It is not uncommon to find the following diagram in a biology textbook:

What is depicted is cell division, where a large cell divides into two smaller cells which each subsequently divide into two smaller cells. The result are four cells each containing 1/4 of the oringinal cell's material.

The interesting aspect of this diagram is that when the mother cell divides into two parts, the subsequent cells are shown to have half the radius of the large cell. Similarly, when these two intermediate cells divide, the radius of the daughter cells is half that of the intermediate cells (and 1/4 of the diamter of the mother cell).

A quick look at the diagram reveals a peculiar fact. Imagine fitting the four daughter cells into the mother cell - a relatively simple task. In fact, there apears to be lots of room to spare. However, these four daughter cells are each supposed to contain 1/4 of the matter (equals area) of the mother cell. Thus, it should be the case that the four daughter cells should completely fill the mother cell. It seems there is some problem with halving the radius to form the daughter cells!

Assuming that the radius of the mother cell is 1, we find the following area for the mother and a daughter cell:

And so it turns out that each daughter cell is 1/16 NOT 1/4 of the mother cell and subsequently the four daughter cells are made of only 1/4 of the material. To the question arises of how large the radius of the intermediate cells (and also the daughter cells) must be in order the represent an accurate area division. In short, we must solve the following equation:

And we find that which is approximately 0.7071. Compare that to the original diagram where the radius of the intermediate cell was 0.5. This means that the daughter cells would have a final radius of

How strange! The intermediate cells in our original drawing are actually the correct 2-dimensional representation of the daughter cells. Thus a more correct picture should look something like this.

In reality, a cell is not a 2-dimensional object. That is, our renovated digram while being appropriate for a textbook is still inadequate.

A brief examination reveals that extending the original (incorrect) diagram to three dimension only worsens the problem. Check it out below.

So, it would take 64 daughter cells (not 4) to fill the mother cell. Yikes! So, what would the radius of the 3-dimensional intermediate (and daughter) cells need to be in order to produce an accurate diagram? We'll need to solve the following equation:

Thus the radius of the intermediate cell is approximately 0.7937. Very interesting that at this higher dimension, the radius of the intermediate cell is greater than in 2 dimensions (0.7937 compared to 0.7071). In the end we find that the radius of the daughter cells should be approximately 0.6300 (or 0.7937^2). Very different from 0.0156 (or 1/64)!