Proving Impossibility Through The Universe of Possibility

How does one demonstrate that something cannot be achieved? Surprisingly, this task can be more straightforward than it appears. In mathematics, many impossibility proofs stem from exploring the realm of what is possible: we define all potential outcomes and then verify if a proposed attribute or method fits within those parameters.

One of the simplest examples of such a proof is demonstrating the impossibility of constructing a Platonic Solid—specifically, a convex polyhedron in three-dimensional Euclidean space with, for instance, fifty or a thousand faces. By applying the Euler Characteristic, which must equal 2 for such a solid, and incorporating a linear equation that connects a unique number of edges to the number of regular faces, we can formulate a Diophantine equation that delineates all potential solids. Through basic algebra, we reveal that only five integer pairs of edges and faces can exist: the tetrahedron, hexahedron (cube), octahedron, icosahedron (twelve faces), and dodecahedron (twenty faces). These five represent all the regular polyhedra possible in Euclidean three-dimensional space.

Selena Ballerina delights in soft objects and enjoys hugging her squishy companions. Therefore, I prefer the topological proof I mentioned earlier, which utilizes the Euler Characteristic. Euclid's original proof differs significantly and can also be found on Wikipedia!

Thus, any regular polyhedron (to borrow Maxwell's charming misnomer from his childhood: "I have made a tetra hedron, a dodeca hedron and two more hedrons that I don’t know the right names for") must exist within this exhaustive universe of possibilities. While regular pentahedra do not exist, irregular ones certainly do—take the grand structures in Giza as an example! Therefore, our task isn't to evaluate every possible number of sides; instead, we only need to examine the realm of possibilities.

I have always been captivated by such characterizations of possibility, including topics like Galois Theory, which illustrates why there is no general solution to quintic equations using radicals and basic field operations (though one exists involving Jacobi Elliptic Functions), and the Lie Correspondence, which is particularly relevant to my work as a physicist dealing with waveguides.

Trisecting Angles and Higher Dimensional Numbers

Why can the real numbers only be extended to number "fields" of dimensions 1, 2, 4, and 8? What implications does this have for...

Ruth Moufang's lesser-known contributions also highlight how prejudice can obscure significant figures in mathematics. Her gender played a role in her relative obscurity, akin to Gödel's struggles, who, despite his brilliance, battled with paranoia and relied heavily on his supportive wife, Adele Numbursky Pockert, for grounding.

Gödel's mental health challenges and the perceived status of his wife drew derision at Princeton, reflecting the snobbery toward both his genius and her unassuming role. Society often devalues mental illness and womanhood—an unfortunate truth that many still uphold.

The stigma towards Gödel's wife, a dancer, led to derisive remarks at Princeton. I, too, share a passion for both science and the arts, finding solace in Burlesque and Comedy Performance. My work as a BDSM assistant Domina provides connections with dear friends, explaining my unexpected presence in this realm, especially as someone who identifies as asexual.

I often assert that the women I meet in Berlin's sex work scene possess depth far beyond those who, despite their academic superiority, look down upon me. This world fosters a vibrant Selena who radiates creativity and joy. My concern for our planet's future weighs heavily on me, but with my supportive friends, we can confront these challenges with humor, ensuring I remain a pillar of support for my children.

Man is Not The Measure of All Things

Science has drawn me in precisely because it reveals that humanity is not the center of existence. There exists a world of beauty independent of our perceptions or existence. As a teenager struggling to conform, I found solace in science—an intimate connection with Nature that quieted my ego.

This affinity for mathematics, which reveals limitations and categorizes possibilities, resonates deeply with me. Mathematics, the "Queen of Sciences," empowers us to understand its constraints without being altered by its own might.

I empathize with both Adele Numbursky Pockert and Gödel, who held steadfast to a Platonist belief in the existence of truth beyond human comprehension. This perspective allowed him to pursue his incompleteness theorems with unwavering conviction, asserting that Platonic Truth could be discerned beyond the axioms he utilized.

Limitations in Number Systems

Now, let's revisit the exploration of limitations within mathematical constructs: specifically, how far can we extend number systems? Notably, two mathematicians illuminated these boundaries: Adolf Hurwitz in the 19th century and Ruth Moufang in the 20th century, particularly through her work on Moufang Loops and Octonions.

Moufang's investigation into the Octonions stemmed from her examination of Desargues’ Theorem, which led to her discovering a remarkable counterexample. This theorem holds relevance in specific projective spaces, and in 1933, she unearthed a fact that, while intriguing in its own right, held profound implications for the foundations of Quantum Mechanics.

Quantum Mechanics, Desargues' Theorem, and the Octonions

After 1926, especially post-1928, physicists wrestled with the peculiar algebra stemming from the Dirac Equation, making projective geometry a hot topic. In my previous article, I delve into the Dirac Equation's implications on understanding rest mass.

In essence, quantum states are represented as rays in projective space, with normalized quantum states residing on a unit radius hypersphere. Quantum superpositions arise from algebraic operations on these projective rays. Dirac's algebra revived interest in 19th-century work by William Kingdon Clifford, with three prominent scientists—Eugene Wigner, Pascual Jordan, and John von Neumann—seeking to define the algebra of quantum mechanics. At one point, they considered using Desargues’s Theorem as an axiom, hoping it could guide their understanding of projective space within quantum algebra.

However, Ruth Moufang's revelation that the Projective Octonion Plane fails to satisfy Desargues’s Theorem thwarted their efforts to axiomatize quantum mechanics using these concepts. Unaware of her findings, Jordan, Wigner, and von Neumann continued their research for several years until they stumbled upon their own limiting theorems. For each noncommutative algebra of quantum operators, a commutative but non-associative Jordan Algebra can be defined, leading to insights about the structure of Octonions. Ultimately, in 1949, they independently recognized Moufang's projective octonion plane, effectively stalling their program sixteen years after she first demonstrated the existence of non-Desarguian projective planes. Today, octonions remain an area of interest for quantum researchers seeking insights into the elusive nature of quantum algebra.

As we explore the mathematical and structural reasons behind the octonions' positioning within number theory, we uncover a narrative that highlights the beauty of mathematics recognizing its own limitations—an exquisite instance of sublime imperfection.