In the hyperbolic group pages, we met the subgroup H(2) of I(2) consisting
of elements which map C (the unit circle) to itself, and D0
(the interior of
the unit circle) to itself. Observe that such elements must also map
D1
(the exterior of the unit circle) to itself.
We defined the hyperbolic group H(2) as the group of restrictions of elements
of H(2) to D0. This gives a model of
hyperbolic geometry defined on D0.
If, instead, we restrict the elements of H(2) to D1, then we obtain a geometry
defined on D1. This is another model of hyperbolic
geometry.
To see this in a formal way, let h0 denote inversion in the circle C.
Then h0 maps
D0 to D1, (and D1 to D0).
Thus, we have a one-one correspondence
between subsets of D0 and subsets of D1.
For example, an h-line is a set H = LnD0,
where L is an i-line orthogonal to C. Since L and C are orthogonal,
h0 maps L to L.
Thus h0(H) = LnD1, i.e. is the part of
L lying outside C. Since h0 preserves angles,
the angle between two h-lines is the same as the angle between the corresponding
subsets of D1. Also, we can check that the function D, defined earlier on D0, is
invariant under h0, so we have a distance function in our "new" geometry, and this
corresponds to hyperbolic distance in D0.
Taking the Klein view, we have two geometries (with sets D0 and D1).
The groups
are obtained by restricting elements of H(2),and each is isomorphic to H(2). The
geometries have identical properties and theorems.
Definition
Geometry G with set S and group G, and geometry G* with set S* and group G*
are isomorphic if there is a function f mapping S to S*
such that G* = fGf-1.
Clearly, isomorphic geometries have identical properties and theorems.
We generally think of them as different models of the same geometry.
We have met this situation before, with complex and vector models of
euclidean and similarity geometries.
Here, we consider yet another geometry derived from H(2). This has a
quite different flavour. This demonstrates that the properties of any
geometry depend on the action of the group on the set, rather than
simply on the structure of the group.
This geometry does not appear in the literature - many of its properties
are quite strange. We shall call it weird geometry.
Observe that the elements of H(2) map C to C.
Definition
The weird group W(2) consists of the restrictions of elements of H(2) to C.
The group W(2) defines weird geometry on C.
In the other geometries derived from H(2), we defined the "lines" as the
intersections of the set with i-lines orthogonal to C.
Definition
A w-line is a set of the form LnC,
where L is orthogonal to C.
Of course, an i-line orthogonal to C meets C exactly twice, and, for any
two points P,Q on C, there is a unique i-line through P,Q orthogonal to C.
From these observations, we have:
Theorem
(1) The w-lines are the pairs {P,Q ε C: P ≠ Q}.
(2) Two w-lines meet at most once.
(3) Any two distinct points of C define a unique w-line.
Since inversive geometry has the concept of angle, any geometry whose
group is a subgroup of the inversive group will have the notion of angle.
In hyperbolic geometry, we defined the angle between h-lines meeting at P
as the angle at P between the corresponding i-lines. If we adopt the same
definition in weird geometry, then, since the relevant i-lines are orthogonal
to C, they meet at angle zero at the point on C. Thus, in weird geometry,
all angles are zero.
As usual, a triangle consists of the points of three intersecting lines:
Definition
A w-triangle is a set of three distinct points of C.
From our observation about angles:
Theorem
The sum of angles of a w-triangle is zero.
In hyperbolic geometry, we met the concept of hyperbolic distance. This was
derived from the function D(.,.) which is invariant under the group H(2) acting
on the set D0. If we consider the action on C,
there is an unexpected problem.
The function D(z,w) = |z-w|/|zw*-1| is defined unless zw* = 1.
If z,w both belong to D0 (or both belong to D1)
we cannot have zw* = 1.
Now suppose that z,w belong to C.
Observe that zw* = 1 if and only if z = 1/w* = w (as |w| = 1).
Also, if z ≠ w, then |zw*-1| = |(z/w)-1| = |z-w|/|w| = |z-w| (as |w| = 1),
so that D(z,w) = |z-w|/|zw*-1| = 1.
Definition
For z,w ε C,
D*(z,w) = 0, if z = w, and
D*(z,w) = 1, if z ≠ w.
From the definition, it is easy to see that D* is invariant under W(2).
We observe that this definition can be applied to any geometry, and
gives a function invariant under the group. Although it clearly satisfies
the conditions D1, D2 and D4 for a distance function, it will generally
fail D3. Here, however, it does satisfy D3 for a rather devious reason.
A w-line PQ consists of just two points, so there are no points lying
between P and Q. Thus, the equality clause of D3 is vacuously true.
In hyperbolic geometry, the invariance of the hyperbolic distance function
shows that we cannot in general map a pair of points of D0 to another
pair. Here, we have a far more interesting fundamental theorem:
The Fundamental Theorem of Weird Geometry
If L = (A,B,C) and L' = (A',B',C') are lists of distinct points of C,
then there is a unique element of W(2) mapping L to L'.
Proof of the fundamental theorem
This can be restated in terms of w-triangles as follows:
Theorem
Any two w-triangles are w-congruent.
Our geometry has angle and distance, so we might expect congruence conditions,
like (SSS), (AAA) and so on. The above result includes all of these, so we do
have analogues of all of the corresponding results in hyperbolic geometry.
|