The Sheet That Knows Its Geometry

Draft for the author's reading and rewriting. Established science throughout; the framing and the leaps are the author's.

What is perception, as created through evolution, but a method which aids pure survival?

For humankind, part of the free wild cunning intelligence which is a marvel of sensory evolution.

Our human perception has been attenuated by evolution to serve the needs of pure survival, perhaps at the sacrifice of what other living beings may perceive.

What is this finely tuned set of instruments which we now use to far more of an extent than simply survival?

What purpose such richly endowed sensory capacity for the roughshod cunning of pure survival?

Survival does not require the finely tuned hands, brain, hearing and proprioception of a great musician, does not require the fine senses which create a great chef or winemaker.

Where our lack of sensory capability is most noticeable is in our ability to perceive what is actually going on beneath the surface of things – and out into the cosmos where our senses cannot reach.

Our transformation now into the Quantum Paradigm, now makes perfectly clear where our limitations of proprioception lie.

A point in your visual field can be written as a distance and an angle from wherever you are looking. The primary visual cortex receives that point, to first approximation, as the logarithm of the distance paired with the angle — the complex logarithm of the retinal position. Schwartz worked out the correspondence in 1977, and it has held as a good first description ever since [1, 2]. The result is that the curved polar grid of the retina is ironed flat onto the cortex: circles of constant distance become parallel straight lines, rays of constant direction become the orthogonal family, and local angles and shapes survive the ironing intact.

Two hard motions become two easy ones

The payoff is immediate. A rotation about the point of regard — something the world does every time you turn your head — becomes a rigid slide along one cortical axis. A change of scale — something looming toward you — becomes a slide along the other [1]. The two transformations the visual world performs most often, both of them complicated curves on the retina, arrive at the cortex as the two simplest motions a sheet allows: shifts. The brain has factored its two most common problems into addition.

And the same sheet does more. Orientation preference — which direction of edge a cell responds to — is laid out across the identical surface in a repeating pinwheel pattern, woven through the grid without tearing it [3, 4, 5]. One continuous sheet carries position, scale, rotation, and orientation at once.

Advantage explains that; it does not explain this

The standard account of perception is a survival account: our perceptions are a useful compression, whatever kept the ancestors alive. The account is true, and it is silent on the only interesting question. Advantage explains that a map exists. It says nothing about why the map has this form. Nothing about survival demands a logarithm. Evolution could have wired the retinal sheet to the cortex a thousand tangled ways, and many of them would have worked well enough.

What selection actually found is a conformal map — the one coordinate change in all of mathematics that preserves local angles, that turns rotation and scaling into pure translation, that lets a single sheet carry several kinds of information without tearing. That is not a kludge that happened to survive. It is a piece of geometry that mathematicians name and study on its own merits, arrived at by wetware long before anyone wrote it down. The structure was there to be found, and selection found it the way water finds the shape of the valley.

The distinction this book needs

Here is the turn the chapter exists for. Perception is a simplification — but it is not a careless one. The brain's compression is dimensionally intelligent: it knows which relations to preserve (angles, shapes, the slide-ability of rotation) and which to let go, and it carries the ones it keeps in its wiring. The failure this book tracks is not simplification as such. The failure is simplifying without knowing what was dropped — and then mistaking the flat sheet for the territory. The visual cortex never makes that mistake. Only the flatland reader does.

So there are two kinds of flat reading. One is an instrument: a compression that knows its geometry, that chose its coordinates so the dropped dimensions do not matter for the work at hand. The other is a habit: a reading that does not know it dropped anything, and so cannot feel the missing dimension when it moves.

Why this becomes critical now

Until now the distinction has been instructive. In the quantum paradigm it becomes load-bearing. There, the reading is not a passive description of the system; the act of measurement is part of the event, and the choice of what to ask — which basis, which observable, which coarse-graining — decides what kind of answer exists to be found. A lens that does not know what it drops will not merely miss the extra dimensions; it will report their absence as a fact about the world, and build on the report. The next chapter is about exactly such reports: the placeholders, the vacuum mismatch, the hundred-and-twenty decades of dropped dimension named as a number. The cortex is the proof that biology already knows how to do this correctly — choose the sheet deliberately, keep the relations that carry, and never confuse the map's flatness with the territory's.

References (to be checked against the originals)

1. Schwartz, E. L. (1977). Spatial mapping in the primate sensory projection: analytic structure and relevance to perception. Biological Cybernetics 25, 181–194.
2. Schwartz, E. L. (1980). Computational anatomy and functional architecture of striate cortex: a spatial mapping approach to perceptual coding. Vision Research 20, 645–669.
3. Blasdel, G. G. & Salama, G. (1986). Voltage-sensitive dyes reveal a modular organization in monkey striate cortex. Nature 321, 579–585.
4. Bonhoeffer, T. & Grinvald, A. (1991). Iso-orientation domains in cat visual cortex are arranged in pinwheel-like patterns. Nature 353, 429–431.
5. Hubel, D. H. & Wiesel, T. N. (1974). Sequence regularity and geometry of orientation columns in the monkey striate cortex. Journal of Comparative Neurology 158, 267–293.

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