Following the release of the star-studded mystery thriller “The Number 23” in 2007, a significant number of individuals began to perceive occurrences of the titular number in various aspects of their lives. I was a student at the time, and some of my peers would react with unease whenever the number 23 made an appearance. This phenomenon, often referred to as the “frequency illusion” or the Baader-Meinhof phenomenon, intrigued many, as it illustrated how paying closer attention to a specific element, such as a number, could lead to a heightened perception of its presence.

For a considerable period, there was speculation that the late mathematician John McKay experienced a similar phenomenon, fixating on the number 196,884. McKay’s encounter with this number occurred in 1978 while he was exploring a mathematical paper unrelated to his primary field of study. Despite being immersed in geometry, McKay stumbled upon a sequence of numbers within number theory, beginning with 196,884.

This figure sparked familiarity in McKay due to his previous work on a theoretical mathematical entity known as the “monster.” This complex structure aimed to describe the symmetries of a geometric object residing in 196,883 dimensions, only one dimension fewer than the number 196,884. McKay’s discovery of this number seemed serendipitous, as he connected it to the dimensions of the monster structure: 196,883 + 1 = 196,884.

Although McKay’s findings initially garnered little attention from experts who attributed it to coincidental numerical similarities within diverse mathematical contexts, McKay persisted in his belief that geometry and number theory could be intertwined. He even sported T-shirts inscribed with the equation “196,883 + 1 = 196,884” at academic gatherings.

Subsequently, mathematician John Thompson’s exploration supported McKay’s suspicions by establishing a link between higher dimensions of the monster’s symmetries and subsequent numbers in the number theory sequence. Thompson’s calculations revealed a surprising correlation, reinforcing the possibility of a connection between seemingly disparate mathematical domains.

This revelation piqued the interest of the mathematical community, leading to further investigation into the potential link between geometry and number theory. Despite initial skepticism, the concept of “monstrous moonshine,” as termed by mathematicians John Conway and Simon Norton, gained traction as evidence of numerical patterns emerged, suggesting a deeper connection between these mathematical realms.

The theoretical prediction of the monster structure within group theory, resembling a periodic table for finite symmetries, further fueled this inquiry. With nearly all finite groups categorized into 18 classes and 26 outliers, the concept of the monster challenged conventional mathematical understanding, hinting at profound connections yet to be fully comprehended.

The initial outlier among these groups was the “monster,” foreseen by mathematicians Bernd Fischer and Robert Griess in 1973. The moniker derives from the immense scale of this group, boasting over 8 x 10^53 symmetries. To put this into perspective, consider that the symmetry group of a 20-sided “D20” die, or icosahedron, comprises only 60 symmetries, indicating that there are merely 60 conceivable transformations (rotations or reflections) that can be executed without altering the D20’s orientation.

Due to its immense scale, the monster posed significant challenges for mathematicians. “Most people thought it was going to be hopeless to construct it since much, much, much smaller groups required computer constructions at that time,” Borcherds explained in his YouTube video. Even powerful computers struggled with a structure consisting of 8 x 10^53 elements.

However, this pessimistic outlook was ultimately proven incorrect. In 1980, Griess successfully constructed the monster, thereby proving its existence—without the aid of computers.

Number theory primarily deals with integers, which may seem straightforward initially. However, to delve into the relationships between them, experts employ intricate concepts like modular forms. These functions, denoted as f(z), exhibit extreme symmetry. Similar to the sine function, understanding a specific segment of a modular form reveals its appearance across the entire function.

Modular forms are something like trigonometric functions, but on steroids,” mathematician Ken Ono told Quanta Magazine.

However, despite their complexity, modular forms play a pivotal role in mathematics. For instance, Andrew Wiles from the University of Oxford utilized them to prove Fermat’s Last Theorem, while Maryna Viazovska from the Swiss Federal Institute of Technology in Lausanne employed them to determine the densest sphere-packing arrangement in eight spatial dimensions. Due to the intricate nature of modular forms, they are often approximated by infinitely long polynomials, such as:

[ f(q) = \frac{1}{q} + 744 + 19,688q + 21,493,760q^2 + 864,299,970q^3 + \ldots ]

The prefactors preceding the variable ( q ) constitute a number sequence with intriguing properties from a number-theoretical perspective. McKay linked this number sequence with the monster group.

Borcherds first encountered the moonshine conjecture in the 1980s, recalling, “I was just completely blown away by this,” during an interview with YouTuber Curt Jaimungal. Sitting in one of Conway’s lectures at the time, Borcherds learned of the mysterious connection between number theory and group theory, a subject that captivated him thereafter. He embarked on a quest to uncover this suspected connection, eventually publishing his groundbreaking result in 1992, for which he received a Fields Medal six years later. His conclusion suggested that a highly speculative area of physics, string theory, could provide the missing link between the monster group and the number sequence.

String theory aims to unify the four fundamental forces of physics (electromagnetism, strong and weak nuclear forces, and gravity). Unlike conventional theories relying on particles or waves as the universe’s basic constituents, string theory involves one-dimensional structures resembling tiny vibrating threads, generating the familiar particles and interactions observed in the universe.

Borcherds recognized that string theory relied on numerous mathematical principles related to symmetries, including moduli. Closed strings oscillating through spacetime create two-dimensional tubes exhibiting the same symmetry as modular shapes, regardless of their oscillation patterns.

The specific type of string theory Borcherds explored could only be mathematically formulated in 25 spatial dimensions. While our observable universe comprises three visible spatial dimensions, string theorists presume the remaining 22 dimensions are compactified into tiny spheres or toroidal shapes. However, the physics hinges on their precise configuration, known as “compactification.”

Borcherds compactified 24 dimensions into a 24-dimensional toroidal surface and discovered that the associated string theory exhibited the symmetry of the monster group. The fact that only one free spatial dimension remained did not deter him, as he focused on the model’s mathematical properties rather than its applicability to our physical universe.

In this constructed framework, strings oscillate along the 24-dimensional torus, with the monster group dimensions representing the various vibration modes at different energy levels. Thus, at the lowest energy level, there is only one vibration mode, while at higher energy levels, there are 196,883 distinct possibilities, with the resultant trace exhibiting the symmetry of a modular form.

Borcherds successfully demonstrated the connection between the monster group and a modular form. This was not an isolated case; subsequently, mathematicians have linked other finite groups with various modular forms, with string theory serving as the conduit. Hence, even if string theory fails as a description of our universe, it can still unveil entirely new mathematical vistas.