How to make an object build itself? (sort of)

I want to achieve an effect where an object’s faces sort of fold out as the object sort of builds up, and slowly gets bigger and bigger. I’m not really sure what to call this effect, but I was hoping the blender community could help me with it.

I have found a vague example of this, in the first few seconds. I want to do something like this, but where the folding faces build an object.

I think it will be more simple to do the same effect using different objects, keyframing their visibility, and rotating, scaling as needed. You should be able to make is look like it’s one object.