Why Physicists Believe Geometry Holds the Key to All Theories

Can you envision the impression a 4D hexagon might create as it travels through a 3D kitchen table? It might seem implausible, yet some individuals can perceive it.

One such individual was mathematician Alicia Bourstott, daughter of logician George Bourg. In the early 20th century, she devised models of shapes while moving through three-dimensional objects. Years later, when mathematicians could verify her work with computer programs, they found that Stott had an uncanny ability to accurately depict these shapes.

For many of us, geometry recalls images of pencils, rulers, triangles, and circles. It evokes the complex questions posed in school involving parallel lines and angles. However, as Boole Stott’s experience illustrates, scholars have been expanding the scope of geometry for some time.

Geometry can transcend the conventional realm of 2D and 3D shapes. A prime example is Albert Einstein’s theory of gravity, known as general relativity, which intertwines with time to form a four-dimensional stage where the universe unfolds.

Moreover, geometry can also explore dimensions that defy physical reality. Take meteorology, for example. Atmospheric data encompasses multiple “dimensions” such as latitude, longitude, temperature, pressure, wind speed, and more.

Researchers visualize these dimensions as shapes extending into higher dimensions, aiding in understanding atmospheric behavior. “From this, we can implement mathematical models to explain what occurs. [those properties] In numerous dimensions,” states mathematician Snezana Lawrence of Middlesex University in London.

For theoretical physicists, extra dimensions appear to be essential for a complete understanding of the universe, with some suggesting that our reality might be a “projection” from a higher dimension. While this idea might sound peculiar, under certain simplified assumptions, physicists can perform calculations related to fundamental particles and black holes.

Some physicists have even proposed the concept of “all theories,” a curious geometric idea that may lead to a unified explanation of the universe and everything within it. One of these concepts is the “amplituhedron,” introduced by Jaroslav Trnka from the University of California, Davis, and Nima Arkani Hamed at the Institute for Advanced Study in New Jersey. Imagine it as an abstract, multidimensional crystal that offers an alternative perspective on the fundamentals of particle physics.

Another concept is “causal dynamic triangulation,” developed by Renate Roll at Radboud University in the Netherlands. This approach stitches together various geometric shapes to craft an explanation of space-time that seems to embody characteristics of both quantum mechanics and general relativity—two concepts that are traditionally seen as incompatible. She asserts that it serves as a testable reflection of both abstract geometric theories and true properties of the universe, as observed in the cosmic microwave background radiation.

Neither of these ideas has yet been universally accepted in all theories. However, some believe that a fresh perspective on physics is essential for progress. There is a growing consensus that this perspective may be expressed through the language of geometry. While the truth of this notion remains to be seen, it is evident that geometry encompasses far more than just hexagons.