cos (d12/γ) = ± (x1x2 + y1y2 + z1z2 + w1w2) / {(x1² + y1² + z1² + w1²) (x2² + y2² + z2² + w2²)}1/2
(10).
Thus there are two distances between the points, and if one is d12, the other is πγ-d12. Every straight line returns into itself, forming a closed series. Thus there are two segments between any two points, together forming the whole line which contains them; one distance is associated with one segment, and the other distance with the other segment. The complete length of every straight line is πγ.
The angle between the two planes l1x + m1y + n1z + r + 1w = 0 and l2x + m2y + n2z + r2w = 0 is
cos θ12 = (l1l2 + m1m2 + n1n2 + r1r2) / {(l1² + m1² + n1² +r1²) (l2² + m2² + n2² + r2²)}1/2
(11).
The polar plane with respect to the absolute of the point (x1, y1, z1, w1) is the real plane x1x + y1y + z1z + w1w = 0, and the pole of the plane l1x + m1y + n1z + r1w = 0 is the point (l1, m1, n1, r1). Thus (from equations 10 and 11) it follows that the angle between the polar planes of the points (x1, ...) and (x2, ...) is d12/γ, and that the distance between the poles of the planes (l1, ...) and (l2, ...) is γθ12. Thus there is complete reciprocity between points and planes in respect to all properties. This complete reign of the principle of duality is one of the great beauties of this geometry. The theory of lines and planes at right angles is simply the theory of conjugate elements with respect to the absolute. A tetrahedron self-conjugate with respect to the absolute has all its intersecting elements (edges and planes) at right angles. If l and l′ are two conjugate lines, the planes through one are the planes perpendicular to the other. If P is the pole of the plane p, the lines through P are the normals to the plane p. The distance from P to p is ½πγ. Thus every sphere is also a surface of equal distance from the polar of its centre, and conversely. A plane does not divide space; for the line joining any two points P and Q only cuts the plane once, in L say, then it is always possible to go from P to Q by the segment of the line PQ which does not contain L. But P and Q may be said to be separated by a plane p, if the point in which PQ cuts p lies on the shortest segment between P and Q. With this sense of “separation,” it is possible[2] to find three points P, Q, R such that P and Q are separated by the plane p, but P and R are not separated by p, nor are Q and R.
Let A, B, C be any three non-collinear points, then four triangles are defined by these points. Thus if a, b, c and A, B, C are the elements of any one triangle, then the four triangles have as their elements:
| (1) | a, | b, | c, | A, | B, | C. |
| (2) | a, | πγ − b, | πγ − c, | A, | π − B, | π − C. |
| (3) | πγ − a, | b, | πγ − c, | π − A, | B, | π − C. |
| (4) | πγ − a, | πγ − b, | c, | π − A, | π − B, | C. |
The formulae connecting the elements are