Like all other theories in physics, it is highly mathematical-but the necessary mathematics has not yet been invented. In the late 20th and early 21st centuries, string theory was formulated as an approach to unifying gravity and quantum physics into a theory of everything. The cross-fertilization of ideas between the mathematical sciences and theoretical physics continues to this day. Reprinted with permission from Paul Nylander. Ģ4 / A cross-section of a quintic hypersurface. The twist given to the fibers makes the Mobius band topologically different from an ordinary cylindrical band. The physicists’ gauge fields were like individual arrows in the mathematicians’ quiver of arrows.Ģ3 / The Mobius band is the simplest nontrivial example of a fiber bundle. It wasn’t until the 1970s that mathematicians and physicists realized that they were doing the same thing. Physicists started thinking about fields that permeate all of space, which they called “gauge fields.” (Examples include the electromagnetic field and the weak and strong nuclear forces.) Meanwhile mathematicians, for different reasons, became very interested in a new kind of geometric space, called a fiber bundle, which is roughly like a curved space with a quiver of arrows attached at every point (see Figure 23 for an example). In the later 1930s and 1940s, physicists and mathematicians started losing touch with one another. They are the key ingredients in positron emission tomography, or PET scans, which are used to study the workings of the human brain. The equations predicted extra solutions that seemed like positively charged electrons. The discovery of antimatter, in 1932, grew directly out of an attempt to reconcile relativity with the quantum-mechanical description of the electron.

Similar geometric constructions underlie the field theories that describe particle physics. Somewhere, Galileo must be smiling.īut Einstein’s general relativity was only the beginning. To understand any of these ideas fully, you have to learn Riemannian geometry. A long list of profound discoveries followed from the equations that Einstein wrote down in 1915: black holes, the expanding universe, the big bang, and dark energy. Reprinted with permission from Gerard Westendorp. A sphere (left) is positively curved, while the surface on the right, which resembles a six-connection pipe fitting, is negatively curved. But in the early 1900s, Albert Einstein used Riemann’s mathematics as a language to express his theory of general relativity-a theory in which gravity results from the curvature of four-Ī long list of profound discoveries followed from the equations that Einstein wrote down in 1915: black holes, the expanding universe, the big bang, and dark energy.Ģ2 / Two-dimensional manifolds are also known as surfaces.

He called these curved spaces “manifolds” (see Figure 22 on page 50).įor some time these new geometries remained just a mathematical curiosity. Riemann’s geometry also allows space to take on any number of dimensions-two, three, or even more. In the 1850s, Bernhard Riemann took another bold step forward, describing spaces in which the curvature could change from point to point within the space. They may have the positive curvature of a sphere, or they may have negative curvature, which is harder to visualize but may be compared to the frilly surface of some leafy vegetables. By contrast, non-Euclidean geometries are curved. Euclidean geometry is flat-it is the geometry of a tabletop, infinitely extended. In the early 1800s, it was shown that the familiar Euclidean geometry, which has been taught since the ancient Greeks and is still taught in high schools today, is only one of an infinite variety of possible geometries. The secrets of the universe are still written in geometric terms, although the figures Galileo wrote about have now been replaced by more exotic and abstract ones: manifolds, fiber bundles, and Calabi-Yau spaces. Nearly four centuries later, we can only marvel at his prescience, because what he wrote then is even more true today. So wrote Galileo Galilei in 1623, at the dawn of the scientific era. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures.” “Philosophy is written in this grand book-I mean the universe-which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language in which it is written.

Non euclidean geometry art