How to make custom tools on GeoGebra Classic?
A tutorial to create custom tools on GeoGebra Classic.
Euclid's Elements
Week 1 Lecture 1: A first look at Euclid’s
A first introduction to Euclid’s geometry, the axiomatic method, primitive terms,
postulates, models, and Euclid’s first propositions.
Week 1 Lecture 2: Euclid’s rigor, constructions, and angles
A discussion of the six parts of a proposition, Euclid’s rigor in Propositions I.1, I.2,
and I.4, constructions in GeoGebra Classic, angle definitions. Click on activity W1L2
for the GeoGebra constructions we did in class.
Week 2 Lecture 1: “summary diagrams”, “Euclid-style” proofs, and more on postulates
A discussion of proposition “summary diagrams”, “Euclid-style” proofs, stronger
forms of Postulates 1 and 2, and the definition of right angles and Postulate 4.
Week 2 Section Worksheet
A worksheet on Euclid-style proofs, summary diagrams, and constructions from
Book I.
Week 2 Lecture 2: ASA, parallel lines, and angle sums in triangles
A discussion of the ASA and AAS congruence theorems, parallel lines and
Postulate 5, basic properties of parallel lines, the construction of a parallel line
through a point, and the angle sum in a triangle.
Week 3 Lecture 1: Angles in triangles, RASS, and the center of a circle
A discussion of the sum of the angles in a triangle, polygons and convexity, the RASS
congruence theorem, an application to chords in a circle, and the construction of the
center of a circle.
Week 3 Section Worksheet
A worksheet on constructions, the RASS theorem, and inscribed/circumscribed circle
to a triangle.
Week 3 Lecture 2: Tangents and angles in circles
A discussion of tangent lines, tangent circles, the relation between tangency and
perpendicularity to the radius, and the main angle theorems in circles.
Hilbert's Axioms
Week 4 Lecture 1: Hilbert’s axioms, models, and parallel properties
An introduction to Hilbert’s axiomatic approach to Euclidean geometry, the axioms
of incidence, models of incidence geometry, the Euclidean/elliptic/hyperbolic parallel
properties, and isomorphisms of models.
Week 6 Lecture 1
Counting points in affine planes, projective planes, and the first idea of projective
completion.
Week 6 Section Worksheet: Counting in Affine and Projective Planes
In this worksheet, we study several counting arguments in affine and projective
planes. We first prove that any two lines in a projective plane have the same number
of points. We then deduce the standard formula for the number of points in a finite
projective plane. Finally, we prove that in a finite affine plane of order \(n\), every point
lies on exactly \(n+1\) lines.
Week 6 Lecture 2
Projective completion, points at infinity, and models of the real projective
plane.
Week 7 Lecture 1: Visualizing Projective Completions
Visualizing the real projective plane and introducing homogeneous coordinates.
Week 7 Section Worksheet: Homogeneous Coordinates and Projective Completion
The purpose of this worksheet is to make sure you have a solid understanding of
homogeneous coordinates and to fill in some of the gaps from Section 11 of
W7L1.
Week 7 Lecture 2 and Week 8 Lecture 1
Transformations of incidence geometries, permutations, examples, and the group
structure of automorphisms.
Week 8 Lecture 2
Groups of transformations, with a detailed activity on the transformations of the
integer line.
Week 9 Lecture 1
Finishing the transformations of the integer line, then introducing Pappus’ theorem
and the reduction from the affine plane to the projective plane.
Week 9 Section Worksheet: Transformations of Infinite Incidence Geometries
The purpose of this worksheet is to classify all transformations of two infinite
incidence geometries.
Week 9 Lecture 2
Continuing Pappus’ theorem: reducing the affine statement to the projective
statement, introducing projective transformations, and normalizing the two
lines.