programdemos/manifold-tour/manifold-tour.xtl

a ᵘp̲airs b: a and b side by side, one (a, b) row per item (the rows of a map). A general pairing belongs in X_eTaL-libraries: a workaround here until it is asked for (docs/plan.md, G3b).

ᵘr̲ound m: m + 1 angles evenly round a full turn, 0 to 2 pi, the first repeated last: a closed wire line.

n ᵘe̲ach v: every item of v repeated n times, in v's order.

n ᵘt̲imes v: v whole, n times over.

ᵘl̲ength M: the length of the wire line through the rows of M, segment by segment (measured along the wire, not the curve).

p ᵘs̲traight q: the straight-line distance, through whatever surrounds the space.

ᵘt̲wo M: for the two points (rows) of M on the sphere or the circle, the straight line through it and the distance along it.

ᵘl̲onLat P: points of the sphere back onto its map, (longitude, latitude) in degrees: where a drawn arc runs on the flat view.

Two points a tenth of a radian apart, and two 2.5 radians apart: the chord through the disk and the arc along the circle, for each.

The map's wire lines: five parallels (latitudes -60 to 60 degrees, 48 segments round) and twelve half meridians (every 30 degrees of longitude, pole to pole in 24 segments), as (longitude, latitude).

Three points by longitude and latitude in degrees: p, q near it (6 degrees east, 3 north), r far from it; the chord through the sphere and the great circle along it, p to q and p to r; the arcs.

The chord from p to q worked in full: the differences coordinate by coordinate, their squares, the sum (the chord is its square root); and the length of p + q, from which the great circle is 2 atan2(|p - q|, |p + q|).

Six rings round the hole (v every 60 degrees, 48 segments) and twelve round the tube (u every 30 degrees, 24 segments), as (u, v).

The outer ring (v = 0) and the inner ring (v = 180 degrees): the same line of the map, measured along the wire.

The sheet: t from 1.5 pi to 4.5 pi (a turn and a half), h from -6 to 6. Five lines along the roll (60 segments) and thirteen across it (straight, 2 points), as (t, h); and the map they unroll to, (the length along the sheet, h).

Three points of the sheet: a, b a little farther round, c one full turn farther round at the same height, so that it sits right above a's turn: through the air and along the sheet, a to b and a to c.

The rim (not part of the disk: points of the disk never reach it), and the disk's ruler laid along a diameter: the points 0, 1, 2, 3, 4 and 5 apart from the middle by the disk's distance.

Two points near the rim, 20 degrees apart round it (at the top, off the ruler), and the same two drawn in toward the middle: on paper and by the disk's distance, for each; and the geodesics between them.

Three more geodesics, to see their shape: a diameter, and two arcs.

Each space's intrinsic dimension (how many coordinates its map has: the circle's map is a list of angles, the others' are pairs) and its ambient dimension (how many coordinates the space it is drawn in has), read off the arrays' shapes: one row per space, the circle, the sphere, the torus, the roll, the disk.