It bothers me that Rocky and Bramble, though not obviously neurotic, don’t share the joyful manner of their predecessor Pillow (2005–2010?), who gave the impression of expecting fun around every corner. What am I doing wrong?
It now occurs to me: at least part of the difference must be that, as Rocky and Bramble never go outside, I never see them trotting.
My old calculus book gives a formula for the curvature of a parametric arc in the plane — that is, an arc defined by two functions (x(t),y(t)) of one variable. For thirty years I didn’t think about the derivation of that formula. Just now it hit me (and I did the algebra to confirm) that, in terms of the complex plane (z=x+iy), the curvature formula is equivalent to
Im(z″/z′) / |z′|
This should improve my cubic approximations to transcendental curves.
To throw for maximum distance (on an infinite plane in a vacuum), you aim at 45° elevation; in other words, split kinetic energy evenly between vertical and lateral velocity. (I dimly remember having proven that, but am not awake enough to do it again now.)
It follows that the kinetic energy of a perfect vertical throw is twice the vertical kinetic energy of a perfect distance throw.
The arc of the latter is a parabola, of course, and its height is easily found to be a quarter of its length. Double that, because with double the energy you get double the altitude; and the result is consistent with Munroe’s estimate.
What about air resistance? The vertical throw has a moment of zero speed, but the distance throw’s minimum speed is 1/√2 the maximum; so it seems to me that the vertical throw suffers less air resistance (not even considering the thinner air up there), and therefore the altitude estimate given above is low.
A new father asked:
Are you charmed or annoyed to learn that your 3d geometric object is making a great toy for [baby, 8 mo.]?
I replied:
I have cats, so I’m not a bit surprised!
On a related note, if I had any knowledge of Web programming I’d make something analogous to this.
If you don’t remember George Carlin’s speech that contains the phrase “six sins for one feel”, go ahead and look it up.
When I first heard it, the phrase “feel her up” was new to me, but its meaning was obvious. Thirty-some years later, I reckon I must have known some synonymous expression, but cannot recall what it was!
I had assumed that Pixar did not use ray-tracing because it could not provide certain desired lighting effects. Now Dad tells me that Monsters University is Pixar’s first ray-traced feature, which implies that the speed wasn’t available until now.
I’m re-reading To Your Scattered Bodies Go and, of course, pondering the arrangements.
The premise is that all humans who ever died (for some convenient definitions of ‘human’ and ‘ever’) are simultaneously resurrected (for purposes unknown to them) on an artificial planet whose surface is one long and twisty river valley. In each neighborhood along the river there’s initially a majority of people from one region (spatial and temporal), but also a large minority of random others. Why?
How would I arrange them? Perhaps by date of birth. Arranging by date of death would be more likely to bring enemies together.
Alternately, having fallen in love with topological coordinates, I’d use a kinship graph: each person is directly linked to parents and children. I’d be interested to see the overall shape of this graph, as embedded in ℝⁿ. If enough generations are involved, the longest axis of this embedding is close to the birthdate axis: your parents might have no common ancestors within a thousand years, but they can’t have been born a thousand years apart. But what are the other axes like?
Turning away from mathematical nerdery now — One minor character says he’s especially pleased to regain the leg he lost in a road accident at age 50. That’s consistent with the apparent policy of restoring adults to age ~25 (and children to their age at death). But what is the Revivers’ policy on birth defects, genetic and otherwise? If you grow up with a damaged brain, can your mind be installed in a normal brain? If you live to adulthood without legs and are then revived in an adult body with legs, how easily can you learn to use them? How many limbs do Brittany and Abigail Hensel get?
2022 Nov 04: I made a more stable page collecting thoughts from this thread.