David Jones
4/14/02
Math 300
Hyperbolic Quadrilaterals
Quadrilaterals
are amazing things. They seem like such
a simple idea, yet historically they have provided immense amounts of
information. It is a commonly known
fact that in hyperbolic geometry a rectangle can not exist. This has been verified by proofs that,
assuming hyperbolic geometry is consistent, prove that rectangles can not exist
in that geometry. But a rectangle is
only one form of a quadrilateral, and quadrilaterals can and do exist in other
geometries. This paper will examine two
special cases of quadrilaterals, the Saccheri Quadrilateral and the Lambert
Quadrilateral and their particular contributions to mathematical history.
A
Saccheri Quadrilateral is a figure consisting of two right base angles and two
congruent sides opposite each other and adjacent those base angles (Figure
1). A Saccheri Quadrilateral is also
known as an isosceles quadrilateral or isosceles birectangle because of the similarities
to an isosceles triangle. Each figure
has congruent sides and base angles.
The Saccheri Quadrilateral first arose from attempts by Omar Khayyamin
and Nasir Eddin al-Tusi to create a rectangle in neutral geometry, but was made
famous by a Jesuit priest named Girolamo Saccheri.[1]
Figure 1: A Saccheri
Quadrilateral
Saccheri proposed that there
were only three cases for the summit angles of said quadrilateral:
Case 1: The summit angles
are right angles.
Case 2: The summit angles are obtuse angles.
Case 3: The summit angles are acute angles.
Saccheri felt if he could
show that the first case was the only possible, reasonable case that he would
have proved the Euclidean Parallel Postulate which, paraphrased, says that
there can only be one line parallel to another through a given point. It is fairly simple to show that the first
case is possible. If one assumes Euclid
V to be correct, then lines with a common perpendicular are parallel. (Figure
2) This would make the quadrilateral a
rectangle, which was what Saccheri wanted to achieve.
Figure
2: A Saccheri Quadrilateral with right summit angles becomes a rectangle
Saccheri then showed the second case to be
impossible. He found that if the summit
angles were obtuse then this implied Euclid V, which implied right angles and
led then to a contradiction with the hypothesis.[2] The last case, however, stumped Saccheri,
who in the end concluded that: “The hypothesis of the acute angle is absolutely
false, because it is repugnant to the nature of the straight line.”[3] What Saccheri did not realize was that he
was on the fringes of hyperbolic geometry and could therefore never prove his
third case to be either true or false.
Saccheri actually
concluded that his acute angle hypothesis showed that there exist two straight
lines that merge into one straight line at infinity and have a common
perpendicular at infinity (Figure 3).[4] However, he was too bound by tradition to
consider a non-Euclidean geometry, and as a result did not realize he had
virtually uncovered spherical geometry.
While this figure seemed absurd to him, he was not convinced by his
proof that Euclid V did not hold absolutely.
Figure 3: Saccheri’s conclusion of two straight
lines that merge into one straight line at infinity.
Approximately 40 years after Saccheri, another
attempt was made to create a rectangle in neutral geometry. Johann Lambert used the idea of an Egyptian
Scientist ibn-al-Haytham to create a quadrilateral with three right angles to
try and force the fourth angle to be right as well (Figure 4). Using this quadrilateral Lambert was able to
disprove an obtuse fourth angle, but, just as Saccheri, was foiled by the proof
that acute angles could not occur. His
observations of the acute angle hypothesis led him to the conclusion that on a
sphere of imaginary radius the acute angle could be verified.[5] Sommerville defines a sphere as: “In space
of three dimensions the locus of a point which is equidistant from a fixed
point.”[6] When that fixed point is at infinity, the
sphere becomes a horosphere. When the center
is an ideal point, the sphere is called an Equidistant surface. While geometry on a sphere with a real
vertex is elliptical, Lambert’s results helped show that geometry on an
equidistant sphere is hyperbolic.[7] But, Lambert wanted to prove create the
rectangle in two dimensions using Neutral geometry, and couldn’t. Although he didn’t realize it at the time,
he was studying hyperbolic geometry and his discoveries, published
posthumously, would fuel later research in the area.[8]
Figure 4: A Lambert Quadrilateral
has three right angles and the fourth angle is undesignated.
While they didn’t realize it at the time, both Saccheri
and Lambert were entering into the realm of hyperbolic geometry, a realm where
parallelism was not the same as it had been for the two thousand years since
Euclid presented his geometry. Both of the presented quadrilaterals take on new
and interesting properties in hyperbolic geometry.
What distinguishes
hyperbolic geometry from Euclidean geometry is a single parallel axiom. In Euclidean geometry, Euclid’s fifth
proposition holds. This proposition is
equivalent to an axiom presented by David Hilbert in the late 1800’s stating
that there can be at most one line through a point not on a given line that is
parallel to the original line.
Hyperbolic Geometry is based on the negation of Hilbert’s axiom; more
than one line through a given point parallel to the original line can exist.[9] When one considers this change in the basis
of geometry, many interesting and intriguing deductions make themselves
apparent. The Saccheri and Lambert
quadrilaterals are no exception to this rule.
As stated above, in hyperbolic geometry a rectangle can
not exist.[10] Yet, Lambert and Saccheri did everything in
their power to try and force one to exist.
While he did not know he was doing so, Saccheri deduced many of the
theorems that became foundations for non-Euclidean geometry.[11]
With regards to his quadrilateral (refer back to Figure 1), Saccheri made some
discoveries that went against many previously concluded theorems. If one were to construct an isosceles
quadrilateral ›ABCD with right base angles ÐA and ÐB, and then construct acute
summit angles, segment CD must then be longer than segment AB, and segments AB
and CD will only have a common perpendicular at the midpoint (Figure 5). This idea of a single common perpendicular
flies in the face of anyone used to Euclidean Geometry.
While interesting things can be derived from the Saccheri
Quadrilateral, even more can be learned from the Lambert Quadrilateral. If there can not be a rectangle in
hyperbolic geometry, then by
the law of the excluded middle the fourth angle in a Lambert Quadrilateral
(refer back to Figure 4) must be either acute or obtuse. Using the same logic that Lambert used to
show that the angle could not be obtuse in Euclidean geometry one can show that
the angle can not be obtuse in hyperbolic geometry.
Figure 5: A Saccheri
Quadrilateral with acute summit angles.
Therefore, the angle must be acute. But, when one looks at the angle of the diagonal to the non-right angle, one learns that the fourth vertex does not exist, but is an ideal point at infinity (Figure 6).[12] This is consistent with Lambert’s discovery of the sphere of imaginary radius. The two sides adjacent to the undetermined angle act as asymptotic rays to the diagonal and only come into contact at infinity. It is from this conclusion that Lambert discovered the Equidistant surfaces. In Euclidean geometry a triangle formed by the arcs of a great circle has an angle sum greater than 180°. The excess above 180° is proportional to the area of the triangle, with the constant of proportionality being r2, or the radius of the sphere. To find triangles with less than 180° Lambert speculated that if r were replaced by ir (where i is Ö-1) then the proportionality constant would be negative. From this conclusion the sphere of imaginary radius was deduced.[13] The point at infinity had at that time not been considered in geometry, and that is why Lambert found the idea embarrassing and did not publish his discoveries.[14]
Figure 6: A Lambert
Quadrilateral with the fourth vertex at infinity.
The historical progression of hyperbolic geometry was a
2000-year process. With the foundation
set by Euclid, Geometers around the world worked to solve the mysteries of the
Universe. Unfortunately for them and
fortunately for us the foundation that Euclid set could not solve all the
mysteries. Lambert and Saccheri were
two men whose studies furthered the science of mathematics through both their
successes and their “failures.” The
quadrilaterals that they studied have become an integral part of the modern day
study of geometry. As such, they should
be revered in the annals of history as partial founders of non-Euclidean
Geometry, and their respective quadrilaterals looked at as launching points for
the study of such geometries.
References Cited
Duncan
M. Y. Sommerville, The Elements of
Non-Euclidean Geometry, G. Bell and Sons, London, UK , 1914.
H.S.M. Coxeter, Non-Euclidean Geometry, Sixth Edition,
Mathematical Association of America, Washington D.C., 1998.
Harold E. Wolfe, Introduction to Non-Euclidean Geometry, Holt,
Rinehart and Winston, Inc., New York, NY, 1945.
Marvin Jay Greenberg, Euclidean and Non-Euclidean Geometries,
Third Edition, W.H. Freeman and Co., New York, NY, 1993.