Plato's Cave and Calabi-Yau Manifolds
Copyright © 2006, David A. Epstein.
All Rights Reserved.

December 18, 2006

One of the more fascinating allegories to be found in the annals of Western philosophy or literature is the story of Plato's Cave. The basic premise is that inside of a large cave, a group of prisoners are shackled to some seats and their heads are firmly braced in place, preventing them from turning around and seeing what is behind them. They can only look straight ahead and view what is projected onto a large wall. Unbeknownst to the group is a large parapet (a raised walkway) located directly behind them. Along the walkway, some people, who for all practical purposes are puppeteers, hold up figures of people, animals, and objects. The light from a fire in the back of the cave is projected upon these figures and their shadows are cast upon the wall. The prisoners see these shadowy movements and mistakingly interpret them to represent reality.

The allegory was used by Plato to demonstrate the dichotomy between the World of Everyday Reality (the shadows) and the World of Ideal Forms (the figures held up on the stage). He argued that the World of Everyday Reality is illusory, just like the Hindu mystics argued that the world of Maya is an illusion. The shadows might appear to be real, for they can be seen by humans; but they are mere projections cast from the true objects inhabiting the higher world. This everyday reality would include objects and beings discernable by the senses, but are mere approximations of the transcendental objects and beings in the World of Ideal Forms. Plato (via Socrates) amply describes what would happen if one of prisoners managed to escape and see these figures, the people holding up these figures, and the fire itself: there would have been disorientation, denial, anger, fear, and a host of other emotions. Eventually, though, he would acclimate to the new reality and would learn to understand and appreciate its true nature.

The duality between the "true" and "projected" worlds is also found in Gnostic philosophy. The contrary nature of "true" and "illusory" reality is certainly an illustrative outcome of Plato's allegory; but this is not the only outcome, for there is another equally profound conclusion to be drawn from consideration of the allegory: it demonstrates that a higher dimensional world is projected onto a lower dimensional screen. The projected world (what appears on the screen) will have lower dimensionality (two dimensions in this case) than the real marrionette world (three dimensions in this case). The message that Plato was conveying was that true reality exists in an illuminated higher dimensional world (containing more than three dimensions) that is projected onto a 3-dimensional screen (the day-to-day world as we know it).

Indeed, higher-dimensional worlds theorized in scientific thought might not have been directly inspired by the allegory of Plato's Cave, but they are compatible with its philosophical message. Minkowski and Einstein's space-time coordinate system calls for the existence of a four-dimensional universe (3 dimensions of space, and 1 of time). The Kaluza-Klein model necessitates the existence of a five dimensional universe based upon Einstein's four-dimensional space-time continuum. Kaluza-Klein extended this to include a hidden, curled-up circular fifth dimension (light). When light is emitted due to an excitation of electrons, which result in their jumping to higher orbitals, this hidden dimension becomes revealed for all to see, for light is "projected" onto our known 3-dimensional world! Kaluza-Klein theory unifies the electromagnetic force (represented by Maxwell's field equations) and gravity (represented by Einstein's field equations), and this can explain some of the interactions between light and our known physical world.

Finally, String theories necessitate the existence of ten and twenty-six dimensional worlds for the vibrating strings to inhabit (we'll disregard the M-Theory version which requires 11 dimensions; this theory unifies the other 5 string theories). Superstrings rotating clockwise exist in ten-dimensions (this is the Fermion string theory), while strings rotating in the counterclockwise direction exist in twenty-six dimensional space (this is the Bosonic string theory). The heterotic string theory is basically a combination of the fermionic and bosonic string theories.

Where do the numbers 10 and 26 come from? They are derived from a special type of mathematical construct known as the modular function. Modular functions map complex numbers to other complex numbers and have a number of interesting properties including the preservation of symmetry relationships on the surfaces for which these functions are defined. They describe the modes of the string movements across Riemann Surfaces, where each mode is a distinct pattern of motion (i.e. with a signature vibrational frequency). Strings moving across these surfaces must satisfy a property known as conformal invariance. This is a type of symmetry requiring all of the points along a string's pathway of motion (its worldsheet) to be analysed, but the order of analysed points is not important, for the string equations must be invariant for any ordering. The conformal symmetry property appears in the special modular function known as the Ramanujan's Modular Function which describes the string motions in space-time. This function is a 24-degree polynomial equation (it has 24 analytical solutions which correspond to the 24 string modes), and contains this term: 1- (D - 2)/24, where D is the dimensionality of the domain space. In order to satisfy relativistic and other conditions, this term is required to equal 0. This means that D must equal 26. In the generalized version of Ramanujan's Modular Function, which is an 8-degree polynomial equation, the term equals 1 - (D - 2)/8; in order for this term to equal 0, D must equal 10.

According to superstring cosmology theory, the original universe consisted of a supersymmetric ten-dimensional space-time continuum. The Big Bang caused a symmetry breaking that split the universe into two parts: an expanding four-dimensional space-time universe (the universe we inhabit) and a contracting six-dimensional universe (represented by the curled-up compactified six dimensions of the Calabi-Yau manifold). Each of these six dimensions is the size of a superstring (approximately 10 ^ -33 of a centimeter). This occurs because a Calabi-Yau manifold is the topological representation of the movements of a superstring. The string can be viewed as projecting its motions onto this six-dimensional landscape. Here, we see the direct link to the Allegory of Plato's Cave, for the superstring worldsheet existing in the higher dimensional space casts its shadows upon the wall of the lesser dimensional Calabi-Yau manifold, and its confined residents would believe that they are viewing the actual superstring world.

In fact, the ratio of actual  to projected dimensions for the two realms is very close. In Plato's allegory, the 3-dimensional world is projected onto a 2-dimensional wall; the ratio is 3:2 = 1.5. In superstring theory, a 10-dimensional superstring is projected onto a 6-dimension landscape; the ratio is 10:6 = 1.666.

Yet this Calabi-Yau manifold is not an ordinary receptive "wall". In fact, if the shape or size of their dimensions changed, the "wall" would not only have different contours, and hence the vibrational shadows cast on its surface would be viewed differently, but the "wall" itself would affect the patterns of the projections upon its surface. In superstring theory, any change to the size or shape of the six curled-up dimensions would change the vibrational patterns of the superstring. This, it turn, would alter the properties of the particles which arise from this vibrating string. In this regards, the analogy to Plato's Cave breaks down. If the wall changed size and shape in the Cave, the shadows would change accordingly, but the wall would not affect the nature of the objects held up for projective view.

Or would it? Maybe the potential effects from altering the Calabi-Yau manifold reveals a more general pattern in nature, a point further exemplified by the mapping of this manifold onto 3-dimensional space. Perhaps if the dimensions that comprise Plato's wall were changed (not changing the wall's length, height, and shape, but changing the size and shape of the actual two dimensions from which the wall is constructed), this indeed would alter the properties of the objects that are casting its shadows. The nature of the light reflecting off the screen might equally be altered, perhaps by modifying the wave/particle composition of light. This light might then interact with the raised figures along the walkway and cause them to vibrate at different frequencies to such an extent that it would change at least some of their physical properties.