Other Geometries
There are MANY geometries other than the familiar Euclidean. We will look at several of them. Please pay attention to what is similar and what is different about each as we progress.
Finite Geometries
Incidence Geometry
Taxicab Geometry
Spherical Geometry
Hyperbolic Geometry
Introduction to Axiomatic Systems
In studying any geometry, it is important to note the axiomatic framework of the geometry and keep it in mind. Often students are so challenged by the details that they forget that there is a structure to geometry. Each geometry has a framework called its axiomatic system. An outline of a typical axiomatic system is below.
Any axiomatic system has four parts:
undefined terms
axioms (also called postulates)
definitions
theorems
The undefined terms are a short list of nouns and relationships. These terms may be visualized but cannot be defined. Any attempt at a definition ends up circling around the terms and using one to define the other. These are the basic building blocks of the geometry. It is usually a good idea to have a mental image of the undefined terms – a visualization of the objects and how they relate.
Axioms (or postulates) are a list of rules that define the basic relationships among the undefined terms and make clear the fundamentals facts about a system. Axioms are always true for the system. No deviation from the facts they state is permitted in working with the system.
Definitions and theorems build on the axioms and undefined terms, clarifying relationships and auxiliary facts.
We will be using, with slight modification, the set of undefined terms and axioms developed by The School Mathematics Study Group during the 1960’s for this module. This list of axioms is not as brief as one that would be used by graduate students in a mathematics program nor as long as some of those systems in use in middle school textbooks. One definite advantage to the SMSG list is that it is public domain by design. We will be using the Cartesian coordinate plane as our visualization of the undefined terms of Euclidean geometry.
Once we have spent time learning the axioms, some definitions and a few theorems we will move to the second module on Euclidean Topics and look at geometric shapes and proofs that require using the axioms, definitions, and theorems in concert
In the following geometries:
In the following two examples of finite geometries, each has exactly one model and neither has an alternative model with more or fewer points. The axioms are quite specific and controlling on this issue.
Note that the axioms are quite specific about which undefined terms are “incident” or bearing upon one another in all three geometries.
Then we will explore another type of geometry is called an Incidence Geometry.
The axioms for an Incidence Geometry are specific about a couple of things but do allow at least two distinctly different models.
In TCG, EG, SG there are only one model. HG has several 2D and a 3D model!
In Euclidean Geometry, there is exactly one line through a given point not on a given line that is parallel to the given line. In the following geometries, some have NO parallel lines, others have more than one through that point. It’s a variable kind of thing!
The Three Point Geometry
Undefined terms:point, line, on
Axioms:A1There are exactly three distinct points.
A2Two distinct points are on exactly one line.
A3Not all the points are on the same line.
A4Each pair of distinct lines is on at least one point.
Model
Three Point Geometry exercise:
There’s only one basic model for this geometry. Sketch it here:
Possible Definitions:
Collinear Points – points that are on the same line are collinear.
Theorems:
Theorem 1:Each pair of distinct lines is on exactly one point.
Theorem 2:There are exactly 3 distinct lines in this geometry.
Proof of Theorem 1
Theorem 1:Each pair of distinct lines is on exactly one point.
Suppose there’s a pair of lines on more than one point. This cannot be because then the two lines have at least two distinct points on each of them and Axiom 2 states that
A2Two distinct points are on exactly one line.
Thus our supposition cannot be and the theorem is true.
QED
This type of proof is called a proof by contradiction. It works like a conversation.
Someone asserts something and someone disagrees and contradicts them. The assertion is the theorem and the contradiction is the sentence that begins with “Suppose…”.
Then the first person points out why the supposition cannot possibly be true…which has the handy property that it proves the theorem.
The proper contradiction to an assertion that “exactly one” situation is true is to suppose that “more than one” is true.
Note: we have no need of a distance formula or a way to measure angles with this simple little geometry. These things MUST be defined in the axioms for you to have them!
Given a line and a point not on that line, how many lines are there through that point and parallel to the given line?
The Five Point Geometry
Undefined termspoint, line, on
AxiomsA1There are exactly five points.
A2Any two distinct points have exactly one line on both of them.
A3Each line is on exactly two points.
Models
Points:
Lines:
Note that the lines crossover one another in the interior of the “polygon” but DO NOT intersect at points. There are only 5 points! LINE STUFF!
Possible Definitions
Triangle -- a closed figure formed by 3 lines. An example: is a triangle.
How many triangles are there?
Quadrilateral – a closed figure formed by 4 lines. An example: is a quadrilateral. How many quadrilaterals are there?
Parallel lines – two lines are parallel if they share no points.
Note that is parallel to line. So are and …
Let’s look at a given line and a given point…
Other possible definitions:
Collinear points
Interior
Plane
Theorem
Each point is on exactly 4 lines.
An Incidence Geometry
Undefined terms:point, line, on
Axioms:A1There is exactly one line on any two distinct points.
A2Each line has at least two distinct points on it.
A3There are at least three points.
A4Not all the points lie on the same line.
Note: no distance, no angle measure
Models:
Two examples follow; there are others.
Definitions – We’ll look at the models and see what makes sense…
Parallel lines
The distance from point one to point two
Intersecting lines
Triangles
Quadrilaterals
Between or interior
Concurrent lines – share a point…more than 2 are “concurrent”
Theorems:
Theorem 1:If two distinct lines intersect, then the intersection is exactly one point.
Theorem 2:Each point is on at least two lines.
Theorem 3:There is a triple of lines that do not share a common point.
Theorems must be totally true in EVERY MODEL!
An Incidence Geometry, continued
A six point model:
The ONLY points are the 6 dots that are labeled. Note that in the interior of the “polygon” there are NO intersections of lines at points.
Imagine the points are little Styrofoam balls and that the lines are pipe cleaners…where two pipe cleaners lay on top of each other there’s no intersection only a “crossover”. Only at the ends where the ends are stuck into the balls is there a point and an intersection.
The points are: A, B, C, D, E, and F.
The lines are:
ABBCCDDEEF
ACBDCEDF
ADBECF
AEBF
AF
A1There is exactly one line on any two distinct points. See the list
A2Each line has at least two distinct points on it. See the “endpoints”.
A3There are at least three points. There are 6 which is “at least 3”.
A4Not all the points lie on the same line. See the list.
Theorem 1:If two distinct lines intersect, then the intersection is exactly one point.
Theorem 2:Each point is on at least two lines.
Theorem 3:There is a triple of lines that do not share a common point.
Theorem 1:For example: lines BF and BE intersect only at B.
The “crossovers” in the interior are not intersections.
Theorem 2:Each point is on 5 lines which is “at least 2”.
Theorem 3:All you have to do with Theorem 3 is show one triple:
AB, CF, and ED do not share a common point.
Let’s look at the situation with respect to parallel lines.
We will use the definition that two lines parallel lines if they share no points.
In Euclidean Geometry, if you have a line and a point not on that line, there is exactly one line through the point that is parallel to the given line.
Let’s check this out:
Take line AC and point B. These are a line and a point not on that line.
Now look at lines BF, BE, and BD. Both of these lines are parallel to line AC.
(recall that the lines that overlap in the “interior of the pentagon” do NOT intersect at a point – there are only 6 points in this geometry).
So there are exactly THREE lines parallel to a given line that are through a point not on the given line. This is certainly non-euclidean!
An Incidence Geometry, continued
A Model with an infinite number of points and lines:
Points will be , the interior of the Unit Circle, and lines will be the set of all lines that intersect the interior of this circle.
So our model is a proper subset of the Euclidean Plane.
Model:
Note that the labeled points (except H) are NOT points in the geometry. A is on the circle not an interior point. It is convenient to use it, though.
H is a point in the circle’s interior and IS a point in the geometry.
We cannot list the number of lines – there are an infinite number of them.
Checking the axioms:
A1There is exactly one line on any two distinct points.
This model is a subset of Euclidean geometry and the axiom holds.
A2Each line has at least two distinct points on it.
Each line has an infinite number of points by Euclidean Axioms.
A3There are at least three points.
The unit disc has an inifinite number of points.
A4Not all the points lie on the same line.
True
Definitions:
Parallel lines: lines that share no points are parallel.
In Euclidean Geometry, there is exactly one line through a given point not on a given line that is parallel to the given line.
Interestingly, in this geometry there are more than two lines through a given point that are parallel to a given line.
Let’s look at lines GC and GB. They intersect at G…which is NOT a point in the geometry. So GC and GB are parallel. In fact, they are what is called asymptotically parallel. They really do share no points.
Now look at P1P2. It, too, is parallel to GC. Furthermore both P1P2 and GB pass through point H.
P1P2 is divergently parallel to GC.
Not only is the situation vis a vis parallel lines different, we even have flavors of parallel:
asymptotic and divergent. So we are truly non-euclidean here, folks.
Theorem 1:If two distinct lines intersect, then the intersection is exactly one point.
Inherited from Euclidean Geometry.
Theorem 2:Each point is on at least two lines.
Each point is on an infinite number of lines.
Theorem 3:There is a triple of lines that do not share a common point.
FE, GC, and AD for example.
The SMSG Axioms for Euclidean Geometry
Undefined Terms:point, line, and plane
We take as our beginning point the undefined terms:
point, line, and plane.
Most people visualize a point as a tiny, tiny dot. Lines are thought of as long, seamless concatenations of points and planes are composed of finely interwoven lines: smooth, endless and flat.
Think of undefined terms as the basic sounds in a language – the sounds that make up our language for the most part have no meaning in themselves but are combined to make words.
The grammar of our language and a good dictionary are what make the meaning of the sounds. This part of language corresponds to the axioms and definitions that you will find next in the module.
From there the facts, flights of fancy, and content-laden sentences are built – these are the theorems and definition in an axiomatic system.
The conventions of the Cartesian plane are well suited to assisting in visualizing Euclidean geometry. However there are some differences between a geometric approach to points on a line and an algebraic one, as we will see in the explanation of Axiom 3.
SMSG Postulates for Euclidean Geometry:
A1.Given any two distinct points there is exactly one line that contains them.
A2.The Distance Postulate:To every pair of distinct points there corresponds a unique positive number. This number is called the distance between the two points.
A3.The Ruler Postulate:The points of a line can be placed in a correspondence with the real numbers such that
A.To every point of the line there corresponds exactly one real number.
B.To every real number there corresponds exactly one point of the line,
and
C.The distance between two distinct points is the absolute value of the difference of the corresponding real numbers.
A4.The Ruler Placement Postulate:Given two points P and Q of a line, the coordinate system can be chosen in such a way that the coordinate of P is zero and the coordinate of Q is positive.
A5.A.Every plane contains at least three non-collinear points.
B.Space contains at least four non-coplanar points.
A6.If two points line in a plane, then the line containing these points lies in the same plane.
A7.Any three points lie in at least one plane, and any three non-collinear points lie in exactly one plane.
A8.If two planes intersect, then that intersection is a line.
A9.The Plane Separation Postulate: Given a line and a plane containing it, the points of the plane that do not lie on the line form two sets such that
A. each of the sets is convex, and
B.if P is in one set and Q is in the other, then segment PQ intersects theline.
A10.The Space Separation Postulate: The points of space that do not line in a given plane form two sets such that
A.each of the sets is convex, and
B.if P is in one set and Q is in the other, then the segment PQ intersects
the plane.
A11.The Angle Measurement Postulate: To every angle there corresponds a real number between 0 and 180.
A12.The Angle Construction Postulate:Let be a ray on the edge of the half-plane . For every between 0 and 180 there is exactly one ray with in such that .
A13.The Angle Addition Postulate:If is a point in the interior of , then .
A14.The Supplement Postulate:If two angles form a linear pair, then they are supplementary
A15.The SAS Postulate:Given an one-to-one correspondence between two triangles (or between a triangle and itself). If two sides and the included angle of the first triangle are congruent to the corresponding parts of the second triangle, then the correspondence is a congruence.
A16The Parallel Postulate:Through a given external point there is at most one line parallel to a given line.
A17.To every polygonal region there corresponds a unique positive number called its area.
A18.If two triangles are congruent, then the triangular regions have the same area.
A19.Suppose that the region R is the union of two regions R1 and R2. If R1 and R2 intersect at most in a finite number of segments and points, then the area of R is the sum of the areas of R1 and R2.
A20.The area of a rectangle is the product of the length of its base and the length of its altitude.
A21.The volume of a rectangular parallelpiped is equal to the product of the length of its altitude and the area of its base.
A22.Cavalieri’s Principal:Given two solids and a plane. If for every plane that intersects the solids and is parallel to the given plane, the two intersections determine regions that have the same area, then the two solids have the same volume.
Taxicab Geometry
- uses the SAME undefined terms and axioms as EG up to SAS A15.
- is based on the Cartesian Plane. We will use the SAME points in the plane as we use for Euclidean Geometry and the same lines. We will use the same way of measuring angles, too. BUT we will use a DIFFERENT formula to measure distance.
Let’s first review the old distance on a line formula: in one dimension and in two dimensions:
Where Euclidean Geometry then uses the Pythagorean Theorem to get distance, we will simple ADD the two distances:
Do you see WHY it’s called “Taxicab”?
HANDOUT 6.1 A
Let’s find some Taxicab distances and some Euclidean distances and see what’s going on.
Find the distances between the following points:
Points / Distance E / Distance T(0, 0) to (0, 4)
(1, 1) to (3, 4)
(−1, 2) to ( −5, −2)
(0, 0) to (2, 2)
(0, 0) to (5, 0)
What do you notice?
Another interesting point to note is that the Euclidean distance is always less than or equal to the Taxicab distance (p. 211).
Let’s explore a triangle or two. BOTH geometries have triangles, but Taxicab Geometry has some surprises in theirs.
HANDOUT 6.1 B
Let’s study some triangles. Grab a piece of graph paper and get started:
Here’s one:
Find the vertices (1, 2), (3, 2) and (2, 3). Now let’s list the Euclidean characteristics.
Side lengths (Euclidean):
Angle measurements:
Now those angle measurement stay the same in TCG.
Side lengths (TCG):
Oops what did we find?
Here’s another:(5, 4), (5, 1) and (9, 1)
It’s a right triangle in both geometries, isn’t it?
Let’s measure the sides.
EG:TCG:
Ooops! What theorem is NOT true in TCG?
Here’s a third one:
Let’s move into Quadrant 2: Put on the base: (3, −5) and (3, −8). Let’s use TRIG to find the coordinates of the apex angle:
Recall:
we can solve for the height with this!