Fat Shellable Spheres

The fatness of a 4-polytope or 3-sphere is defined as $(f_1+f_2-20)\mathrm{/}(f_0+f_3-10)$. We construct arbitrarily fat, strongly regular CW 3-spheres that are both shellable and dual shellable. These spheres have $f$-vectors $(\mathrm{\Theta }(n),\mathrm{\Theta }(n\alpha (n)),\mathrm{\Theta }(n\alpha (n)),\mathrm{\Theta }(n))$, where $\alpha$ is the inverse Ackermann function.