Also known as Archimedes's constant and Ludolph's number, π is the ratio of a circle's circumference to its diameter, in Euclidean geometry, or the ratio of a circle's area to the area of a square whose side is the radius. π is irrational, meaning it can't be written as the ratio of two integers, and transcendental, meaning there's no polynomial with rational coefficients of which π is a root. The latter means that π is not constructible: because the coordinates of all points that can be constructed with a compass are constructible numbers, it is impossible to square the circle: it is impossible to construct with a compass a square whose area is equal to the area of a given circle. But the best definition I've ever seen is this: a circle with a diameter of one has a circumference of π. Meaning that it takes exactly π rotations to end up where you started, working your way around the wheel. Irrational and transcendent, but there you have it: what should be easy, matching a static square -- four hearts, two sets of twins -- to a circular journey is impossible, and yet you end up where you started nonetheless, and there's never been a circle that wasn't stronger than this fact, thanks to π. But what you can't construct can still be deconstructed, even if it takes thirteen episodes: On the other side of Hell, we burn off what doesn't work, and that amounts to 4 minus π: just a smidge less than one whole heart.
Cybermen bring Crane before Lumic, calling him "unprogrammed." Lumic whines that he thought Crane was "one of the faithful," and Crane lies: "My EarPod must've malfunctioned. For which I apologize." Lumic stares at him, one corner of the set dressing sticking out between his teeth. Crane steps closer: "I'd like to request an upgrade, sir. I've seen the future, and it's copyright Cybus Industries." Nice! "Sign me up." Lumic reapprises him: "A willing volunteer." I love how the Doctor came up with that same defense, and it's working even on Lumic. "You've known me a long time, sir," Crane tells him. "Quite a team, you and me." I don't know if it's pain, or craziness, or more bizarre acting, but Lumic's face twists in a mockery of a smile. "I've been with you all the way, but believe me, sir. Knowing you so well, I know exactly what to do." Crane lunges forward and rips out Lumic's breathing tubes, against weak protest. He smashes Lumic's life-support, and parts of Lumic's wonderful chair explode. Lumic begs the Cybermen for aid, and they jump to it, smacking Crane across the room. "Die, you!" says Crane, but they don't. A Cyberman snaps his neck and tosses him away. And Lumic continues to beg for help. And the genie outside the bottle says, "You are in pain. We can remove pain forever." Temptation as usual, for this show: remove pain, stop feeling, go to sleep, change your universe for the worse. Lumic protests that he's not ready, wheezing and gasping, but they aren't designed for that kind of thinking: "We will give you immortality." And Lumic, at the site of his own murder, screams weakly at them: "I told you, I will upgrade only with my last breath!" And the Cyberman solves the equation for π: "Then breathe no more." All the way back around to inhuman again. The Cybermen wheel Lumic out, his shouts growing weaker.