The Klein View of Geometry

The Klein View

Klein defines a geometry as

a set S of objects - the points of the geometry, and
a group G of transformations of S.

Definitions

a geometrical figure is a set of points, i.e. a subset of S.
a geometrical property (g-property) is a property invariant under G.
geometrical figures P and Q are g-congruent if there is an element of G
mapping P to Q.
Note that, if P and Q are g-congruent, then, as g-properties are invariant under G,
P and Q have identical g-properties.
The study of the geometry defined by G consists of

the description of the g-congruence classes,
the identification of g-properties, and
theorems about g-congruence classes and g-properties.
There are no axioms - just theorems. This has the advantage that such a "geometry"
is consistent. With the axiomatic approach, it is always possible that the axioms lead to a
contradiction. In such a case, there can be no "model" for the geometry.
As an example, euclidean geometry may be described as having

the set R², and
the group E(2) generated by translations, rotations and reflections of R².
Since each such transformation preserves distance between points, distance is a
geometrical property of euclidean geometry.
We can use the group to prove that each of Euclid's axioms is a theorem of this geometry.
The other advantage of Klein's approach is that it allows us to relate different geometries.
Suppose that G is a group of transformations of the set S, and that H is a subgroup of G.
Then H is also a group of transformations of S, and so defines another geometry on S.

Theorem K1
Suppose that G is a group of transformations of a set S, and H is a subgroup of G.
If two figures are h-congruent, then they are g-congruent.

Proof
Suppose that P and Q are h-congruent.
Then there is an h in H with h(P) = Q.
As H is a subgroup of G, hεG so P and Q are g-congruent.
It follows that each g-congruence class is a union of h-congruenced classes, so that,
in the geometry defined by G, there are fewer (but larger) congruence classes.
Example
The euclidean group E(2) may be extended to the similarity group S(2)
by adding transformations which scale all distances by a constant factor.
These will be discussed in detail later. Here, we will consider just the respective
congruence classes which contain the circle C : x²+ y² = 1.
Suppose that tεE(2). As t preserves distance, t(C) will be the circle
with centre t(0,0) and radius 1. On the other hand, if the circle C' has centre P
and radius 1, then the translation which maps O to P will map C to C'.
Thus the euclidean class containing C consists of all circles of radius 1.
In euclidean geometry, there is a class of circles of radius r for each positive r.
Now suppose that sεS(2). Since scaling maps circles to circles, we see that
s(C) is a circle with centre s(0,0) and radius k, the scaling factor of s.
If the circle C' has centre P and radius r, then we may map C to C' by
first mapping O to P (by a translation), then dilating by factor r about P .
Thus the similarity class containing C consists of all circles.
Note that the similarity class of C contains the euclidean class of C.
It consists of the union of all euclidean classes of circles.

Theorem K2
Suppose that G is a group of transformations of a set S, and H is a subgroup of G.
If D is a g-property, then it is an h-property.

Proof
Suppose that D is a g-property, so is invariant under g for all g in G.
For h in H, hεG as H≤G, so that D is invariant under h.
Thus, the geometry defined by G may have fewer properties.
Example
We again consider the groups E(2) and S(2) acing on R².
Since translations, rotations and reflections preserve distances and size of angles
so does any element of E(2) since these transformations generate E(2). Thus distance
and angle size are euclidean properties.
Since scaling by a factor k maps a triangle to a similar triangle, each element of S(2)
preserves angle size. But if k ≠ 1, the scaling alters the length of each side, so does not
preserve distance.
Thus, angle size is a property of both geometries, but distance is only a euclidean property.

main klein page

Klein defines a geometry as a set S of objects - the points of the geometry, and a group G of transformations of S. Definitions a geometrical figure is a set of points, i.e. a subset of S. a geometrical property (g-property) is a property invariant under G. geometrical figures P and Q are g-congruent if there is an element of G mapping P to Q. Note that, if P and Q are g-congruent, then, as g-properties are invariant under G, P and Q have identical g-properties. The study of the geometry defined by G consists of the description of the g-congruence classes, the identification of g-properties, and theorems about g-congruence classes and g-properties. There are no axioms - just theorems. This has the advantage that such a "geometry" is consistent. With the axiomatic approach, it is always possible that the axioms lead to a contradiction. In such a case, there can be no "model" for the geometry. As an example, euclidean geometry may be described as having the set R², and the group E(2) generated by translations, rotations and reflections of R². Since each such transformation preserves distance between points, distance is a geometrical property of euclidean geometry. We can use the group to prove that each of Euclid's axioms is a theorem of this geometry. The other advantage of Klein's approach is that it allows us to relate different geometries. Suppose that G is a group of transformations of the set S, and that H is a subgroup of G. Then H is also a group of transformations of S, and so defines another geometry on S.
Theorem K1 Suppose that G is a group of transformations of a set S, and H is a subgroup of G. If two figures are h-congruent, then they are g-congruent.
Proof Suppose that P and Q are h-congruent. Then there is an h in H with h(P) = Q. As H is a subgroup of G, hεG so P and Q are g-congruent. It follows that each g-congruence class is a union of h-congruenced classes, so that, in the geometry defined by G, there are fewer (but larger) congruence classes. Example The euclidean group E(2) may be extended to the similarity group S(2) by adding transformations which scale all distances by a constant factor. These will be discussed in detail later. Here, we will consider just the respective congruence classes which contain the circle C : x²+ y² = 1. Suppose that tεE(2). As t preserves distance, t(C) will be the circle with centre t(0,0) and radius 1. On the other hand, if the circle C' has centre P and radius 1, then the translation which maps O to P will map C to C'. Thus the euclidean class containing C consists of all circles of radius 1. In euclidean geometry, there is a class of circles of radius r for each positive r. Now suppose that sεS(2). Since scaling maps circles to circles, we see that s(C) is a circle with centre s(0,0) and radius k, the scaling factor of s. If the circle C' has centre P and radius r, then we may map C to C' by first mapping O to P (by a translation), then dilating by factor r about P . Thus the similarity class containing C consists of all circles. Note that the similarity class of C contains the euclidean class of C. It consists of the union of all euclidean classes of circles.
Theorem K2 Suppose that G is a group of transformations of a set S, and H is a subgroup of G. If D is a g-property, then it is an h-property.
Proof Suppose that D is a g-property, so is invariant under g for all g in G. For h in H, hεG as H≤G, so that D is invariant under h. Thus, the geometry defined by G may have fewer properties. Example We again consider the groups E(2) and S(2) acing on R². Since translations, rotations and reflections preserve distances and size of angles so does any element of E(2) since these transformations generate E(2). Thus distance and angle size are euclidean properties. Since scaling by a factor k maps a triangle to a similar triangle, each element of S(2) preserves angle size. But if k ≠ 1, the scaling alters the length of each side, so does not preserve distance. Thus, angle size is a property of both geometries, but distance is only a euclidean property.