Download 09 Neutral Geometry I

Introduction to neutral geometry
We give the axioms and basic definitions that will be true through almost all the geometry
we want to study.
OK, now that we’ve seen a couple weird geometries, let’s take a look to see what we
really want out of geometry. We will give some very basic things that we really want to be
true, and then investigate the ramifications of these basic notions.
Euclid is the master. The Elements has been used as a standard geometry text up until
this century—over 2000 years! After the bible it is the second most printed book of all time
in western civilization (Mao’s Little Red Book is the second-most printed overall). We will
vary a little from Euclid’s approach, but not much.
Euclid started with giving a list of definitions. In modern formalism, we realize that we
can’t quite define all the things the way Euclid did, and that some things have to be left
undefined. Euclid’s attempts at definitions of these terms will indicate how we intend to use
them more than determining their properties. So let’s get started!
Definition 1 Point: that which has no part.
Now today, we use “point” as an undefined term. All this really indicates is that points
do not have any constituents—they are indivisible, like atoms.
Definition 2 Line, curve, plane, surface.
We also leave these undefined in a modern treatment, though Euclid tried to explain
them. Actually, Euclid allowed general curves, and called them “lines” and singled out
“straight” lines separately. His curves were always finite in extent, having endpoints which
he called “extremities” and which he specifically noted were points. We do not allow curve
to cross themselves. A straight line was a curve where all the points lay strictly between the
endpoints. Similar reasoning applied to planes (which were also finite) and their extremities,
which were lines.
We will use the following notations. A line should refer to the infinite extent in both
directions object and can be denoted either by a single letter (usually script) as in l, or by
←→
the double-arrow overline notation: AB. The ray which starts at A and continues through
−→
B will be denoted AB and the line segment with endpoints A and B is AB. The distance
from A to B is undecorated AB.
Definition 3 Two rays with the same initial point, are called an angle.
Actually, there are two angles associated to such a pair of rays, depending on which
portion of the plane is decided to be the interior of the angle. (It is actually an assumption—
an axiom—that we will state later that angles do divide the plane into two parts. Except
for zero angles, of course!) We will allow straight angles and zero angles.
The next definition requires us to measure something. It is implicit in Euclid’s work that
everything has some kind of measure—segments have length, angles have measures, regions
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have area, etc. In modern developments of geometry, there are usually axioms that support
this. For instance, the Ruler postulate states that for every two points there is a unique
distance between them. The Angle measure postulate says that every angle has a measure,
and that two angles are declared congruent if and only if they have the same measure.
Inherent in these is the idea of a rigid motion. The ideas is that geometrical objects are
rigid, and you can move them about and the don’t “change shape.” And when you move
on on top of another and the line up perfectly, we same they are the same in some way.
This both does and does not correspond to our experience. We are used to moving objects
around and having them not change. But think about perspective drawing—as an object
moves away from us it appears to shrink. So we do need to be a little careful!
Definition 4 If two lines meet so that adjacent angles are equal, these angles will be caller
right angles, and the lines will be called perpendicular.
Though it looks obvious, we do actually need to prove that if one pair of adjacent angles
are equal, all four pairs are equal!
Definition 5 An obtuse angle is greater than a right angle. An acute angle is smaller
than a right angle.
Definition 6 A circle is a curve in a plane where the distance from each point on the curve
to a fixed point of the plane are equal. The fixed point is called the center of the circle. A
diameter is a line segment whose ends lie on the circle and which contains the center of the
circle.
Of interest to note: from this definition, it is not clear that circles really exist! There
are two conditions: that it is a curve, and that all the points are equidistance from a fixed
point. It is not clear that these two are compatible! We’ll actually need an axiom to take
care of this.
Definition 7 A polygon is a curve that is made from a finite collection of line segments.
It is implicit in this definition that the line segments meet only at their endpoints. These
endpoints will be called vertices, and the edges can only meet at vertices. Each vertex is
on exactly two edges. The segment between two vertices that are not on the same edge are
called diagonals. A polygon is called convex if all its diagonals are contained in its interior.
Definition 8 A triangle is a polygon composed of three line segments.
Definition 9 An equilateral triangle is one whose three sides are equal (in length). An
isosceles triangle is one that has at least two equal sides.
Finally,
Definition 10 Two lines in the same plane are parallel if they do not meet.
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