Three-dimensional normal maps XYZ + 3 Primary Colours RGB ... coincidence?

In lecture 77 on terrain textures, Rick mentions that normal maps work by assigning (red green blue) => (x, z, y) vector directions for light-mapping purposes. But my question is a physics/optics/perception question: is this a coincidence that there are exactly THREE primary colours we use to define all other wavelengths in the human-visible specturum as well as THREE spacial dimentions that we perceive, or do our eyes/brains view the world in RGB-colour-mode for some kind of similar biology reason as we view the world in 3D.

Or let me phrase my question like this: most animals are colour blind, even many new world monkeys, whereas most great apes (like us) and old world monkeys have trichromatic (literally “three colors”) vision.

This raises the question: why 3? Could this be related to how great apes tend to move in three dimentions, running on the XY plane of the ground and climbing trees along the X; whereas other mammals such as dogs and hyenas are dichromatic (they only see in two colours, which makes sense with my theory because they do most of their perceptual work on the ground-level-only).

Fish and birds are often quadchromaic: four pigments are used for vision. This makes sense because they need the 3 for the 3D world, plus a forth to quickly see heat from a distance (for birds to guess where to hunt, or how far it’s safe to fly) or to see ultra-violet colour (useful for fish to discern the depth of the water). Humans don’t really have much need for this because we have advanced heat-sensors in our brains and blood, and can usually tell the depth of where we are by looking around the trees.

Just an idea…? Any thoughts?

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