__Table of Contents__
Module 7
Undefined Terms:
Hyperbolic Postulates
A4 If two intersecting lines cross at adjacent
congruent angles, then each of these is congruent to any other angle created the same way.
This is necessary because Euclid’s proof of
SAS used the notion of moving parts of one triangle over another to achieve coincidence or
superposition.
It is necessary to define “moving”
objects and the act of defining movement opens the door to our next module:
Transformational Geometry. For now, keep “moving” out of the picture.
Hyperbolic Axioms
We may also use any theorems we proved (or proved
by anyone!) that don’t implicitly or explicitly use the existence of parallel lines.
Models
Klein-Beltrami Model
This model is excellent for illustrating some
startling hyperbolic facts about parallel lines.
In Euclidean geometry,
Why does mathematics have such a term? (ie Why do
we care?)
Notice that two lines can be parallel and are NOT
equidistant.
They could be parallel and share a boundary point (
this is called asymptotically parallel) since boundary points are not in the model.
So, we have kinds of parallelism* and we’ve
lost the transitivity of parallel lines.
The distance formula for the Klein-Beltrami model
PPT Slide
Imagine what a triangle looks like in the
Klein-Beltrami model. It looks normal, Euclidean…and we’ve already agreed the
the sum of the interior angles is less than 180 degrees…so the image of a triangle is
a problem.
A more visually intuitive model, the Poincare disk,
comes with a more difficult visualization of what lines are, but the distance formula is
slightly simpler.
Poincare Disk
Let’s look at a sketch of orthogonal and
non-orthogonal circles on the overhead.
Back to the model:
The distance formula is
Two segments, AB and CD, are called
Poincare-congruent if D(A,B) = D(C,D).
Upper Half-Plane Model
A line will be any portion of a circle whose center
is on the x axis.
The algebraic formula for a line is
The Upper Half-Plane model is an unbounded model.
It is a proper subset of the Euclidean plane.
The formula for distance is
Cosh(x) is a function that is more formally known
as hyperbolic cosine. Arccosh(x) is the inverse to hyperbolic cosine. This function has
uses on its own -- it describes curves that happen in nature and was known before its use
here.
The curve of a power line suspended from two light
poles has a shape defined by f(x) = a cosh(x). The shape is also known as a catenary. The
hyperbolic cosine formula is
One of the original researchers who learned how to
work with hyperbolic sines and cosines was Johann Lambert, whose extension of the Saccheri
quadrilateral is called a Lambert quadrilateral.
The types of parallel lines still show very nicely
in this model. In fact, this model is quite easy to set up in Sketchpad or on graph paper.
Minkowski Model
Points are points on the surface of half of a
two-sheeted hyperboloid…this is a special shape in three dimensions.
The distance formula requires even more knowledge
of mathematics -- and angle measure is a trigonometric function of a vector quantity.
Calculus III is a minimum requirement to work on the Minkowski model.
I’ve included it to show you that, like with
Spherical geometry, geometry can be defined on surfaces.
Notebook Problem
Properties
2. There are no triangles with the sum of the
interior angles is 180 degrees.
4. There are no lines that are everywhere
equidistant.*
6. No rectangles exist.
11. There are triangles that cannot be
circumscribed.*
As one of my favorite authors* wrote:
Connections to Real Numbers
In Group V, Hillbert deliberately introduced the
notion of length and tied in both rational and irrational numbers.
In some versions of the axioms of Euclidean Plane
Geometry you will find a “Ruler Postulate” (SMSG Postulate 3):
3b. To every real number there corresponds exactly
one point of the line.
Using the Klein-Beltrami model and the simplest
line that there is: the diameter that is the x axis, I want to review that distance
formula with pictures and numbers.
You will see some versions of the distance formula
using the natural logarithm “ln(x)” instead of the base 10 logarithm
“log(x)” … this creates some slight numerical changes but doesn’t
change the ability of the function to map the whole real line into a bounded space.
I’ll work this example on the overhead.
Theorems
definition
definition
An example
Area
Are the area formulas in the non-Euclidean
geometries really different?
“Area” in geometries
It is, of course, more than unsettling to realize
that a collection of objects with area equal to zero has a non-zero real number
representing the collective area…but that’s an issue for real “foundations
of mathematics” buffs.
Theorem 2
So, as in Euclidean geometry, area is additive.
Decomposition is the technique of choice for figuring out the areas of polygons…what
do you suppose a regular pentagon looks like in the Poincare model?
How do you suppose that you would calculate the
area of the pentagon? Remember that k is fixed, so triangles with the same defect have the
same area no matter what they look like...
In Euclidean geometry, we used the definition of
the area of a rectangle to get to the definition of the area of a triangle.
Theorem 3
Theorem 4
First I’ll outline the construction and then
I’ll show you on the overhead. (note: this process is a picture, not a proof)
Next draw perpendiculars to this line from each
vertex of the triangle.
I’ll be using the Klein-Beltrami model because
it’s the easiest for these pictures. Please keep in mind the inaccuracy of the image
-- the shapes will look Euclidean and they’re really, really not.
Before sketching the proof, it is necessary to
realize that AAS is a congruence condition in Hyperbolic geometry. It was a situation
peculiar to Spherical geometry that caused AAS to be unreliable: the existence of poles.
As all the cases are reasonably similar, I’ll
only outline Case I:
By similar reasoning,
This makes ACRP a Saccheri quadrilateral. (Claim 1)
Now, PQ = PM + MQ and since
Now notice that ? MAP ? ? MBQ. This means that
For similar reasons,
m?MAP + m?BAC = m?MBQ + m?BAC
Adding both of these equations gives
The left hand side adds angles 1,2, 5, and 6 giving
the sum of the summit angles (the angles that are acute, the angles where the defect
occurs.)
So, we’ve shown that this construction creates
an associated quadrilateral for any triangle, with base measurement twice that of the line
segment joining the chosen midpoints and with the same defect as the triangle.
Do you think that this idea of having an associated
Saccheri quadrilateral translates (within reason) to
Biangles revisited
Hyperbolic geometry has biangles, too. (And
hyperbolic biangles have alternate names; it’s author-dependent.)
definition
Saccheri Quadrilaterals
Saccheri quadrilaterals are called rectangles in
Euclidean geometry.
The line segment joining the midpoints of the base
and summit is perpendicular to both of them.
In Hyperbolic geometry, Saccheri quadrilaterals are
called just that.
The summit of a Saccheri quadrilateral is longer
than its base.
The base and summit are ultra parallel, as are the
other two sides.
In Spherical geometry, a Saccheri quadrilateral is
called that.
The line joining the midpoints of the base and
summit is perpendicular to them both.
Neutral properties
Biperpendicular quadrilateral --
Geometry Specific
Called another name in Euclidean geometry
(rectangle)
Compare/Contrast
2. Two lines perpendicular to the same line are
3. Parallel lines are
4. Sum of interior angles of a triangle:
5. Area of a triangle is
6. A triangle of maximal area
7. Two triangles with congruent corresponding
angles are
8. Biangles
9. The intersection of two distinct lines is
10. Straight lines are
11. “betweenness” makes sense
12. If a line intersects one of two parallel lines,
it
13. The summit angles of a Saccheri quadrilateral
are
14. Exhibits plane separation*
Anyone who chooses to do the Mobius Strip project
will be working with plane separation issues. A line in Euclidean geometry separates the
plane into two regions that share the line as a boundary…a Mobius strip can help
model a way to connect 2 points without crossing the line.
Many theorems depend on having an interior and
exterior to a shape. In a geometry that lacks this property, none of those theorems hold.
Working on Mobius strips emphasizes the difference
between infinite and boundless, too.
Changing gears a bit
What are some equivalent statements to this?
The summit angles of every Saccheri quadrilateral
are 90o (P/2 in radian measure).
This is a list of the reasons to look for in
theorems when you are trying to use a familiar Euclidean one in a non-euclidean geomety.
If these appear, you’re warned that the proof depends on Euclidean parallel lines.
So now we’ve looked at lists of properties
that are the same or are different.
Neutral Geometry
Euclid did this, too…note that 29 of his
propositions (theorems) were proved without the fifth postulate.
*
**a major caveat here:
We’re ok with Euclid’s work up to Book I,
Proposition 28.
Our undefined terms:
Incidence Axioms* from SMSG
P7 Any three points line in at least one plane, and
any three non-collinear points lie in exactly one plane.
Definitions:
Equivalence relation:
Given set elements: a, b, and c and a relationship,
R, on S
and symmetric means
and transitive means
A relationship that has all three qualities is
called an equivalence relation.
N Theorem1
N Theorem 2
N Theorem 3
N Theorem 4
N Theorem 5
Proof:
Since L is a line in the plane so that it separates
the plane into two sets.
Since L doesn’t intersect CB, then B and C are
on the same side of L.
Theorem 6
Theorem 7
Have proofs of these in mind for the upcoming test.
Definition:
Theorem 8
Proof
Let E be the midpoint of side BC.
?AFC ? ? BFE and ? ACF ? ? FBE
Overview
N Theorem 9
Logical language
The conditional
Conditional, cont.
N Theorem 6:
N Theorem 10
N Theorem 11
N Theorem 12
All of these lead to a
N Theorem 13
Proof: Consider DABC. We must show that AC + CB
> AB.
Select the point on ray AC called S such that
This creates an isosceles triangle CBS by
construction. Note that
Since C is in the interior of ?ABS
Using AC + CS = AS
N Theorem 14
N Theorem 15
One theorem found in many texts is:
An amazing aside
In Euclidean geometry, the length of the segments
joining two midpoints of the sides of a triangle is 1/2 the length of the other side.
They fixed the observers head to avoid motion
parallax and used three lights in a triangular pattern (symmetric, low intensity in a dark
setting, along a eye-level plane). A fourth light is introduced and the observer as asked
to place the light on the left, equidistant from the two endpoints and then a fifth light
on the other side. This is locating those midpoints.
After some additional instruction and practice, the
experiment began and the location of “equidistant” was recorded numerous times
for seven subjects.
Source:
Now |