Two Proofs of Constant A/V Ratio Cell Growth and Division

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 = ….


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10 Responses to Two Proofs of Constant A/V Ratio Cell Growth and Division

  1. garyk30 says:

    Seems very logical and since there is math involved, I am not surprised to see no replies. This lack of math understanding is a shame.

    What has always seemed curious to me is not the ‘how’ of cell division; but, the ‘why’ of it all.

    From conception on, my cells divided at a tremondous rate and, by the age of about 18, I had gone from very-very small to 6′ 1″ and 200 pounds.

    Then my cells seem to have gone into a ‘maintain mode’ and for 50 years I have gained neither height nor weight.

    But, nail and hair cells(those that I still have) continue to grow as when I was younger.

    What happened and why?
    DNA instructions,I guess.

    Why do some strange cell divisions cause cancerous growths and some cause people to be 8 feet tall?

    Plants have the same growth patterns, are we related to plants?

    Cells could just get bigger and bigger, why must they divide at all?
    That,I think, is a function of gravity and other laws of physics.

    Too many questions to list, actually.

    • Frank Davis says:

      There are few comments because few people are reading it. Not many people are interested in cell growth and division.

      I’m impressed if you’ve actually followed the logic.

      Cells divide, in my opinion, because they maintain a constant A/V ratio, as I argue above.

      • garyk30 says:

        To have identical cells, the nucleus must divide also.

        Perhaps, the nucleus dividing is the driving force behind the cellular division?

        What makes the nucleus decide it is time to divide?

        • harleyrider1978 says:

          Takes wild guess,genetics.

          I havent got a clue but the math problem I can follow. I just dont understand its answer.

  2. Frank Davis says:

    The first proof is OK, but the second one sucks. And I have annotated it accordingly.

  3. Frank Davis says:

    I’ve now replaced the defective second proof with another proof, and adjusted the text accordingly.

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