A new type of algebra for Minkowski space‐time is described, in terms of which it is possible to express any conformally covariant or Poincaré covariant operation. The elements of the algebra (twistors) are combined according to tensor‐type rules, but they differ from tensors or spinors in that they describe locational properties in addition to directional ones. The representation of a null line by a pair of two‐component spinors, one of which defines the direction of the line and the other, its moment about the origin, gives the simplest type of twistor, with four complex components. The rules for generating other types of twistor are then determined by the geometry. One‐index twistors define a four‐dimensional, four‐valued (``spinor'') representation of the (restricted) conformal group. For the Poincaré group a skew‐symmetric metric twistor is introduced. Twistor space defines a complex projective three‐space *C*, which gives an alternative picture equivalent to the Minkowski space‐time *M* (which must be completed by a null cone at infinity). Points in *C* represent null lines or ``complexified'' null lines in *M*; lines in *C* represent real or complex points in *M* (so *M*, when complexified, is the Klein representation of *C*. Conformal transformations of *M*, including space and time reversals (and complex conjugation) are discussed in detail in twistor terms. A theorem of Kerr is described which shows that the complex analytic surfaces in *C* define the shear‐free null congruences in the real space *M*. Twistors are used to derive new theorems about the real geometry of *M*. The general twistor description of physical fields is left to a later paper.

## REFERENCES

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*r*dimensionality in the

*real*sense.

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*M.*An explicit construction in terms of incidence of null lines is given in Sec.IX, but in any case, the geometric (and conformally invariant) nature of a Robinson congruence is already implied by Sec.IV and V.

*C*would be the same if we specified only that $(T\beta \alpha )$ be proportional to the inverse of $(t\beta \alpha )$ since the $X\alpha $ are

*projective*coordinates for

*C*. However, the stronger requirement (7.2) is adopted here since the factor of proportionality of a twistor is required when the more complete representation in accordance with Sec. V is used.

*essential*restriction on the way in which the twistors are represented is that the signature of the form

*X*$X\u0304\alpha \alpha $ must be (+ + − −) [cf., (6.1)].

*not*analytic in the real sense emerge here simply as the system of null lines meeting a

*nonanalytic*curve in

*M*.

*P*as a parallelism on

*M*(with torsion; left‐handed if $P\u2282C+$) which is closely related to Clifford parallelism on S

^{8}. The sets of

*null*directions which are to be regarded as parallel are those of the Robinson congruences represented by the points of the line

*P*in

*C*(i.e., by planes through

*P*̄). A transitive four‐parameter group of (conformal) motions of

*M*preserves this parallelism, namely that given by twistor transformations (7.1) for which the line

*P*is left pointwise invariant. This group is readily seen to be the group of unitary $(2 \xd72)$ matices and leads to Uhlmann’s representation (see Ref. 18) of the points of

*M*in terms of such matrices.

*M*defines both a point in

*C*and a plane in

*C*through this point, the configuration of Fig. 6 is represented in the

*C*picture as a pair of mutually inscribed and circumscribed tetrahedra—a configuration familiar to geometers. We may note that the

*full*complexification of a null line in

*M*leads, in the

*C*picture, strictly to a point in

*C together*with a plane through it. This gives a five‐complex‐dimensional system as we would expect.

*O*meeting

*U*,

*V*, and

*W*which lies on a null cone through

*U*.