Let us now take a simple example. Define a triangle as follows, dispensing entirely with any attribute concerning measurement: A triangle is determined either by any three points in a plane, but not in a line; or by any three lines in a plane, but not in a point. In other words, it matters not where any of the members (points or lines) of a triangle are positioned in the plane in which it lies, whether in the finite or infinitely distant. Dispense with measurement, but include the concept of the infinite, and a triangle is a triangle wherever its parts are located in the plane, even if one of its points or one of its lines is in the infinite and it does not therefore look like a conventional triangle; in thought it nevertheless is one. We are led to realize that points or lines at infinity function just as any other points do, and in fact are indispensable to the whole. Old distinctions resting on measurement cease to have any great significance; right-angled, isosceles or equilateral triangles are all included in the archetypal idea of a triangle.
Projective geometry rests on beautiful and harmonious truths of coincidence and continuity, and provides us with a realm of unshakable mathematical validity in which to move about freely in thought. The creative process can be continuous and sustained, passing through the infinite point or plane and returning again. The very fact that the creation of forms in this geometry is not dependent in the first place on any kind of measurement, but is concerned primarily with relationships between geometrical entities, gives access to processes of creative thought entirely different from those used, for example, in building a house. In the latter case, a beginning is made from, say, a cornerstone, some predetermined measure is laid out from corner to corner and so the building grows. In projective geometry, on the other hand, the actual measures of the once completed form may be quite fortuitous and appear last, not first; yet the form, however it may arise—however it may, so to speak, crystallize out in the process of construction, and however bizarre its appearance—will always remain true to the archetypal idea underlying it.
Let us take as an instance the theorem of Pascal (1623-1662) concerning the hexagon inscribed in a conic: The common points of opposite pairs of sides of a hexagon inscribed in a conic are collinear (i.e., lie on the same line).3
3 A conic is any curve which is the locus of a point which moves so that the ratio of its distance from a fixed point to its distance from a fixed line is constant. (Mathematics Dictionary, G. & R. C. James, eds.)
The theorem deals with six points in a plane and the remarkable fact that, provided all six points are points on a conic (any conic will suffice), then the “opposite” pairs of sides of the hexagon determined by these six points— three pairs of them—will each have a common point, and all three of these points will always be in line! Of course, the term “opposite” is a spatial one and correctly refers to the regular, Euclidean hexagon in the circle. This turns out to be a special case of the far more general, projective form, and the three points common to its pairs of opposite sides are all infinitely distant. The inclusion of the concept of points at infinity provides the clear thought that, in this special case, the three points are points of the infinitely distant line of the plane in which hexagon and circle lie. The variety of possibilities in the positioning of the points on the conic and also the choice of the cylic order in which to join them, added to the fact that the conditions still hold for a hexagon inscribed into any of the projective forms of a circle, fills the mind with wonder at the mobility of the spatial interpretation of the all-relating concept as expressed in the statement of the theorems. (See figures 1-4.) In this example, the line of three points or Pascal line, as it is called, may be regarded as the Absolute of the configuration, in the sense in which this term has been used above.
Because this is a realm which contains the archetypes and also the actual laws of forms as yet uncreated, it is important to understand and to become creative in the development of projective geometrical relationships. It is a realm of clear mathematical thought with its own remarkable laws, among which the laws of Euclidean geometry are one of the particular cases. The British scientist George Adams called the domain of protective transformations “Archetypal Space”; Louis Locher-Ernst used the German word Urraum to describe what we might conceive to be an ever-moving, fluid continuum of projective possibilities, containing the seeds or potentialities of all types of form.
In addition to the all-pervading element of movement in projective geometry, there is a fundamental principle of great significance—the principle of Duality or Polarity, the ideal and mutual equivalence of point and plane. Polar to the geometry of points and lines in the plane there is a geometry of lines and planes in the point; the two mutually complement one another. As Cremona wrote, “There are therefore always two correlative or reciprocal methods by which figures may be generated and their properties deduced, and it is in this that geometric Duality consists.”1 Laws at work in the “extensive” two-dimensional field of the plane are found again in polar opposite form in the “intensive” field of the point. So, too, the three-dimensional constructions of positive Euclidean space have their polar counterpart in a world of forms held in a point. These forms, and the laws according to which projective transformations take place among them, are of significance for a field of research which seeks a deeper approach to the living kingdoms of nature and man.
- CHAPTER VII – ‘Always Stand by Form’
- CHAPTER VIII – Dynamics versus Kinetics