Re Pi=4 (or
making a square room a round one... No, because you'd still not have constant radius. If Pi!=~3.14 then you live in obviously non-Euclidean space where orbiting a point takes longer (or shorter) than it should. (Although in a spherically curved geometry, the effective value of Pi varies according to radius, and indeed would end up as zero when r==the UniversalPi*UniversalR... Ditto in a hyberpolic/poincaré geometry then scale shows differences, with it being locally closer to Pi but on larger scale (and, in some geometries, further from the privileged centre) Pi enlarging. Part of the reason why we think sending up satellites to do precise laser-interferometry upon each other at long distances could show what curvature of space we might have in the Universe, once we account for local distortions and any gravitational waves that wander along.)
Now talk to me about rooms with Rotational Symmetry of 0.5
(Sometimes I set up Doom (v1) maps with such rooms, back in the day. All you need is a central pillar to hid the discontinuity, however thin, and it makes for fun deathmatches!)
The doctor built his own screwdriver, so their was likely only ever one sonic screwdriver, more likely so after the time lords all died. The Daleks also constantly adapt [what's left of them anyways] much like the doctor changes his screwdriver, they would have prepared for their only last enemies [the Doctor and stairs].
I do believe he went off to get a new one, at at least one point. Although undoubtedly heavy personal customisation occurred for each iteration.