Plato’s Allegory of the ‘Cave’ and Hyperspaces: Sonic Representation of the ‘Cave’ as a Four-Dimensional Acoustic Space via an Interactive Art Application

Abstract

Mathematician and philosopher Charles Howard Hinton posited a plausible correlation between higher-dimensional spaces, also referred to as ‘hyperspaces’, and the allegorical concept articulated by the Ancient Greek philosopher Plato in his work, Republic, known as the ‘Cave.’ In Plato’s allegory, individuals find themselves situated in an underground ‘Cave’, constrained by chains on their legs and neck, perceiving shadows and sound reflections from the ‘real’ world cast on the ‘Cave’ wall as their immediate reality. Hinton extended the interpretation of these ‘shadows’ through the induction method, asserting that, akin to a 3D object casting a 2D shadow, the ‘shadow’ of a 4D hyper-object would exhibit one dimension less, manifesting as a 3D object. Building upon this conceptual framework, the authors posit a correlation between the perceived acoustic space of the bounded individuals within the ‘Cave’ and the characteristics of a 4D acoustic space, a proposition substantiated mathematically by scientific inquiry. Furthermore, the authors introduce an interactive art application developed as a methodical approach to exploring the hypothetical 4D acoustic space within Plato’s ‘Cave’, as perceived by the bounded individuals and someone liberated from his constraints navigating through the ‘Cave.’

1. Introduction

The exploration of hyperspaces offers a rich interdisciplinary terrain for philosophical inquiry, mathematical elucidation, and artistic expression. Our primary contribution lies in advancing Hinton’s proposition of a correlation between Plato’s ‘Cave’ and higher-dimensional spaces, particularly in the realm of acoustics. Building upon Hinton’s inductive approach, we delve into the resonance between the perceived acoustic space within the ‘Cave’ and the properties of a 4D acoustic environment, substantiated by mathematical inquiry. Furthermore, we introduce an innovative interactive art application, serving as a methodical tool to navigate and comprehend the hypothetical 4D acoustic space depicted in Plato’s allegory. Through this interdisciplinary endeavor, we aim to deepen our understanding of the profound interplay between philosophical narratives, mathematical abstractions, artistic representations, and the potential implications for human perception and creativity.

1.1. Brief History of Hyperspaces

The initial exploration of higher dimensions finds its roots in ancient Greece, notably with Plato’s allegory of the ‘Cave’ [1,2]. Pioneering philosophers and scientists such as Bernhard Riemann and Peter Ouspensky raised fundamental questions about the dimensionality of the ‘real’ space and the geometry that defines it. Although new spaces and dimensions were postulated, definitive proof of their existence is still elusive [3,4,5].

The scientific discipline of mathematics, particularly topology, underwent significant development from the eighteenth century onward. During this period, Irving Stringham, Esprit Jouffret, Henri Poincaré, Charles Hinton, and Claude Bragdon delved into the examination of hyper-solids and the interaction of hyperspace with our familiar 3D space [6,7,8,9,10,11,12,13].

By 1920, artists began grappling with the concept of the fourth spatial dimension, marking a common feature in the movements of modernity throughout the first three decades of the twentieth century. Literature, influenced by Edwin Abbott’s Flatland and H. G. Wells’ works, and painting, particularly through the Cubists and Russian artists, actively sought to visualize the fourth spatial dimension [3,14,15,16,17,18].

However, physics brought both evolution and setbacks to the concept of higher dimensions, with significant shifts occurring since the late nineteenth century. The emergence of new phenomena, such as electrons, X-rays, and electromagnetic waves, sparked interpretations of an invisible world, complemented by the Theory of Ether. Notably, Einstein’s Theories of Relativity in the early twentieth century challenged the existence of higher spatial dimensions, dampening general interest [3,19,20,21,22].

The resurgence of hyperspaces at the end of the century came with String Theory, proposing a ‘Theory of Everything’ explaining our universe as vibrations of strings in eleven dimensions. Advancements in computer graphics capabilities by the end of the twentieth century facilitated the simulation of hyper-solids’ projections into 3D space. Collaboration between artists and scientists, fueled by computer and virtual reality technology, rekindled interest in the subject [3,23,24].

In the realm of music, French composer Edgar Varèse endeavored to capture hyperspatial sound using the available tools of his era. Although he was cognizant of the interest expressed by cubist painters in Paris regarding the fourth spatial dimension, his shift towards a new philosophy on music occurred in New York. It was in this setting that he composed notable works such as Amériques in 1920–1921, Offrandes in 1921, and Hyperprism in 1922–1923, drawing influence from Claude Debussy, Igor Stravinsky, and Arnold Schönberg. The term ‘Hyperprism’ alludes to the geometry of hyperspaces and the formation of ‘sound masses’ [3].

Varèse articulated his views on 4D sound by delineating the attributes of a hypothetical new musical instrument:

“[…] liberation from the arbitrary, paralyzing tempered system; the possibility of obtaining any number of cycles or, if still desired, subdivisions of the octave, and consequently the formation of any desired scale; unsuspected range in low and high registers; new harmonic splendours obtainable from the use of sub-harmonic combinations now impossible; the possibility of obtaining any differentiation of timbre, of sound-combinations; new dynamics far beyond the present human-powered orchestra; a sense of sound-projection in space by means of the emission of sound in any part or in many parts of the hall, as may be required by the score; cross rhythms unrelated to each other, treated simultaneously, […] all these in a given unit of measure or time which is humanly impossible to attain”

[25].

Varèse himself attributed Josef Hoene-Wroński and Hermann von Helmholtz as sources from the scientific world whose work influenced his perception of music as a spatial phenomenon. Wroński defined music as the embodiment of the intelligence concealed in sounds, while Varèse was also inspired by Helmholtz’s studies on sirens and the physiology of sound. Varèse’s experimentation with the sound of sirens, symbolizing hyperbola and parabola as spatial concepts, was influenced by Helmholtz’s work. Malevich, in advising modern Russian composers, and Bragdon, envisioning 4D music creating geometrical figures in the air, similarly supported the concept of sound masses in space [3].

1.2. An Abbreviated Mathematical Examination of the Fourth Spatial Dimension

The predominant mathematical definition of spatial dimensionality posits that the number of coordinates distinguishing one point from another within a given space is equivalent to the dimensions of that space. Consequently, in 4D hyperspace, each point can be uniquely identified by four coordinates relative to others [26].

Generating the ‘Shadow’ of a Hypercube

A comprehensible four-step approach to conceptualizing the fourth spatial dimension involves progressing from the zeroth dimension to the fourth spatial dimension. The initial step entails beginning with a point (zeroth dimension) and transitioning it into a line segment, thereby entering the first dimension (Figure 1a). Subsequently, in the second step, the line segment is shifted in a perpendicular direction, giving rise to a square within the second dimension (Figure 1b). The third step involves further shifting the square perpendicularly to its two axes, resulting in the formation of a cube within the third dimension (Figure 1c). Finally, the cube undergoes a perpendicular shift relative to its three axes, producing the hypercube, a 4D object characterized by a hyper-surface comprising eight cubes. This directionality, however, is beyond our familiar comprehension, limited as we are to perceiving only 3D space. Consequently, what is observed in Figure 1d is the projection (shadow) of the hypercube onto 3D space [27].

The projection, as an interaction of a 3D solid with a 2D plane, may be employed, when applying the inductive method, to understand the interaction of a 4D hyper-solid, e.g., the hypercube, with 3D space [27].

By holding a semitransparent 3D cube so that the light casts its shadow on a flat surface, it is possible, while turning it, to project different 2D shadow patterns on this flat surface. If the light comes at a vertical angle from a point near the cube, as shown in Figure 2a, then a 2D shadow is formed. Correspondingly, Figure 2b shows the ‘shadow’ of a specific 3D projection of a hypercube, the hypersurface of which consists of eight cubes [27].

1.3. Hinton’s Inductive Approach to Plato’s Allegory of the ‘Cave’

Charles Howard Hinton, a mathematician and philosopher specializing in hyperspaces, expounded on the potential connection between higher-dimensional spaces and our perceived world in Chapter IV of his book The Fourth Dimension (1904). He asserted that ancient thinkers like Greek philosopher Parmenides and Asiatic scholars had initially conceived of such a relationship. Moreover, Hinton proposed a plausible link between Plato’s renowned allegory of the ‘Cave’ and hyperspaces [1].

Plato introduced his ‘Cave’ allegory in the seventh book of Republic. Within sections 514a–515c, Socrates, engaging in a dialog with Glaucon, envisions humans confined in an underground dark cave, tethered with chains on their legs and necks. Restricted from turning their heads, these individuals can only perceive the cave wall before them. A fire positioned at a distance and at a higher-level casts light behind them, projecting the shadows of humans, statues, and animal figures onto the cave wall in front of the chained observers. These shadows constitute the only images accessible to the captives, leading them to construct their perceived reality based on these fleeting impressions [2].

Hinton expanded the meaning of the ‘shadows’ via the induction method: as the shadow of a 3D object is a 2D shape, similarly, the ‘shadow’ of a 4D hyper-object is one dimension less; that is, a 3D object, and posited that Plato described a hyperspatial ‘real’ world, the shadows of which give rise to our perceptual environment [1].

2. Materials and Methods

Within section 515b in the seventh book of Republic, Socrates, while elucidating the allegory of the ‘Cave,’ includes a reference to the auditory perception of sounds from the ‘real’ world by the individuals constrained within the ‘Cave’: “What if the prison had an echo from the wall in front of them? Every time one of the people passing by spoke, do you suppose they would believe the source of the sound to be anything other than the passing shadow?” [2].

We advance further along the line of Hinton’s thought and propose that the manner in which sound reflects off the walls of the ‘Cave’ and is perceived as a 3D acoustic space by bounded individuals shares numerous similarities with the properties of 4D acoustic space.

2.1. Propagation of Sound Waves in 4D Space

When a wave is propagated in the familiar 3D space, we observe that if a source, a lamp, for example, is lit for exactly three seconds, an observer standing at some distance from the source shall see the lamp’s light lasting for three seconds. The same goes for waves of any other type, as in our case, a sound wave. This is a property of waves that features are valid only in spaces with an odd number of dimensions. A wave in these dimensions is propagated at a steady velocity and brings about the aforementioned result. In the second spatial dimension, if a wave is generated, for example, by a pebble thrown on the surface of a still pond, one would expect for a radial wave to be created that would move away from the source, from the point where the pebble fell, without any distortion on the surface before or after the wave. Our experience, however, suggests that the water surface will not be still within the circle. That is, the surface of the pond enclosed by the radial main wave will be distorted by waves of smaller amplitude that follow the main wave and attenuate some time later. In terms of 4D space, we expect that, just as in the case of the second dimension of space, some initial stimulation may cause a distortion, which will be propagated in the form of a main wave followed by secondary waves of lower intensity (Figure 3). Accordingly, in 4D space, sound distortions manifest as primary soundwaves, followed by secondary waves of reduced intensity—a phenomenon reminiscent of tunnel acoustics. Also, this sound propagation could be conceptualized as a longitudinal volume, where the physics of sound factors are influenced by coupled effects with double slope decay. Therefore, the theoretical model of 4D sound corresponds to our case, analogous to the echo presented in a cave as described by Plato. [28,29,30].

2.2. Sonic Representation of the ‘Cave’ as a 4D Acoustic Space via an Interactive Art Application

In order to facilitate a clearer comprehension of the aforementioned concept, we have developed an interactive art application that provides a representation of Plato’s ‘Cave’ as a 4D acoustic space. In this space, individuals who are bounded, experience an echoing effect when hearing authentic voices due to the propagation of sound waves through 4D space.

An additional characteristic of hyperspatial sound that we must encompass in our application is the heightened decay of intensity with distance from the source. In an acoustic space, the sound intensity received by the surface of our eardrum depends on the amplitude of the oscillation of the sound wave and the distance from the source. Specifically, if a spherical wave propagates in a N-dimensional space, sound intensity is inversely proportional to the surface area of the N-dimensional spherical surface that anticipates a decrease following the inverse proportionality of R^N−1, where R denotes the distance from the source and the radius of the N-dimensional sphere. In our conventional 3D environment, with a point source, sound intensity is inversely proportional to the square of the distance from the source. In a 4D space, we expect that sound intensity that is generated from the same source follows an inverse proportionality to the exponent three of this distance. It is proved, however, that for spaces with dimensionality higher than three, the decrease is even greater. This heightened reduction is attributed to the amplitude containing terms inversely proportional to higher exponents of the distance from the source [31].

Description of the Interactive Art Application

The depiction of the ‘Cave’ space and the design of user interaction were composed using ‘Processing’ open-source programming language and development environment and ‘Beads’ library. Adhering to Plato’s depiction, a fire was positioned at the bottom of the screen. In proximity to the fire, three spheres are positioned, symbolizing 4D entities: Socrates, a musician, and a man accompanied by two dogs. These entities, illuminated by the fire, cast their shadows on the ‘Cave’ wall positioned at the top of the screen. The user, portrayed as a sphere in its bounded 4D nature, is approximately placed in the center of the screen.

Successively appearing on the screen based on the user’s choice, Socrates, the musician, and the two dogs each generate their own sound disturbance in the space. Socrates recites a portion of Plato’s Apology of Socrates, the musician performs a Seikilos song called Epitaph with the Lyre of Apollo, and the barking of dogs resonates with the animal as described by Plato in his allegory of the ‘Cave’ (Figure 4) [32,33].

What would be the outcome if one of the individuals bound by constraints were to break free and move through the cave? This concept is also alluded to by Socrates in the seventh book of Plato’s Republic.

“[…] Think what their release from their chains and the cure for their ignorance would be like. When one of them was untied, and compelled suddenly to stand up, turn his head, start walking, and look towards the light, he’d find all these things painful. Because of the glare he’d be unable to see the things whose shadows he used to see before. […] ”

[2].

Our interactive art application enables the unrestricted movement of the unchained humans, allowing them to navigate in 4D ‘Cave’ space at variable speeds. This movement results in an auditory experience where sounds are echoed and shifted towards higher playback speeds as the humans approach the source and towards lower up as they move away from the source. These functions align with a theoretical 4D auditory system.

Specifically in our familiar space, the 3D waveforms of sound actuate our 2D membranes, namely our eardrums. After processing primarily within the cerebral framework, this information culminates in the cognitive apprehension of the 3D acoustic environment. Employing the inductive approach, we posit the possibility of detecting 4D sound through 3D auditory sensory organs. The auditory stimuli contained within a 3D eardrum have the potential to analyze and process it, allowing traversal in any direction, including backwards. In addition, this process replaces the accustomed 1D linearity of a sonic stimulus with the 2D nature of writing. In this context, navigating the sonic stimulus becomes akin to reading the pages of a book. If sound acquires the properties of writing, it gains an additional dimension and can therefore be considered hyperspatial. [34].

Within our interactive art installation, users can control their on-screen positioning through an input device, such as a computer mouse. This control allows them to navigate the acoustic space generated by three distinct virtual sound sources: Socrates, the musician, and the barking dogs. Each source creates its own individualized acoustic space, forming a unique sound environment.

As users move through the ‘Cave’ space, every point they reach corresponds to a specific point on the sound waveform, dependent on the temporal progression of the sound. This setup enables users to perceive the sound emitted by each source at varying paces—standard, decelerated, or accelerated—based on their velocity, movement direction, and proximity to the source. For example, if a user moves towards the musician’s sound source, they perceive the music as playing faster. Conversely, moving away from the barking dogs may cause the barking to sound more prolonged, even backwards. Additionally, users can interact with the acoustic environments of the other sound sources simultaneously. This interaction considers the user’s relative speed, direction of movement, and distance from each sound source, creating a distinctive blend of auditory experiences.

The installation offers an immersive experience where users’ movements directly affect their auditory perception, enabling a deeper connection between physical navigation and sound interaction. It highlights the spatial properties of sound and emphasizes the role of user interaction in shaping auditory experiences, providing a dynamic and engaging exploration of acoustic environments. Our development of the interactive art application provides a novel platform for users to engage with and navigate the theoretical 4D acoustic space and extends the boundaries of traditional acoustic theory, offering insights into the potential properties of higher-dimensional auditory environments. These methodological innovations not only enhance our understanding of hyperspaces but also pave the way for future interdisciplinary research at the intersection of philosophy, mathematics, and art.

3. Results

This article delves into the convergence of Plato’s philosophical concepts and the notion of hyperspaces, as expounded upon by the mathematician and hyperspace philosopher Charles Howard Hinton. It suggests a deeper connection between Plato’s allegorical narrative of the ‘Cave’ and the concept of hyperspaces. By exploring the interrelations between Plato’s allegorical representation, mathematical theory, and artistic expression, this study aims to enhance our understanding of the profound impact of higher-dimensional spaces on human perception and creativity, particularly in the realm of auditory experiences.

The phenomenon of sound propagation within a cave, where sound waves interact with the cave walls to create a 3D acoustic environment, shares notable similarities with the characteristics of a 4D acoustic space, as evidenced by mathematical findings. Our developed audiovisual interactive art application provides users with the opportunity to immerse themselves in the auditory manifestations of hyperspace, such as echoes and variations in sound intensity, presented in an engaging visual format. By offering a tangible representation of abstract mathematical concepts within an artistic context, this innovative approach introduces a completely new auditory experience, suggesting its potential effectiveness as a tool for enhancing the cognitive ability to discern a theoretical 4D auditory system.

In addition to Hinton’s contributions, there are numerous similarities between the user’s interaction with our interactive art application and Varèse’s depiction of 4D sound through his hypothetical musical instrument. Furthermore, Varèse’s exploration of electronic techniques in his musical compositions, particularly his experimentation with playing recorded sounds in reverse, as demonstrated in his composition Intégrales, orchestrated for eleven wind instruments and four percussionists during 1924–1925, bears resemblance to the techniques utilized in our interactive art application. Here, positive and negative playback speeds enable an acoustic experience reminiscent of navigating a 4D acoustic space with varying directions and velocities [35].

Our research is subject to a notable limitation concerning the absence of empirical evidence supporting the existence of hyperspace. Consequently, assertions regarding the exact nature or approximation of hyperspace sound remain speculative, as they lack empirical validation. The transition from theoretical mathematical models to practical outcomes, as demonstrated in our study, represents a significant advancement. However, it is essential to recognize that these outcomes are unprecedented and cannot be directly compared with existing research due to the nascent nature of the field. The user experience within the interactive application provides a novel auditory encounter but cannot be equated with a genuine auditory perception of hyperspace, given the absence of prior experiential reference points. Analogously, the interpretation of a 3D cube drawing by a 2D being illustrates the limitations of perspective based solely on mathematical descriptions. Thus, while our artistic application represents a tangible manifestation of mathematical theory, its interpretation remains open to scrutiny and excitement, reflecting the inherent complexities of exploring theoretical concepts through practical means.

4. Discussion

Can the human brain apprehend images and sounds of a 4D hyperspace, or are we confined within our familiar 3D spatial realm?

The human brain engages in the processing of incomplete information from visual, auditory, and tactile stimuli emanating from 3D objects to construct a comprehensive 3D model of the surrounding space. This developmental process initiates in early childhood and persists into adulthood [36]. Neuroscientific investigations into the evolution of our 3D perceptual and sensorimotor capacities, utilizing functional magnetic resonance imaging (fMRI), indicate that activations in various brain regions, including the entorhinal cortex and the visual system hierarchy, contribute to the extraction of 3D information [37]. According to neuroscientists, these areas might also play a role in the representation and integration of 4D information. In a separate study, neuroscientists contend that the human brain possesses the capability to compute positions and distances and devise spatial routes within a simulated 4D environment [38]. This suggests a potential evolutionary trajectory for our cognitive abilities towards the perception of hyperspaces.

Numerous researchers have undertaken to comprehend hyperspace through visual means [39,40,41,42]. However, the auditory ‘visualization’ of hyperspace emerges as a potentially more fruitful avenue. This could be attributed to parallels between the characteristics of sound and hyperspatial gravitational waves.

Specifically, in a parallel manner to an accelerating charge radiating electromagnetic waves, including light and radio waves, Albert Einstein, through his General Theory of Relativity, postulated that impactful events involving large masses or, more precisely, distributions of matter and energy undergoing acceleration, generate wave distortions akin to a pulse in the fabric of spacetime. These distortions are referred to as gravitational radiation or gravitational waves. A distinctive feature of gravitational waves is their ability to traverse space without significant interaction, in stark contrast to electromagnetic radiation like light, radio waves, X-rays, and gamma rays, which can be influenced by other masses and charges, as well as magnetic or electric fields [21,22]. Furthermore, according to String Theory, gravitational waves possess the capability to travel in higher dimensions [43,44]. Consequently, if there were waves detectable by a hyperspatial sensory organ, gravitational waves would be prime candidates. Intelligent hyperspatial beings could potentially perceive their surrounding space based on these gravitational waves. Although human sensory organs are incapable of directly detecting gravitational waves, evidence suggests that they exhibit behavior reminiscent of sound waves [45]. Therefore, the auditory centers in our brain are better equipped for the perception of hyperspace, prompting a more effective outcome through a sound-centric approach.

5. Conclusions

In this study, we explored the intriguing connection between Plato’s allegory of the ‘Cave’ and hyperspatial sound, building upon Charles Howard Hinton’s inductive approach. Hinton’s mathematical insights suggested that the shadows in the ‘Cave’ could be projections of 4D entities onto 3D space, expanding our understanding of Plato’s profound metaphor.

Our investigation delved into the auditory dimension of the ‘Cave’ allegory, proposing that the way sound reflects off the walls within the ‘Cave’ shares parallels with the properties of a 4D acoustic space. Utilizing mathematical evidence, we demonstrated that in 4D space, sound distortions manifest as primary soundwaves followed by secondary waves of reduced intensity, akin to echoes resonating within a cave.

To enhance the comprehension of this theoretical framework, we developed an interactive art application simulating Plato’s ‘Cave’ as a 4D acoustic space. The application allows users to experience echoing effects as they move through the ‘Cave,’ providing a unique auditory encounter aligned with the theoretical 4D auditory system.

Our exploration has established significant connections with Varèse’s concept of 4D sound, as demonstrated by his description of the characteristics of a hypothetical new musical instrument. In our application, considering the user’s relative speed, direction of movement, and distance from each sound source, it can produce any number of cycles or subdivisions of the octave, creating sub-harmonic combinations and simultaneously treating cross rhythms that are unrelated to each other within a given unit of measure or time. These attributes reinforce the parallels between our findings and the experiences conveyed through hyperspatial sound in the realm of music.

In the discussion section, we questioned the capacity of the human brain to perceive 4D hyperspace and presented evidence suggesting an evolutionary trajectory towards hyperspace perception. We proposed that the auditory ‘visualization’ of hyperspace, particularly through the characteristics of sound and gravitational waves, might offer a more fruitful avenue for exploration.

In conclusion, this study contributes to the interdisciplinary dialog between philosophy, mathematics, art, and neuroscience, offering a unique perspective on the allegorical richness of Plato’s ‘Cave’ and its potential connections to hyperspaces and sound. The interactive art application serves as a practical manifestation of our theoretical framework, inviting users to explore the perceptual nuances of a 4D acoustic space. As we continue to unravel the mysteries of higher dimensions and their potential impact on human perception, this study opens avenues for further research at the intersection of philosophy, art, mathematics, and sensory experience.

Augmented reality technology now enables the projection of virtual environments directly onto our 3D environment. This prompts a direct comparison of the esthetic quality between virtual and real environments, challenging our cognitive perception. The integration of scientific, technocratic, and artistic knowledge gives rise to a new professional, the ‘Technartist’, capable of developing advanced audiovisual applications, including 3D interactive graphics. Technartists may further lead the way in creating interactive audiovisual 4D representations of exceptional esthetic value, allowing us to perceive a 4D hyper-environment [46].