Minds as Hyperspheres; the equal-extension thesis and its implications for the framerate of consciousness


There’s a traditional joke among physicists about “spherical cows”:

> Milk production at a dairy farm was low, so the farmer wrote to the local university, asking for help from academia. A multidisciplinary team of professors was assembled, headed by a theoretical physicist, and two weeks of intensive on-site investigation took place. The scholars then returned to the university, notebooks crammed with data, where the task of writing the report was left to the team leader. Shortly thereafter the physicist returned to the farm, saying to the farmer, “I have the solution, but it works only in the case of spherical cows in a vacuum.”

This joke hides practical wisdom: if we don’t know the shape of something our default guess should be spherical, i.e. that the object has equal extension across all dimensions.

Likewise, there’s an operation in physics called a “wick rotation” which can be described as rotating a system’s coordinates such that the time dimension becomes a spatial dimension, and one of the spatial dimensions becomes the time dimension. Wick rotations are basically a restatement of a core thesis in General Relativity — that time and space are in an important sense equivalent and interchangeable.

I think wick rotations and the assumption of equal extension across dimensions can be combined into a tool which says something fresh about the framerate of consciousness.


Various hypothetical devices have been proposed to display objective information about consciousness — e.g. “qualiascope” (Trujillo 2003), “consciousness meter” (Chalmers 1996), “psychoscope” (Baars 1998). We can think of a hypothetical ‘qualiascope’ as something that allows us to perceive the most interesting structural elements of consciousness, much as microscopes, spectroscopes, telescopes, etc all highlight features at various scales in the physical world.

If you look through a qualiascope, one of the most striking features you might see is the boundaries of experiences*. While boundaries in the physical world are useful conventions that allow clean simplifications and tractable calculations, in the phenomenological they are (probably, in a specific technical way) clean fundamental facts**. Each moment of experience is a certain sort of logic crystal, which may (or may not — interesting scissor point) be thought of as its own closed universe. A “stream of experience” is a sequence of these crystals.

*Another property a good qualiascope should show is the experience’s valence.

**See also Bayesian Mechanics’ treatment of boundaries as fundamental

My version of Strong Monism suggests these “qualia crystals” are everywhere, existing in parallel with the physical world — but mostly they exist as fine dust, tiny bits of primordial qualia. Human minds combine a lot of this dust together into big crystals, each of which constitutes one experience. We don’t know for sure what shape or size these crystalline chunks of human consciousness are, but it seems reasonable to claim they’re roughly spherical — the brain is roughly spherical, probably the mind is too.

Importantly, these chunks are four-dimensional, not three — that is, they have extension in both space and time. And so if we’re evaluating the shape of a crystal, we also need to infer its extension in time. I’ll suggest that the most reasonable assumption here is to assume uniformity — that the most likely shape of an experience is that it has equal extension in both time and space. I.e. until we learn otherwise, let’s assume that experiences are hyperspheres (the fancy mathematical term for 4-dimensional spheres).

An ongoing question in consciousness research is “what is the framerate of consciousness?” — this is equivalent to asking “how long are these 4d chunks of experience in the time domain?” The “WRHS” (wick rotation + hypersphere) model I’m introducing today suggests that (1) however big they are in space is how big they’ll be in time, and (2) any constraints we can infer about how big an experience is in either time or space can be translated into its complement.


To apply this model and translate space into time, we need two things: the distance and the speed of information propagation.

1. What do we know about the physical size of the mind?

  • The brain is generally viewed as the physical counterpart of the mind, and we know how big the brain is: just a little smaller than the skull. The brain is not a perfect sphere, but if we average the dimensions we get a radius of 6.6cm.
  • The brain’s magnetic field drops off with the cube of the distance, which means it doesn’t extend much further than the brain itself. We can estimate it as having a radius of 7cm.
  • The brain’s electric field drops off with the square of the distance, which makes it extend much further, perhaps 20cm from the skull (at which point it’s absolutely swamped by background EM noise). We can put the radius as 25cm.
  • Insofar as consciousness is electromagnetic, and the heart has a significantly stronger (4x-10x) electromagnetic field than the brain, we should derive the estimates above for the heart as well. Roughly speaking the average human heart has a physical radius of 4.2cm, a magnetic field of radius ~5cm, and an electric field of radius 50cm (though once again, electric field radius is extremely rough)
  • The thalamus is often identified as the seat of consciousness due to two factors: it centrally integrates many sorts of information flows, and any electrical perturbation of the thalamus generally makes people lose consciousness. The radius of the thalamus is roughly 2cm.

The “real” size of the mind may be a complex integration across these and other estimates, but in general most of these estimates are within an order of magnitude — a promising sign. As a naive placeholder average we can use for the purpose of explaining this method, we can estimate the physical size of an average human mind is a sphere with radius of 5cm.

2. What do we know about the physical speed of information propagation in the mind? I’ll suggest three models:

Approximation #1: information propagation at the speed of light

  • Light travels at different speeds in different materials based on the material’s refractive index. The speed of light (photons) in wet fatty tissue like the brain is approximately ~73% of the speed of light in vacuum
  • It takes approximately 2.2825 × 10^-10 seconds (228.25 picoseconds) for light to travel 5 cm in human tissue
  • The frequency corresponding to this time period, and thus the upper bound of phenomenological refresh if photon interactions are the primary binding agents / mediators of consciousness, is approximately 4.38ghz

Approximation #2: information propagation at the speed of electrons in axons

  • The EM field may propagate at the speed of light, but nerve impulses move much more slowly: the speed of electrons in unmyelinated fibers is ~1m/s, in myelinated fibers ~3-120m/s
  • It takes approximately .42ms for electrons to travel 5cm @120m/s (assuming a straight line and no relay neurons)
  • The frequency corresponding to this time period, and thus the upper bound of phenomenological refresh if electron movements are fundamental mediators of consciousness, is 2.4khz

Approximation #3: information propagation at the speed of signal propagation through a connectome

  • We can adjust approximation #2 by taking the connectome of an actual thalamus and tracing how long it takes for a neural signal to propagate halfway through. I.e. looking not just at the speed of electrons in axons, but how many neurons are in the path of a signal and the average latency added by each hop
  • Qiu et al. 2015 simulated signal propagation in the brain, and got an estimate of .1m/s; Muller et al. 2018 suggest 1-10m/s. Selen Atasoy’s CSHW work suggest a wave’s propagation speed is proportional to its frequency, which could explain some of this variance
  • The minimum refractory period of neurons is ~5ms, which makes speed estimates at magnitudes close to this rather noisy/chunky
  • It takes 50ms to travel 5cm at .1m/s, and 5ms to travel 5cm at 1m/s
  • The frequency corresponding to this time period, and thus the upper bound of phenomenological refresh if integration across neuron firing patterns is the primary binding mechanism of consciousness, is somewhere between 20hz-200hz

Unlike our spatial estimates above, our propagation speed estimates are spread across 8 orders of magnitude. This means we have to be opinionated about which estimate is the best one. However, I suggest we can collapse part of the variance by separating “the framerate of consciousness” into two quantities:

  1. The “framerate of phenomenal consciousness” which might be either near (1) or (2) (and less likely to fall in the middle);
  2. The “framerate of cognition” which is more determined by neural firing speeds, connectome structure, heart rate and metabolism, cognitive and harmonic differentiability, task-specificity, and so on, and is plausibly much slower (~20-200hz).

In short, I expect consciousness has both a “real” framerate and a “reportable” framerate, and the former is likely much faster than the latter. I find it physically plausible that the real framerate of experience is above 1khz, perhaps significantly so. However, human experience might exhibit punctuated equilibria where we have hundreds or thousands of nearly identical experiences in a row, until the much slower cognitive processes periodically pump new information into the system.

All that said, the purpose of this writeup is mostly to offer a new method. The most useful hypotheses are “big if true; also big if false” — it would be a big result if we can establish that human experiences are roughly hyperspheres, because then we have a clean way of turning spatial constraints into temporal constraints, and vice versa. It would also be a big result if we could establish they aren’t — if they tend to be lopsided in some way.

And if experiences are spatio-temporally lopsided? Macroscopic topology is a huge constraint on possible symmetries, and so if there does turn out to be large variance in how spherical our moments of experience are this is likely a core factor in their relative emotional valence. Several research threads follow, e.g.

  • Presumably we could radically improve the valence of our experiences if we evened out their macroscopic shape. Maybe this is part of how meditation and somatic practices help.
  • If human experiences are not ideal hyperspheres, I expect both the spatial and temporal extension of experiences to vary substantially from task to task, age to age, and energy level to energy level. It seems statistically unlikely that such conditions will always push spatial and temporal extension equally — e.g. unpleasant tasks likely make our qualia crystals oblong, and oblong qualia crystals make our tasks unpleasant.
  • If we make synthetic minds in the future, let’s make them hyperspherical.

The scenario that would make the above method most useful would be (1) experiences are pretty close to being hyperspherical (so we can use the above method to convert spatial and temporal observations into the other), but (2) insofar as experiences are not hyperspheres, this factors heavily in their valence (so we can use this method to debug why certain experiences are unpleasant).

Author’s note, March 1, 2024: the approximations I have identified for the boundaries of the mind, and the speeds of information propagation, are meant to help illustrate my thesis that assuming equal extension leads to interesting translation of constraints from space to time, and vice-versa. I chose them because they’re accessible and intuitive, not because they are characteristic of my particular hypotheses on the Binding/Boundary Problem.

March 16, 2024: although I’ve phrased the equal-extension hypothesis in temporal-spatial terms, it may be more precise to assume equal/uniform extension in branchial space, out of which spacetime (probably) arises. See also the “MDBP” hypothesis in Principia Qualia, Appendix E.