Free Hyperspace by Michio Kaku, Robert O'Keefe Page B

this problem were continually thwarted by grinding poverty. His successes did not translate into money. He suffered another nervous breakdown in 1857. After many years, he was finally appointed to Gauss’s coveted position at Göttingen, but it was too late. A life of poverty had broken his health, and like many of the greatest mathematicians throughout history, he died prematurely of consumption at the age of 39, before he could complete his geometric theory of gravity and electricity and magnetism.
    In summary, Riemann did much more than lay the foundation of the mathematics of hyperspace. In retrospect, we see that Riemann anticipated some of the major themes in modern physics. Specifically,
1. He used higher-dimensional space to simplify the laws of nature; that is, to him, electricity and magnetism as well as gravity were just effects caused by the crumpling or warping of hyperspace.
2. He anticipated the concept of wormholes. Riemann’s cuts are the simplest examples of multiply connected spaces.
3. He expressed gravity as a field. The metric tensor, because it describes the force of gravity (via curvature) at every point in space, is precisely Faraday’s field concept when applied to gravity.
    Riemann was unable to complete his work on force fields because he lacked the field equations that electricity and magnetism and gravity obey. In other words, he did not know precisely how the universe would be crumpled in order to yield the force of gravity. He tried to discover the field equations for electricity and magnetism, but he died before he could finish that project. At his death, he still had no way of calculating how much crumpling would be necessary to describe the forces. These crucial developments would be left to Maxwell and Einstein.
Living in a Space Warp
     
    The spell was finally broken.
    Riemann, in his short life, lifted the spell cast by Euclid more than 2,000 years before. Riemann’s metric tensor was the weapon with which young mathematicians could defy the Boeotians, who howled at any mention of higher dimensions. Those who followed in Riemann’s footsteps found it easier to speak of unseen worlds.
    Soon, research bloomed all over Europe. Prominent scientists beganpopularizing the idea for the general public. Hermann von Helmholtz, perhaps the most famous German physicist of his generation, was deeply affected by Riemann’s work and wrote and spoke extensively to the general public about the mathematics of intelligent beings living on a ball or sphere.

    Figure 2.5. A two-dimensional being cannot eat. Its digestive tract necessarily divides it into two distinct pieces, and the being falls apart
.
     
    According to Helmholtz, these creatures, with reasoning powers similar to our own, would independently discover that all of Euclid’s postulatesand theorems were useless. On a sphere, for example, the sums of the interior angles of a triangle do not add up to 180 degrees. The “bookworms” first talked about by Gauss now found themselves inhabiting Helmholtz’s two-dimensional spheres. Helmholtz wrote that “geometrical axioms must vary according to the kind of space inhabited by beings whose powers of reasoning are quite in conformity with ours.” 9 However, in his
Popular Lectures of Scientific Subjects
(1881), Helmholtz warned his readers that it is impossible for us to visualize the fourth dimension. In fact, he said “such a ‘representation’ is as impossible as the ‘representation’ of colours would be to one born blind.” 10
    Some scientists, marveling at the elegance of Riemann’s work, tried to find physical applications for such a powerful apparatus. 11 While some scientists were exploring the applications of higher dimension, other scientists asked more practical, mundane questions, such as: How does a two-dimensional being eat? In order for Gauss’s two-dimensional people to eat, their mouths would have to face to the side. But if we now draw their digestive tract, we