Fractals, similarity, intermediate asymptotics

Classical mathematical physics dealt only with perfectly smooth constructs; but today one works with curves having no tangent at any point, surfaces having no area, and other important objects once thought useful merely for tricky exam questions. Several aspects of these rough entities are described, such as the fractional-dimension concept, intermediate asymptotics, and the differential calculus of functions lacking derivatives, based on the Holder index. Applications are possible in many branches of physics, including fluid dynamics, general relativity, and cosmology.