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I would first ask the questions "why is the cerebrum of many animals divided into hemispheres". My answer (it's a guess) is that our deep ancestors would have had bilateral symmetry & a bilateral nervous system follows from that & anything resembling a brain would have been a more recent add-on which followed the developmental pattern. Both halves of the brain would need to talk to each other & this occurs at the base of split where the nerve fibres gather together.
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Why do the functions cross over ? I don't think the answer is known yet. It could be something to do with the evolution of the relationship of the head to the rest of the body. Somehow the brain got twisted through 180 degrees relative to the body over time. For example in some groups of flat fish the fish is born 'normal', but the skull flattens/twists in later life & one of the eyes migrates to lie on the top next to the other eye [this happens in the lifetime of the fish]
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A cephalopod counter example: Some modern octopus brains are as large as a mammals. Its neural architecture is completely different, which is not surprising given that the common ancestor of the vertebrates and the molluscs probably had no brain at all. The octopus nervous system is far more distributed than that of a vertebrate: as much as half of it is in the arms and suckers. And where a vertebrate has a single spinal cord, an octopus has dual nerve cords dotted with several pairs of ganglia—clumps of nerves capable of limited information processing. The octopus brain is in a DOUGHNUT shape that surrounds the mouth (beak). Imagine being an octopus ~ none of that hemisphere architecture nonsense that we are encumbered with !
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Michael [The above is from memory & after a few xmas 'straighteners' from the cocktail cabinet ~ corrections welcome]
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