If a cell maintains a constant ratio of surface area to volume as it grows, there are a variety of ways in which it can grow and divide.
For a cell, the advantage of maintaining a constant A/V ratio is that it only needs to produce surface membrane and internal material in a constant ratio, equal to the initial A/V ratio, throughout the cell cycle. It is probably also the Least Action variant of cell growth and division, since the cell naturally and effortlessly separates in two as a function of its limiting geometry.
Most likely, normal cell growth consists of the two hemispheres of an initially spherical cell expanding to become enlarged segments of a sphere, joined by a fold (or notch), until both segments have become two spheres with the same radius as the original cell.
Here’s a mathematical proof relating segment height h to notch width b through the range h=R to h=2R.
In a second type of growth, which seems to typify some cancer cells, the notch widens to form two cones which connect the two hemispheres of the growing cell. The resulting cells, although they have the same A/V ratio as the parent cell, generally do not have the same volume.
Here’s a mathematical proof for the growth and division of one cube to form two cubes the same size as the initial cube.
When the notch width is greater than zero, these cells form truncated double cones or pyramids between their two halves, which results in a greater extension along the growth axis. Many cancer cells appear to have such double cones, and what would appear to make ‘cancer’ cells dangerous is that, as their cones lengthen, they are able to push past adjacent cells, and spread into other tissues.
Why some cells grow ‘normally’, and others take on the ‘cancerous’ form, is not clear. But in general, since it will require less physical work for normal cells to grow than ‘cancer’ cells, it should be expected that normal cells will predominate.
It is highly unlikely that the ‘cancerous’ form of growth and division is caused by smoking tobacco.
Note: This is a modified version of the original post, which contained an error in one proof. Edited 10 Dec 2012 to remove surplus bracket from cubical proof at c = ….