Line Designs
Julius Plücker (1801–1868) is not a household name among present-day mathematicians, but in the nineteenth century he carved himself a niche in geometry where few others had ventured before. He realized that a curve need not be regarded as a set of points; it can just as well be described as a set of tangent lines. The idea was not entirely new. It had been known for more than a century that certain formal statements about points and lines remain valid when the words point and line are everywhere interchanged. For example, just as two points determine a unique line, so do two lines determine a unique point—their point of intersection.1 This principle of duality became the centerpiece of a new kind of geometry, projective geometry, in which dual relations such as “two points determine a line” or “two lines determine a point” became the main focus, rather than metric properties such as the length of a line segment or the area of a polygon.
But while the principle of duality was well known in Plücker’s time, he gave it a new formulation that placed the subject squarely in the realm of analytic geometry: he showed that a curve can be generated from a set of lines obeying a line equation, in much the same way as the traditional view of a curve as a set of points obeying a point equation. In other words, lines can be used as building blocks of geometric figures just as much as points. Plate 40.1 shows a parabola generated entirely from its tangent lines; not a single point was used in its construction (the plate also shows the parabola’s reflecting property, discussed earlier in chapter 29). Plate 40.2 goes even further, showing a Star of David–like design made of 21 line parabolas.
Plücker’s career took him through strange twists. His major work in geometry was published in two volumes in 1828 and 1831, and it was in the second volume that he gave the analytic formulation of the principle of duality. Yet his work was not favorably received by the two leading geometers of the time, Jacob Steiner and Jean Victor Poncelet, whose synthetic geometry was more in line with the classical geometry of Euclid. It didn’t help that Plücker’s academic position at the University of Bonn was not in mathematics but in physics. This in itself was not unusual (Gauss held the position of director of the astronomical observatory at Göttingen), but it was used by Plücker’s adversaries to claim that he was not a true physicist. To prove them wrong, he abandoned mathematics and for the next 18 years devoted himself to physics, making contributions in optics, magnetism, and spectroscopy. It was only toward the end of his life that Plücker returned to his first love, geometry, where he made several more discoveries. After his death his work was completed by the influential German mathematician Felix Klein (1849–1925), who, like Plücker, was as much versed in algebra and geometry as he was in physics.2
1. If the lines are parallel, the point of intersection recedes to infinity and is known as a “vanishing point.”
2. For a fuller discussion of line equations, see Maor, The Pythagorean Theorem: A 4,000-Year History, chapter 10.