# Weird Geometry

 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'. 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.