2 The axiomatic approach

In order to avoid the many choices for the axiom of parallels
we look at those axioms that are satisfied by a *convex* region of
space (see Figure 1) in *any* axiomatic
geometry--Euclidean or not. The basic undefined entities are points,
lines and planes and the undefined relations among them are incidence
(to-lie-on) and separation (between-ness). The entities and relations
are made clear by the following axioms.

We begin with the axioms of incidence:

- Any two points lie on exactly one line and any line has at least two points on it.
- Any three points not all on one line lie on exactly one plane and any plane has at least three non-collinear points on it.
- If two points of a line are on a plane then every point of the line is on the plane (we then say that the line is on the plane).

Then we have the axioms of dimension:

- There are at least four points which are non-collinear and non-coplanar.
- If two planes meet then they meet in at least a pair of points.

The axioms of order and separation:

- If a point
*B*is between the points*A*and*C*then*A*,*B*,*C*are collinear and*B*is between*C*and*A*. - If
*A*,*B*,*C*are distinct collinear points then exactly one is between the other two. - If
*A*and*B*are distinct points there is at least one point*C*so that*B*is between*A*and*C*and a point*D*so that*D*is between*A*and*B*. - If
*A*,*B*,*C*are non-collinear points and*l*is a line in the plane of*A*,*B*,*C*so that none of these points lie on*l*, then if*l*contains a point between*A*and*B*then it must contain a point between*A*and*C*or a point between*B*and*C*.

Finally, we have the Archimedean least upper bound principle which gives us the least upper bound property for points on a line:

- Given a sequence of points
*A*_{n}and a point*B*such that*A*_{n + 1}is between*B*and*A*_{n}for all*n*, there is a point*C*such that*C*is between*B*and*A*_{n}for all*n*and for all points*D*lying between*B*and*A*_{n}for all*n*,*D*is between*B*and*C*.

A much briefer list of equivalent axioms was provided by Veblen using
only the notion of points and the relation of between-ness; we use the
symbol [*ABC*] to denote *B* lies between *A* and *C*.

- 1.
- There are at least two distinct points.
- 2.
- For any two distinct points
*A*and*B*there is a point*C*so that [*ABC*]. - 3.
- If [
*ABC*] then*A*,*B*and*C*are distinct. - 4.
- If [
*ABC*] then [*CBA*] but not [*BCA*]. - Defn.
- We say that
*C*lies on the*line**l*(*AB*) if*C*=*A*or*C*=*B*or [*ABC*] or [*ACB*] or [*CAB*]. A pair of lines is said to meet if they have a point in common. If*A*,*B*,*C*all lie on a line we say that they are collinear. - 5.
- If
*C*and*D*are distinct points on the line*l*(*A*,*B*) then*A*lies on*l*(*C*,*D*). - 6.
- There is at least one point not on
*l*(*A*,*B*). - 7.
- If
*A*,*B*,*C*are non-collinear and*D*,*E*are points so that [*BCD*] and [*AEC*] then there is a point*F*so that [*AFB*] and*F*lies on the line*l*(*D*,*E*). - Defn.
- If
*A*,*B*,*C*are non-collinear points then a point*D*is said to be*coplanar*with*A*,*B*,*C*if it lies on a line which meets two out of the three lines*l*(*A*,*B*),*l*(*B*,*C*),*l*(*A*,*C*). The locus of all such points is called the*plane*determined by the three non-collinear points*A*,*B*,*C*. - 8.
- There is at least one point not on the plane determined by
three non-collinear points
*A*,*B*,*C*. - 9.
- Any two planes which meet have at least two distinct points in common.
- 10.
- Let the points of a line be divided into two disjoint classes each
of which satisfy: if
*A*and*B*lie in the class the so does every point*C*such that [*ACB*]. Then there is a point*O*on the line and a pair of points*P*and*N*such that [*NOP*] and the two classes consists of all points between*N*(respectively*P*) and*O*. (In addition*O*lies in one of the classes.)

Now we will outline how the points, lines and planes of
such a geometry can be realised as the points, lines and planes of a
*convex* region in coordinate 3-space preserving all the relations
of incidence and between-ness; that is to say we have an *embedding* of our geometry into that of coordinate 3-space.

The first step is to construct a geometry consisting of ``ideal'' points, lines and planes; these would have ``existed'' if our geometry were not confined to a region.

The collection of all lines passing through a fixed point has the following properties:

- Any pair of lines in this collection are coplanar.
- For every point there is a line from this collection that contains it.

- We take any plane
*p*_{1}containing*L*_{1}and a different plane*p*_{2}containing*L*_{2}which meets*p*_{1}. Then we add the line of intersection*L*_{3}=*p*_{1}*p*_{2}to our collection. - Let
*p*be the plane containing*L*_{1}and*L*_{2}and*p'*be any other plane containing a line*L*_{3}constructed as above, such that*p'*meets*p*. We add the line of intersection*L*_{4}=*p**p'*to out collection.

Given a pair of distinct Points
**A** and
**B** we consider
those planes which contain a line each from the collections
corresponding to
**A** and
**B**. The collection of all these
planes determine an ``ideal line'' or Line. Any pair (*p*_{1}, *p*_{2}) of
distinct planes then determines a Line constructed as follows:

- We choose a pair of planes (
*q*_{1},*q*_{2}) such that each*q*_{i}meets each*p*_{j}. - Let
*A*_{i}be the Point determined by the pair of lines (*q*_{i}*p*_{1},*q*_{i}*p*_{2}). - A plane
*p*belongs to our collection if it contains a line in each of the collections*A*_{1}and*A*_{2}.

We say that a Point lies on a Line if every plane that belongs to the
collection determining the Line contains a line that belongs to the
collection determining the Point. This specifies the notion of incidence
for Points and Lines. It is clear that if a Line
**L** is
determined by the line *L'* then
**L** contains the Point
**A** if and only if *L'* lies in the collection corresponding to
the Point
**A**. In particular, if the Point
**A** is that
determined *A'*, then
**A** lies on
**L** if and only if *A'*
lies on *L'*.

Thus we have embedded our given geometry into an ``ideal'' geometry of Points and Lines with relations of incidence and separation. What are the axioms satisfied by this ``ideal'' geometry? We claim that the following axioms are satisfied:

- Any pair of Points
**A**and**B**lie on exactly one Line. We denote this line by**AB**. - Given four Points
**A**,**B**,**C**and**D**, the Line**AB**meets the Line**CD**if and only if the Line**AD**meets the Line**BC**. In this case we call the four Points*coplanar*. - There are five Points so that no four of these are coplanar.
- Given any five Points
**A**,**B**,**C**,**D**and**E**, there is a point**F**on the Line**AB**so that the Points**C**,**D**,**E**and**F**are coplanar.

The relation of separation is a little more intricate. We will try to
define the relation picturised in Figure 7. Given two pairs
(**A**,**B**) and
(**C**,**D**) of Points lying on a
Line
**L**; we say that the pairs *separate* each other if for
some plane *p* in the collection
**L** and some point *O* in the
plane *p*, there is a line *L'* in *p* and four points *A'*, *B'*, *C'*
and *D'* on *L'* so that:

- The lines
*OA'*,*OB'*,*OC'*and*OD'*are in the collections**A**,**B**,**C**and**D**respectively. *B'*and*C'*are between*A'*and*D'*, and*B'*is between*A'*and*C'*.

The axioms satisfied by the relation of separation are:

- If
**A**,**B**,**C**and**D**are collinear and distinct then exactly one of the relations**A****B**||**C****D**,**A****D**||**B****C**and**A****C**||**B****D**holds. - If
**A**,**B**and**C**are collinear then there is a Point**D**so that**A****B**||**C****D**. - If
**A****B**||**C****D**and**O**is a Point, and**L**any Line so that the Points**A**',**B**',**C**',**D**' of intersection of**L**with the Lines**OA**,**OB**,**OC**and**OD**respectively are distinct, then**A**'**B**'||**C**'**D**' holds. - Given a sequence of Points
**A**_{n}and Points**B**and**O**such that**A**_{n}**B**||**A**_{n + 1}**O**holds for all*n*, then there is a Point**C**so that**A**_{n}**B**||**C****O**holds for all*n*; and so for all points**D**so that**A**_{n}**B**||**D****O**holds for all*n*, we have**C****B**||**D****O**

Thus our ``ideal'' geometry is none other than coordinate Projective geometry of of dimension 3 over the field of real numbers. Hence we have an embedding of the given geometry into real Projective space geometry.

The collection of all Points on a Line thus forms a real Projective
line which cannot be totally ordered in a way so as to satisfy the
Archimedean least upper bound principle. Thus for any line there is at
least one Point
**A** which lies on the corresponding Line that
does not correspond to a point. Moreover, if *C* and *D* are points on
the line then every Point
**B** so that
**A****B**||**C****D** corresponds to a point *B*. In
other words the geometry we have is that of a *convex* subset of
Projective space. For any such subset one can find a Projective plane
that does not meet it. Thus the convex subset is actually contained in
real Affine space; in other words we have a coordinatisation of our
geometry.