Two discrete independent events A and B occur during the time interval t = 0 to T. We can represent their joint occurrence by a point in a TxT square, where one axis is the time of occurrence of event A and the other is the time of occurrence of event B. The region of this square that represents the cases in which A and B occurred within d of each other is the region surrounding the diagonal, as shown below. Disregarding the "clipping" at the boundaries, this "region of coincidence" (within +-d of the diagonal) is simply the region swept out by a line segment of length d*sqrt(2) normal to the main diagonal. Therefore, we say the cross-section of the region of coincidences of degree d is a simple line segment of length d*sqrt(2). We will call this the "shape" of 2D coincidences. Coincidences of degree d between THREE events can be represented by the region near the main diagonal of a TxTxT cube. The region of coincidences is swept out by a regular hexagon normal to the main diagonal, so the "shape" of a three-event coincidence is a hexagon, with edge length d*sqrt(2/3). In general, given a unit "cube" in n dimensions with n orthogonal coordinates X1, X2, ...,Xn, we can consider the region of the cube's content ("volume") consisting of points with coordinates [x1,x2,...,xn] such that |xi - xj| < q for all i,j (2) This can be viewed as the region swept out by a particular n-1 dimensional object as it translates along the diagonal of the unit cube. Since the solid is in the (n-1)-dimensional "plane" normal to the diagonal, the coordinates of the solid when it is located at the origin satisfy x1 + x2 + ... + xn = 0 (3) I believe the "volume" of this object is sqrt(n) q^(n-1), although I have proven it only for the cases n= 2, 3, and 4. The n-1 dimensional locus satisfying (2) and (3) can be determined geometrically by considering the vertices. Condition (2) for any i,j with i!=j implies an n-2 dimensional "face" of the convex poly- tope, and the absolute value allows two opposite faces for each such i,j. Thus, the figure has n(n-1) faces. It can also be shown that the number of vertices for the general n-dimensional case is 2^n - 2. Points that lie on more than one face are considered to be on an "edge", and points that are on a maximal number of faces (meaning there is no path from such a point to any other face without leaving one of the faces it is already on) are the vertices of the polytope. The convexity of the polytope with these vertices is easy to see. As an example, the six vertices of the sweeping object in the n=3 case can be expressed in terms of multiples of q/n as follows: [2,-1,-1] [-2,1,1] [-1,2,-1] [1,-2,1] [-1,-1,2] [1,1,-2] By rotating these points into a single 2d plane we arrive at the regular hexagon. Similarly for the case n=4, the fourteen vertices correspond to [3,-1,-1,-1] [-3,1,1,1] [-1,3,-1,-1] [1,-3,1,1] [-1,-1,3,-1] [1,1,-3,1] [-1,-1,-1,3] [1,1,1,-3] [2,2,-2,-2] [-2,-2,2,2] [2,-2,2,-2] [-2,2,-2,2] [2,-2,-2,2] [-2,2,2,-2] These fourteen vertices can be rotated into a 3d space, showing that they form the vertices of a solid with 12 identical diamond-shaped faces. One way of visualizing this solid is to begin with a unit 3d cube and construct an "Egyptian" pyramid on each face such that the faces of the adjascent pyramids are flush. The volume of this solid is 2q^3. This 12-faced solid is known as Kepler's Rhombic Dodecahedron. This fascinating shape has many interesting properties in addition to being the "shape of coincidence" for four events. If your browser supports Javascript, you can see an animated view of this polyhedron by clicking here. It's interesting that this solid can be said to "duplicate the cube", reminiscent of the classical Greek problem of constructing a cube with twice the volume of a given cube. This volume also appears as the correction term in the case n=3. Note that dimensions 3 and 4 are the only two dimensions for which the correction term for one is the sweeping volume for the other.

Return to MathPages Main Menu