*L*/

*T*of a paper to be folded

*n*times and thought I had found out how the formula established by Britney Gallivan, see equ. (1), for the minimal

*L*/

*T*value had been arrived at, namely by neglecting the term +3 in an expression as detailed in the paragraph containing equ. (*) and equ. (8). However, upon further reading on this topic I came across the answer given by MDH to a question on "Britney Gallivan's paper folding formulas" on this web site

http://math.stackexchange.com/questions/742104/britney-gallivans-paper-folding-formulas ,

and I realized that the difference between Gallivan's formula and my equ. (10)—both of which describe the case where all paper is used up in only the folds themselves—is due to the different ways the physics of bending a piece of thin material is described.

My description of the length of paper used in a fold is based on the length of the "neutral axis" in a piece of flat material that is bent with a bending radius of

*r*as illustrated in my first diagram—next to equ. (4)—for the first and the second folds, indicated by the red semicircles. This axis or line is called neutral because it is free of compressive or tensile stress and, therefore, does not change its length in the bending process. For a uniform material the neutral axis lies in the middle between the material's surfaces; thus the radius of the first fold is

*T*/2 and for the second fold it is 3

*T*/2, bringing the length of material used in those folds to π·

*T*/2 and π·3

*T*/2, respectively. The series of

*k*layers in a particular fold then is the sum of the first

*k*odd numbers times

*T*·π/2, or in terms of the length–to–thickness ratio

*L*/

*T*= π/2·(1 + 3 + 5 + .... + 2

*k*–1) = (π/2)·

*k*

^{2 }.

As the first equation in MDH's answer shows, the length of paper used in a fold is calculated from the full thickness

*T*of the material (MDH uses the symbol

*t*for thickness though) as π·

*T*in the first fold and 2π·

*T*for the second fold, and so on, so that for

*k*layers in a particular fold the total length of paper, relative to its thickness, used up there is

*L*/

*T*= π·(1 + 2 + 3 + ... +

*k*) = (π/2)·

*k*(

*k*+1). I assume that Britney Gallivan's approach was exactly this one because it leads to her formula without any further assumptions (or omissions of terms) beyond the first one, namely that the length of paper used in a fold is given by the length of the outer surface of a paper layer rather than by the length of the neutral axis.

The increase in radius from layer to layer in a fold is by the thickness of the paper in each case, the difference lies only in the expression used for the first term, and in the consequence this has for the summation of the individual contributions. It is also seen that Gallivan's formula predicts a slightly larger value than my equ. (10) does for the minimal paper length required for a certain number of folds.

My approach is also physically correct for the general case of describing the bending of a homogeneous piece of material although I cannot say yet whether a piece of thin paper behaves like that; for the first several folds when the paper gets creased, a "neutral axis" may be ill defined but as the stack gets larger the use of the outer paper surface for calculating the length of paper in consumed in each layer of a fold appears to be an overestimate. So the more adequate formula for the minimal

*L*/

*T*ratio needed for a certain number

*n*of foldings is my equ. (10).