*Originally posted by sonhouse*

**So I'm looking at this ordinary bath towel, maybe a meter long and half meter wide and the tufts are about a millimeter apart and a millimeter high on both sides the tweed (little in joke thereðŸ™‚, what would the total surface area be? I assume there would be an additional area from each thread that makes up the individual tufts, but don't know how much that represents. Any ideas what the total surface area is?**

Imagine that the tufts are in a square grid, and each is a cylinder with radius 0.5mm, then:

tufts per towel side = 1000 * 500

surface area of side of each tuft (assuming tufts are smooth) = 2 * pi * 0.0005 * 0.001

the area of the top of each tuft, plus the portion of the towel uncovered by tufts will be equal to the total area of the towel face untufted, so:

total area of tofty towel = untufty_area + tufts_per_side * n_sides * surface_area_of_a_tuft

total_area_of_tufty_towel = 1 + 1000 * 500 * 2 * 2 * pi * 0.0005 * 0.001

= 4.14 m^2

To see what a fractal towel is like, assume that each tuft is itself like a minature tufty towel, then the area of the tufts will increase by a factor of approximately 4.14

As we go down each level of fractality, because we increase the total area by a factor of more than 1, the series will not converge and an infinitely tufty towel will have an infinite area.