  We investigate fractal aspects of elliptical polynomial spirals; that is,
planar spirals with differing polynomial rates of decay in the two axis
directions. We give a full dimensional analysis of these spirals, computing
explicitly their intermediate, box-counting and Assouad-type dimensions. An
exciting feature is that these spirals exhibit two phase transitions within the
Assouad spectrum, the first natural class of fractals known to have this
property. We go on to use this dimensional information to obtain bounds for the
H\"older regularity of maps that can deform one spiral into another,
generalising the 'winding problem' of when spirals are bi-Lipschitz equivalent
to a line segment. A novel feature is the use of fractional Brownian motion and
dimension profiles to bound the H\"older exponents.
