Published on by Catherine Toulsaly


Never fear shadows. They simply mean that there is a light shining somewhere nearby.

Ruth E. Renkel

Ruth Renkel was a deaf writer from Elyria, Ohio, who died at the age of 68 in December 1983. In June 1973,  the Elyria Chronicle Telegram published a piece about her. At that time, she was already a selling writer for 25 years and even wrote hundreds of gags for cartoons. It’s fascinating how words have a life of their own. They resonate inside us and leave breadcrumbs along the way. In my case, the word ‘shadows’ haunts me. Writers fill the role of soulkeepers. They feed on sounds, emotions, flashes of light, dreams piling up on each other so much so that they no longer know where reality lies.


Specters from another life
Will haunt in spite of how I've tried to let this lie
And leave the past behind
But the past will never go
For the things we never show
For what we cannot atone


When we close our eyes on a starry night, we feel the presence of shadows as we listen to stars rustling. Fleeting shadows whispering echoes curve the fabric of spacetime and hint at the faraway presence of unidentified mass as they propagate from their source. There is a game of hide-and-seek between lights and shadows. On the one hand, our expectation of light shining nearby stems from our need to infer light from shadows. On the other, we infer the shadowy afterglow of a disk from the photon region orbiting just outside of an infinite throat.

Black holes are really spectacular creatures
They’ve got some extremely peculiar features
Their gravity’s strong, of course. So strong, in fact
That objects nearby just can not stay intact

A doll floating by would be stretched to spaghetti
As gravity pulls on the legs poor Betty
The name of this process: spaghettification!
The noodles are wrapped up in rapid rotation

A disk is created from torn-apart shreds
Most of the time it’s from stars (not doll heads).
These disks are quite common and often quite bright
They’re kind of like signposts for black holiest night

Sean Raymond, Black holes, Stars, Earth and Mars, 2020

In every phenomenon, there is a part that is manifest and a part hidden from us. When three galaxies collide, what happens to black holes at their center and their dark matter haloes? Will microscopic black holes splash across the resulting object? A study released in December analyzed seven nearby triple galaxy mergers and found one with a single growing supermassive black hole, five with double growing supermassive black holes, and one that is a triple. But on the microscopic level, I still wonder how they behave.  A 2019 paper observes that novel information about the microstructure of black holes could be obtained from a geometric viewpoint. Is everything only geometry, or, more precisely, is everything multidimensional gravity written in geometrical patterns? How do black holes evaporate anyway? Their gravitational collapse offers the picture of their irremediable decay — a thermodynamically irreversible process — although some may just be wormholes in disguise.


Mathematics responds to our need to see the Universe as a structural reality. We draw abstract representations out of the hat of physical structures and peel mathematical layers off physical quantities. Theories keep insiders in and outsiders out. How moonshine study is related to string theory isn’t something that I can wrap my head around,  neither is the difference between versions of superstring theories. Where, then, do we go from here if our paths pull us apart? Geometrical representations, though, captivate me. In my world, there are shadows of wormholes and blackholes in the far end, light beings piercing through tree branches nearby, and the early morning call of white-throated sparrows outside my window.


Stereographic projection in 3D

Words transpose themselves from one field to the next. They swap their cultural or philosophical coat for a mathematical varnish. A faithful representation of a finite group refers to something more abstract than an artistic composition. A shadow function in mathematics isn’t a shadow function in psychology. I wish anyway to form a mental image of algebraic structures and visualize how exactly finite groups and modular objects are related via physical structures. But then it would require me to figure out the complex mathematics of modular functions. 


Among the five fundamental operations of arithmetic, if I can relate to addition, subtraction, division, and multiplication, modular forms appear mysterious as they satisfy a number of internal symmetries. Modular forms, Barry Mazur said, are “functions on the complex plane that are inordinately symmetric… their mere existence seems like accidents. But they do exist.” Shadows of modular forms create a connection in my mind with the spatiotemporal image of every footstep in a lifetime at home and on neighborhood walks, every distance traveled in the air, on the sea, underground, and around the globe to form a web-like monster structure with real-life events as spacetime points. 


Neighborhood graph of Niemeier lattices

In the world of numbers, the classification of finite simple groups was achieved. Mathematicians determined 18 infinite families and 26 sporadic groups, the largest of which is the Fischer–Griess Monster group M whose smallest dimension is 196883.  The Monster contains 20 of the 26 sporadic groups as its subgroups, and these 20, including the five Mathieu groups and the Thompson group, are said to form three generations of a happy family by Robert Griess. It was suggested that the Monster group M has its origin in a gravity theory in 26 + 1 dimensions. The other six sporadic groups are called the pariahs, including the O’Nan group discovered in 1976.


Lucy Reading-Ikkanda, Quanta Magazine

Lucy Reading-Ikkanda, Quanta Magazine

When we say that the Universe is a mathematical structure, I am unsure whether this is based on scientific observation or the unique way we have to connect with it or whether that’s what it is intrinsically. Our brain may be the one with a hierarchy problem. It yearns for structure, naturally numbers, classifies, and hierarchizes. Theories form a labyrinth of nesting boxes through which it appears impossible to find one’s way. Each topic is a box-in-a-box. Each subject slides from thermodynamics to information geometry. Each concept becomes representational. It is transposed, translated, and conjugated to another. Moonshine correspondences are striking because of internal symmetries revealed. The vast numbers involved in the monstrous moonshine make all the more difficult for it to be described as a mere coincidence.


Is “the theory of everything” merely the ultimate ensemble theory? Max Tegmark, 1998

Is “the theory of everything” merely the ultimate ensemble theory? Max Tegmark, 1998

In Thoughtland, where all space-like or time-like dimensions exist beyond our immediate awareness, shadows hint at hidden symmetries and the translational quality of fractals. We are not conscious of extra dimensions because they rule the infinitely small and the infinitely large. For a century, Steven Weinberg writes, physicists have speculated that our familiar four-dimensional spacetime may really be embedded in a higher dimensional continuum. Pockets of inertial frames provide observational proof of an intimate connection between inertia and gravitation. On that same starry night, our eyes wide open, we spin around. Stars rotate with us in a synchronous dance. In that inertial frame of reference in the midst of which we stand, our arms irresistibly draw upward. We feel the immateriality of shadows beyond the four-dimensional realm. 


Realists are wary of the translational aspect of symbols in their search for rigorous proof. Idealists find meaning in bridges between concepts. On the one hand, we could define a ‘dimension’ as nothing more than another coordinate axis, another degree of freedom, a purely symbolic concept. On the other, a dimension may indeed encompass a higher state, open a gate to a whole new Universe. Our four-dimensional spacetime is the tip of the iceberg that conceals beneath extra small dimensions which contract and compactify through traceless routes into our observable reality. The Quantum Universe might give us access to the infinitely small, while fractals of the infinitely large display infinite-dimensional representations on the other end of the spectrum. Beyond the four dimensions, the Kaluza-Klein theory adds a dilaton field. In 2017, Stephane Collion and Michel Vaugon proposed a new approach to the Kaluza-Klein idea and showed that a unified geometrical frame could be set for gravitation and electromagnetism.


Cross section of the quintic Calabi–Yau manifold. Produced using the methods described in Hanson (1994), "A construction for computer visualization of certain complex curves", Notices of the Amer.Math.Soc. 41(9): 1156–1163.

Cross section of the quintic Calabi–Yau manifold. Produced using the methods described in Hanson (1994), "A construction for computer visualization of certain complex curves", Notices of the Amer.Math.Soc. 41(9): 1156–1163.

The investigation into the infinitely small has resulted in the discovery of 59 new hadrons and counting over the past 10 years by the Large Hadron Collider. I imagine higher dimensional theories filled with massless particles. Finding evidence of such particles could be key to understanding how a Universe with 10, 11, 24, 26, and more dimensions operates. Reality, as it stands external to us, isn’t just mathematical but conceptual. Each concept in the four reference circles that define the essence of information could be seen as a whole new dimension.


Edwin Abbott Abbott, Flatland: A Romance of Many Dimensions


Family tree of theories (Tegmark, 2007)

Family tree of theories (Tegmark, 2007)

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