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Despite the intensive research all around the world, as a phenomenon, differentiation is badly understood. For a phyllotaxis model maker, it is possible to separate the problem of pattern formation (phyllotaxis) from differentiation.
The stem tip, or apex, may have a dome-like shape, and that is not a pronounced shape. As long as this is the case, and as long as the primordia arising on it appear sphere-shaped, we may safely regard the pattern building as a geometric problem. At a later point in time after it's arising, each primordium starts to differentiate. Let the primordial differentiation start slowly, then development is obvious, and at last the plant organ has been shaped, so differentiation ends.
The red S-curve shows the differentiation of a single primordium in time. Click on it.