Understanding the relationship between tightness factor and areal density.
Tightness factor (TF) is defined as:
\[
TF = \frac{T^{1/2}}{\ell}
\]
Areal density \( A \) of the fabric is proportional to:
\[
A \propto \frac{T}{\ell^2}
\]
Now, the ratio between TF and areal density is:
\[
\frac{TF}{A} \propto \frac{\frac{T^{1/2}}{\ell}}{\frac{T}{\ell^2}} = \frac{T^{1/2} \ell^2}{T \ell} = \frac{\ell}{\sqrt{T}}
\]
Thus, the ratio is proportional to \( \frac{1}{\sqrt{T}} \) when rearranged.