  We provide three new examples of twisted Hilbert spaces by considering
properties that are "close" to Hilbert. We denote them $Z(\mathcal J)$,
$Z(\mathcal S^2)$ and $Z(\mathcal T_s^2)$. The first space is asymptotically
Hilbertian but not weak Hilbert. On the opposite side, $Z(\mathcal S^2)$ and
$Z(\mathcal T_s^2)$ are not asymptotically Hilbertian. Moreover, the space
$Z(\mathcal T_s^2)$ is a HAPpy space and the technique to prove it gives a
"twisted" version of a theorem of Johnson and Szankowski (Ann. of Math.
176:1987--2001, 2012). This is, we can construct a nontrivial twisted Hilbert
space such that the isomorphism constant from its $n$-dimensional subspaces to
$\ell_2^n$ grows to infinity as slowly as we wish when $n\to \infty$.
