Date
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What we discussed/How we spent our time
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Jan 18
|
Syllabus. Text. Stuff.
We identified the main topics of the course (history of math, logic,
Euclidean geometry, hyperbolic geometry). We briefly listed
the most important milestones of the subject from $\sim$300BC to
$\sim$1935AD.
|
Jan 20
|
We discussed two approaches to axiomatic geometry:
Euclid's approach involving
definitions, postulates, common notions and theorems,
and a more modern approach based on set theory.
The latter starts with the notion of a structure
($\Pi = \langle {\mathcal P}, {\mathcal L}; I, B, \overline{C},
C_{\sphericalangle}\rangle$), a language for geometry
(involving variables, logical symbols, nonlogical
symbols and punctuation symbols), axioms and
theorems. We practiced by writing the first of
Hilbert's axioms in the formal language of
geometry. The following handout
was distributed.
|
Jan 23
|
We discussed some of Euclid's definitions and common notions, and all of Euclid's postulates. We proved the the
first proposition from Elements, and noted a gap.
We then began discussing Hilbert's axioms, and got through all the axioms
of incidence. We drew some pictures of small incidence
geometries.
|
Jan 25
|
The material today comes mostly from Section 6 of Hartshorne concerning Incidence Geometry. We briefly discussed the following (very important) concepts: the satisfaction of statement by a structure, the property of being a model of a set of statements, and independence of statements.
We worked on this handout.
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Jan 27
|
We are still in Section 6 of Hartshorne. We discussed
isomorphism, automorphism, isomorphism invariant,
truth and proof.
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Jan 30
|
We continued discussing truth, consequence and proof.
We introduced the notation $\Sigma\models \sigma$ and compared it
to $\Sigma\vdash \sigma$. Quiz 1.
|
Feb 1
|
We defined ``$\Sigma$ is satisfiable'',
``$\Sigma$ is consistent'',
``$\Sigma$ is categorical'', and
``$\Sigma$ is complete''. We stated Goedel's Completeness Theorem.
We then returned to geometry by introducing the axioms for betweenness
(Section 7). We explained how to prove a stronger version of axiom B2
from the other axioms:
If $A, B$ are distinct points, then there are points $C, D, E$ such that
$C*A*B,\, A*D*B,\, A*B*E$.
|
Feb 3
|
For practice,
we wrote the first betweenness axiom as a first-order sentence.
Then we discussed two types of models of the incidence axioms plus
the betweenness axioms. One was: the Cartesian plane over an ordered field,
and the other was an open disk in a Cartesian plane.
|
Feb 6
|
We defined line segments and sides of lines.
We discussed why the set of points not
on a line $\ell$ must lie on one side or the other.
Quiz 2.
|
Feb 8
|
We discussed plane separation and line separation.
|
Feb 10
|
We defined rays, angles, interior points to angles and triangles.
We argued that a point interior point to two angles of a triangle must also be interior to all three angles. We proved the Crossbar Theorem.
|
Feb 13
|
We summarized some consequences of the Incidence axioms plus the Betweenness axioms.
For example, we showed that
points on a line can be ordered, that the order is a dense order, that $A*B*C$ implies that $\overline{AB}\subseteq \overline{AC}$, and that
no triangle contains a line in its interior.
Quiz 3.
|
Feb 15
|
We introduced congruence of segments and showed that congruence
classes can be added or subtracted.
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Feb 17
|
We discussed the properties of segment ordering,
and worked on this Practice Exercise.
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Feb 20
|
We discussed proof writing, then took this
quiz.
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Feb 22
|
We discussed congruence of angles.
We defined supplementary angles and vertical angles,
and proved things about them.
|
Feb 24
|
We discussed the material on pages 94-95 of the book.
|
Feb 27
|
We proved that all right angles are equal.
We defined neutral geometry.
I circulated this review sheet.
Quiz 5.
|
Mar 1
|
We reviewed for the midterm.
We also worked on these
practice problems.
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Mar 3
|
Midterm! Solutions.
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Mar 6
|
We worked on practice problems.
|
Mar 8
|
We discussed material from Section 10.
|
Mar 10
|
This day was devoted to the Exterior Angle Theorem
and some of its consequences .
|
Mar 13
|
We discussed why the center of a circle is uniquely determined.
Quiz 6.
|
Mar 15
|
We discussed Props 17 and 18 of the Elements (17: the sum
of two angles of a triangle is less than the supplement of the third; 18: the greater side of a triangle subtends the greater angle).
|
Mar 17
|
We discussed Props 19 and 20 of the Elements (19: greater
angle of a triangle is subtended by the greater side;
20: triangle inequality).
|
Mar 20
|
We summarized our progress so far,
and noted that we are now need a method
to create models. We introduced
the definition
and some examples of
ordered fields.
Quiz 7.
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Mar 22
|
We sketched the
main ideas for constructing
a Pythagorean ordered field from a model
of neutral geometry + Playfair's axiom.
Through the course of the sketch we had
to prove the Pythagorean Theorem.
|
Mar 24
|
I announced that there would be
NO QUIZ
on the Monday after spring break.
We sketched the
main ideas for constructing
model of neutral geometry from
a Pythagorean ordered field. Then we spoke
about $\Omega$.
|
Apr 3
|
We reviewed the interpretability of
Neutral Geometry + Playfair's Axiom into
the theory of Pythagorean ordered fields, and
the reverse interpretation. We explained
that one can show that
the Cartesian plane over the Hilbert field $\Omega$
fails the circle-circle intersection property
if one shows that $\sqrt{1+\sqrt{2}}\notin \Omega$.
We discussed how one
shows that a complex number $z\in\mathbb C$
belongs to $\Omega$
(by producing a construction sequence),
and began a discussion of how one shows that
a complex number $z\in\mathbb C$ does not
belong to $\Omega$
(by a symmetry argument). We defined what
an automorphism of $\mathbb C$ is, and gave two
explicit examples: the identity function
and complex conjugation.
|
Apr 5
|
We sketched the reason why $\sqrt{2+\sqrt{2}}\in\Omega$,
but $\sqrt{1+\sqrt{2}}\notin\Omega$.
The latter conclusion shows that the Cartesian
plane over $\Omega$ does not satisfy the
circle-circle intersection property.
We followed this
handout.
|
Apr 7
|
We sketched the verification that the axioms
of neutral geometry hold in the Cartesian plane
over any Pythagorean ordered field.
We worked on this
handout. Solution.
|
Apr 10
|
We discussed
the circle-circle intersection property, and
Euclidean planes and fields.
Quiz 8.
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Apr 12
|
We completed the argument
for the theorem asserting that
the circle-circle intersection
for the Cartesian plane over $\mathbb F$
is equivalent to $\mathbb F$ being Euclidean.
I claimed that there is a smallest Euclidean field, $K$,
called the field of constructible numbers.
I also noted that Exercise 16.14 from the textbook
hints about how to show that $\Omega$ is exactly
the subfield of $K$ consisting of totally real
numbers. I mentioned that $\sqrt{1+\sqrt{2}}$
is constructible, but not in $\Omega$, while
$\cos(2\pi/7)$
is totally real but not in $\Omega$.
Finally, we discussed the link between
the field of constructible real numbers and
the plane figures constructible by straightedge
and compass.
|
Apr 14
|
We further discussed the relationship between
the field of constructible real numbers and
the plane figures constructible by straightedge
and compass, following this
handout.
We discussed how to compute
the Euler phi-function.
|
Apr 17
|
We discussed which regular $n$-gons are constructible.
Quiz.
|
Apr 19
|
We compared Euclidean geometry with projective geometry
and hyperbolic geometry.
Then we described an
ordering of $\mathbb Q(t)$ which makes it
a non-Archimedean ordered field. (Namely,
$(a_mt^m+\cdots+a_0)/(b_nt^n+\cdots+b_0)$ is defined
to be positive if $a_m/b_n$ is positive in $\mathbb Q$.)
|
Apr 21
|
We examined the Cartesian plane over the ring of bounded
elements of a non-Archimedean ordered field, and observed
that Playfair's Postulate fails.
Handout.
|
Apr 24
|
We discussed alternative axioms related to Playfair's postulate. Quiz.
|
Apr 26
|
We discussed this handout.
We then discussed the Poincare model: its points, lines,
incidence relation, betweenness relation, and its
angle measure.
|
Apr 28
|
We discussed inversion through the unit circle.
In particular, we discussed
why the inverse of a line through $O$ is the line itself,
while the inverse of a line not through $O$
is a circle through $O$.
|
May 1
|
More circular inversion!
The most important result
was the statement that if $\Gamma$ and $\gamma$ are circles,
then they are perpendicular to one another if
and only if $\gamma$ contains points $A, A'$
that are inverses of each other relative to $\Gamma$.
|
May 3
|
We discussed why the Poincare model
satisfies some of Hilbert's
axioms.
We discussed how to construct limiting parallels.
Practice problems.
|
May 5
|
Review.
|
