This is a grizzly image but can you calculate the height, area or extent in space that would be occupied if you could connect the first umbilical cords of the first humans, starting from their mother, if there was only one (?), and connect a given size of human to them another level down and then continue through all the generations to the present day? It's tempting to say that the area of the ground floor, i.e. our generation would be the space needed for 6 billion people to stand, which I believe is approx. The Isle of Wight. That doesn't seem enough somehow. None of the males would have any connections coming down in this layout. They would of course have umbilical cords upwards but obviously do not have any downwards.*
Of course the umbilical cord is an aid to visualising the problem, it's not strictly necessary in fact it makes the problem impossible if actual umbilical cord length had to be used, so instead of that we would probably use an average human height, weighted for the change in that over time or just averaged out or anyway taken into account. So instead of joining the people physically, you could just stand them on levels, not equating to generations but equating the the number of people generated by the initial number of people. So from one we maybe get two, or five or whatever it was. Then from those five we get 20 or whatever it was, etc. They're not generations because the ages are all combined and put on the same level. But that should work, should it not or is there a fatal flaw in trying to do that?
So you get something like:
Of course the men in this only have upward connections (the umbilical principle) and no downward ones*. In a way, we could omit the men from the levels and then at the bottom level, just bring all the present day men back in to give us a base approximately equal to The Isle of Wight. Then if we knew the number of levels (but is that even possible, or is there a glitch there?) and an average height, we could work out the height of the whole cone, and then assume it must be equal numbers of men and women on each level, approx, so just connect lines from the base up to the top.
Another point: if we start working back from the present level (everyone alive today) ... ah but here's the glitch: You can't put everyone alive today on the same level, because some of them are grandparents, parents or children of others alive today. That is the glitch.
Why then, if we start from the top does it feel like we can force it into levels? How would it end up to the present stage. Oh I know, it's because all of those alive today is not one of the levels. The levels are different things that don't correspond to temporal locations, they are abstract concepts. The people alive today span several levels, not just one.
If we forget the men and say they are just ballast, then it is women who define generations. Every woman alive today had a mother who had a mother and so on. It's a many-to-one relationship going backwards. Everyone has precisely one mother. Some women have daughters but some don't; not everyone links both ways in this. If we could say that there are more girls with each generation then it would be clear that the population decreases going backwards. This is known or anyway believed to be true. If every woman had precisely one daughter then there would still only be one woman alive plus her mother and grandmother perhaps, maybe even great grandmother so at most four or five women in the whole world.
It's simpler to think of the question as how many generations there have been since the first humans. We could answer if we knew the average number of girls born to each mother. We could extrapolate from the present number of females by using the average increase per generation to work out the total number of generations to date.
We could define the present generation as the set of all females (and males but they are irrelevant) who have not yet had any children. That gives us an unentangled set because they are not the parents of anyone alive at present. Of course the set is changing from minute to minute. Anyway we take a cutoff (no pun intended). Now we can neatly take the mothers of each of these (the present generation) and trace back to an imaginary "level" (not a real time) before any of those mothers had had any children. Repeat and that gives us our generation counter or visualiser.
* Another truism: Everyone has precisely one father and one mother (no clones as yet). So even though there is no umbilical cord to connect a child to its father, the scenario is the same. Everything that could be calculated concerning mothers could be calculated concerning fathers. It is only when combining the two that some tangling occurs. However for a complete model, we would need everyone to have two umbilical cords, one to a father and one to a mother.
But you cannot separate all the progeny "connected by two strings" of the previous level into one next level because some of them might have a relationship upwards and be ancestors of some of the progeny on their own level. That is why it complicates it bringing in two parents. This does not arise in following only the matriarchal or only the patriarchal links.
ANOTHER THOUGHT: We could combine the two models, the male and female "trees" with their perfectly untangled levels but the levels do not map across exactly, so they would look like two cones with their points leaning together. They don't fit but if the male one has less levels the space between them must be stretched so both the models reach from the first humans to the present day, giving us a different average "generation length" for males and females. Model F + model M = everyone who ever lived.
We may tend to assume that population has been increasing steadily but it might have gone down due to problems at various times, for example during the bubonic plague. The amount by which it increases or decreases could vary over time so there must be wavy edges to the pyramid but we know that over the long term it has been increasing.
If archaeology can tell us when the first humans appeared, and we combine with where we are now, which we know about, we could work out a pyramid shape. I think it would be a very tall narrow pyramid. I'm only guessing but it might reach from the Isle of Wight as the base to the Moon, maybe.
I remember reading an article in Atlantic Monthly by a leading mathematician specialising in the maths of heredity, which he said was exceedingly complicated. It turns out that everyone alive today has a common ancestor about 10,000 years ago. Anyone who was reasonably prolific, like Nefertiti or Genghis Khan for example, is most likely ancestor to everyone alive today. This effect leads to the seemingly marvellous result that everyone who researches their ancestry discovers that they are related to some famous forebear. Also he found that 3,000 years from now, everyone alive today will either have no descendants at all or will be ancestor to everyone then alive. (Ref: Joseph Chang, Yale University quoted by Steve Olsen in "The Royal We" Atlantic May 2002 Monthly http://www.theatlantic.com/magazine/archive/2002/05/olson/2497/)