Download "Here`s to Looking at Euclid" A Snapshot of Hyperbolic Geometry

"Here's to Looking at Euclid"
A Snapshot of Hyperbolic Geometry
Nisreen Shiban
Lizzy Bradley
http://jwilson.coe.uga.
Introduction 1. "To draw a straight line from any point to any point."
2. "To produce [extend] a finite straight line continuously in
a straight line."
3. "To describe a circle with any centre and distance
[radius]."
4. "That all right angles are equal to one another."
5. "That, if a straight line falling on two straight lines make
the interior angles on the same side less than two right
angles, the two straight lines, if produced indefinitely,
meet on that side on which are the angles less than the
two right angles."
Introduction 1. A straight line segment can be drawn joining any two
points.
2. Any straight line segment can be extended indefinitely in
a straight line.
3. Given any straight line segment, a circle can be drawn
having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a
way that the sum of the inner angles on one side is less
than two right angles, then the two lines inevitably must
intersect each other on that side if extended far enough.
The Parallel Postulate
If two lines are drawn which intersect a third in such a way that the sum
of the inner angles on one side is less than two right angles, then the two
lines inevitably must intersect each other on that side if extended far
enough.
Proclus' Axiom:
"If a line intersects one of two parallels, it
must intersect the other also."
Playfair's Theorem:
"Through a point not on a given line there exists a
unique line parallel to the given line."
Girolamo Saccheri
D
C
Assume: A
AD = BC
AD BC
B
<ADC = <BCD ; not necess. right <s
D
C
Assume: A
AD = BC
AD BC
B
Assume <ADC and <BCD > 180
1. AB > CD
2. Sum of a triangle's angles > 180
3. A angle inscribed in a semicircle is always
obtuse.
Assume <ADC and <BCD < 180
1. AB < CD
2. Sum of a triangle's angles
< 180
3. Two parallel lines need
not have a common
perpendicular.
4. Parallel lines are not equidistant. When they have a common
perpendicular they recede from each other on each side of the
perpendicular. When they have no common perpendicular, they
recede from each other in one direction and are asymptotic in the
other.
"Out of nothing, I have created a strange
new universe." - Johann Bolyai
"One geometry cannot be more true than
another; it can only be more convenient."
- Poincare
Beltrami proved that inconsistencies existed
simultaneously in Hyperbolic Geometry and in
Euclidian -- ending the debate.
Theorem 1.1
The upper base angles of the Saccheri quadrilateral are equal.
By the SAS (side-angle-side) congruency condition, triangles ABC and BAD are
congruent.
D
C
A
B
Theorem 1.2
The line joining the midpoints of the upper and lower bases is perpendicular
to both. Therefore, the upper and lower bases lie on parallel lines sharing a
common perpendicular.
Theorem 1.3
The angle sum of a triangle does not exceed 180 degrees.
Theorem 1.3 continued
The angle sum of a triangle does not exceed 180 degrees.
Theorem 1.4
Consider a quadrilateral with a lower base that makes right angles with its
two arms.
i) if the upper base angles are unequal, so are the arms.
ii) If the arms are unequal, so are the upper base angles, with the greater
upper base angle opposite the greater arm.
From neutral to hyperbolic geometry
Theorem 1.5
In the Saccheri quadrilateral:
i) the altitude is shorter than the arms, and
ii) the upper base is longer than the lower base
A Theorem to Clarify Parallel Lines
Given two lines, if there exists a transversal
which cuts the lines so as to form equal
alternate interior angles or equal
corresponding angles, then the lines are
parallel with a common perpendicular.
A
Yet Another Parallel Theorem
The distance between two parallels with common
perpendicular is least when measured along that
perpendicular. The distance from a point on either
parallel to the other increases as the point recedes
from the perpendicular in either direction.
Dinosaur Math Jokes are the Best
Quiz time
In hyperbolic geometry, why can there be no
squares or rectangles?
List three real life applications to hyperbolic
geometry
If we define a rectangle as a quadrilateral with four right angles, then this is easily shown by cutting
the rectangle along the diagonal into two triangles. The sum of the angles of the two triangles must
be less than 2*180 = 360, so that we can't possibly have four right angles.
Einstein's Theory Of General Relativity is based on a theory that space is curved. The cause is
explained by the theory itself. Einstein's General Theory of Relativity can be understood as saying
that:
Matter and energy distort space
The distortions of space affect the motions of matter and energy.
If this is true then the correct Geometry of our universe will be hyperbolic geometry which is a
'curved' one. Many present-day cosmologists feel that we live in a three dimensional universe that is
curved into the 4th dimension and that Einstein's theories were proof of this. Hyperbolic Geometry
plays a very important role in the Theory of General Relativity.
Used for navigation around earth by pilots
The Sun causes some medium-scale curvature that - thanks to planet Mercury - we are able to
measure. Mercury is the closest planet to the Sun. It is in a much higher gravitational field than is
the Earth, and therefore, space is significantly more curved in its vicinity. Mercury is close enough
to us so that, with telescopes, we can make accurate measurements of its motion. Mercury's orbit
about the Sun is slightly more accurately predicted when Hyperbolic Geometry is used in place of
Euclidean Geometry.
http://www.youtube.com/watch?
v=ryTGnBe7CjY