The original Russian version The original Russian version of this article was published in 1985 in "Chemistry & Life" magazine of the Soviet Academy of Sciences (No. 1, p. 87). After that the author received a letter from a physicist stating that the idea of representing elementary particles not as points but as lines is not new and is implemented in String theory. In this regard the author would like to notice that the idea of the article was not such a representation but drawing attention to the fact that a type of spatial symmetry exists, which was hitherto overlooked in the theoretical physics.

Points & Straight Lines

Y. B. Karasik

Copyright ©1985-2010 Y. B. Karasik

Taken prisoner of war by the Russians in 1812 the lieutenant of the Napoleon's Grande Armee 24 years old mathematician Jean-Victor Poncelet during his slacking in the Russian captivity in Saratov came to a conclusion that a geometry is possible where point is not a primitive element of space but a complex object ... consisting of straight lines !

This geometry differs from the Euclidean in that it treats the infinitely distant points as any other points. To this end each point on the plane is specified not by two cartesian coordinates X and Y but by three so-called homogeneous coordinates X1, X2, and X3, related to the cartesian ones as follows:

X = X1/X3
Y = X2/X3

Homogeneous coordinates make the infinitely distant points to have finite coordinates. Specifically, the values of X1 and X2 of such points are any finite numbers, whereas X3 is always zero.

Homogeneous coordinates makes the equation of a straight line in the plane self-dual. Indeed, in cartesian coordinates the equation is

A1X + A2Y + A3 = 0

By substituting X and Y with their expressions in homogeneous coordinates we obtain:

A1X1 + A2X2 + A3X3 = 0

This equation is self-dual: if one fixes A1, A2, A3, it becomes the equation of a straight line in point coordinates. But if one fixes X1, X2, X3, then it becomes the equation of a point in linear coordinates. In the first case a straight line consists of points. In the second case a point consists of straight lines. According to Poncelet both points and lines can serve as primitive elements of space.

This allows us to geometrize the relativity of parts and the whole observed in elementary particles physics. As is known, proton can decay into neutron and neutron can decay into proton. This begs a question: which is bigger ? So far the answer to this question was that it is not applicable to elementary particles. But this is not a good answer. Projective geometry allows a better one. As a line can be splitted into points and a point can be splitted into lines similarly any small object can be decomposed into big ones and vise versa. All depends on what is assumed to be the smallest element of space: point or line.

Regardless, homogeneous coordinates introduce a new type of symmetry (between (A1, A2, A3) and (X1, X2, X3)), which never was studied and taken into account in theoretical physics. Taking it into account may lead to discovery of new conservation laws and other physical discoveries.