Clackson scroll formula

The Clackson scroll formula [S.G. Clackson, "The Trinity Church Screen", SCAT Report 1981] ["MSC Craft Based Training - Forging and Hand Skills] , :$l\; =\; pi\; cdot\; s\; cdot\; n^2$is used in blacksmithing for estimating the length $l$ of stock required to produce a scroll of $n$ turns with a spacing $s$ between the turns.

Derivation

The formula can be obtained by approximating the spiral by a series of circles of increasing radius. Namely, as the spiral expands it makes "n" complete turns around the center: in the first turn, it expands from a point (radius zero) to a distance "s" from the center; in the second turn, it goes from distance "s" to 2"s"; and so on, where in the "i"^th turn it goes from distance ("i" − 1)"s" to "is". Without more information about the spiral, it is impossible to say exactly what length of material is needed to accomplish this, but if the length required in the "i"^th turn is "l"_"i", then as long as the spiral doesn't double back, "l"_"i" is between the circumferences of the smaller and larger circles::$2pi(i\; -\; 1)s\; leq\; l\_i\; leq\; 2pi\; is\; quad\; ext\{for\; \}\; i\; =\; 1,\; 2,\; dots,\; n.\; ,$Since "l" = "l"₁ + "l"₂ + ... + "l"_"n", we add these formulas and obtain an upper bound for "l"::$l\; =\; sum\_\{i\; =\; 1\}^n\; l\_i\; leq\; sum\_\{i\; =\; 1\}^n\; 2pi\; is\; =\; 2pi\; s\; sum\_\{i\; =\; 1\}^n\; i\; =\; 2pi\; s\; frac\{n(n\; +\; 1)\}\{2\}\; =\; pi\; s\; n^2\; left(1\; +\; frac\{1\}\{n\}\; ight),$and a lower bound from the same computation::$l\; geq\; sum\_\{i\; =\; 1\}^n\; 2pi\; (i\; -\; 1)s\; =\; pi\; s\; n^2\; left(1\; -\; frac\{1\}\{n\}\; ight).,$That is,:$pi\; s\; n^2\; left(1\; -\; frac\{1\}\{n\}\; ight)\; leq\; l\; leq\; pi\; s\; n^2\; left(1\; +\; frac\{1\}\{n\}\; ight).,$Since 1/"n" is relatively small, if we ignore that term then both bounds agree::$l\; approx\; pi\; s\; n^2,\; ,$which is the Clackson scroll formula.

References