Some years ago, using basic theorems of hyperbolic trigonometry, I worked out a conformal representation of the hyperbolic plane which preserves one line – analogous to the Mercator projection of the sphere, a conformal map which preserves one great circle. (At that time, I had a mistaken notion of the Mercator rule, so I didn’t know until later how strong the analogy is.) Unfortunately I’ve utterly forgotten how I worked it out; and I need to try again; because, now that I’ve learned enough hyperbolic analytic geometry to try making some pictures, there appears to be a flaw in my formula.

Whenever I get settled in my big chair, Pillow (the junior cat) is all over me; but when I’m in bed he almost never comes within reach. So I was surprised when, on waking in the night, I found him sitting on the bed, in the spot most convenient to my hand, waiting to be tickled.

Having spent the previous night in hospital (where many tests found no cause for my chest pain), I thought of Oscar.

A Martian year is 668.6 Martian days; that’s 3.4 less than 24×28. I asked myself, how should the short months be arranged for best ‘balance’? I ran all combinations and this is it:

The three big dots represent the missing days in short months; the smaller dot represents the sometimes-missing day in the variable month; and the diamond, slightly left of center, is the center of gravity of the dots.

Then I thought, what if I were designing a calendar for a world where the number of days in a year is 5¼ off from a multiple of 12?

This may be of interest to only a few: Simplest Canonical Polyhedra of Each Symmetry Type (rotable in Java); in other words, canonical forms of the simplest topologies that allow the specified symmetries and none higher. Any convex polyhedron can be deformed into a “canonical” form whose edges are all tangent to a sphere; if a polyhedron is self-dual, it is canonical.

This collection could be considered a generalization of the set of all fair dice.

I reloaded MacOS, restored my home directory from backup, and was surprised to learn that I have 3e5 files. Most of the bulk is music, but that’s only 7e3 files. Is there a tool analogous to du that gives the number of files in each directory, rather than their aggregate size? —Later: When Apple Mail imported my Thunderbird archives, it made huge numbers of files, but I don’t know yet whether they’re enough to answer the question.

In other news, the medical jargon specimen of the week:

Infant is status post a negative rule out sepsis workup . . .

I guess that means sepsis was ruled out, rather than that it was not ruled out. The weird thing is that “rule out sepsis” is often listed as a diagnosis rather than a procedure.

If I were Pope Gregory’s advisor, I’d urge this: all months to have 30 days until the first (or last) of some month falls on a solstice or equinox; thereafter, alternate 30, 31, 30, 31, 30, 31, 30, 31, 30, 31, 30, 30¼.

Unrelated link: Questioning 7/4

For years I’ve wished to learn enough hyperbolic geometry to write a simple ray-tracer. Recently a post on comp.graphics.algorithms advised me to look up Euclidean and Non-Euclidean Geometry by Patrick J. Ryan; I was pleased to find that its subtitle is An Analytic Approach. Indeed it gives most of what I was looking for; though it relies so much on cross products that extending from two dimensions to three may be difficult.