Babylonian Trigonometry again?

Factual Questions
41

Decimal calculations were well known and taught in schools for centuries, but without calculators people had to do a lot of mental arithmetic.

A pound, being 240 pence, was divisible by
2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 60, 80, 120

It’s not often noted that a guinea, equal to £1 + 1 shilling = 21s = 252d, was also very useful. A guinea was evenly divisible by
2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126

Since a week has 7 days, it was particularly useful for practical purposes that a guinea was divisible by 7, 14, 21, 28.

If you needed to do a calculation in pounds that involved division by 7 or 9 you could do it in guineas, and then make a small approximate deduction of 5% - or a shilling per pound.

e.g. What’s £17 divided by 7? - calculate mentally.
17 guineas divided by 7 is 51 shillings (3 shillings per pound) = £2/11/-
Subtract 1 shilling per pound = 2s 6d approx.
You get the approximate answer of £2/8/6 (2 pounds, eight shillings and sixpence)
The actual answer is £2/8/6.86 - close enough!

42

Aside:

With the other various current threads on NK missiles and SpaceX and such, I keep seeing this thread’s title as “Babylonian Telemetry Again”. Which is a real double-take. :slight_smile:

43

Maps are printed as rectangles and we refer to the left, right, top and bottom sides. With their six-part circle, I wonder if the Babylonians had six direction names, and even if they drew maps as hexagons.

IIRC, a hexagon grid (each point has six equally near neighbors) is used in some military-type board games, for ecology simulations, and perhaps even for some machine vision tasks. (What grid is used in weather simulations? The obvious and ordinary 3D whats-it-called xyz lattice?) If alternate grids have better properties, why are they seldom used? The lack of six direction labels may be a symptom of a general aversion to hexagonal grids. :eek:

I think your (1)-(3) are details which can be varied harmlessly for now. I just want to come up with the six labels. I’d settle for three labels (and a prefix).

You’re reading much more into my alleged musings than was ever there. I just wonder what six good labels would be. I’m ready to give up and would be happy to adopt the Babylonian names if any of the Board’s experts in ancient languages can work them out!

If not, I’ll settle for a direction trio, and a prefix to make it a set of six. But I don’t even know any good trios. :o

There are so many binary word pairs (opposites). But what are some good trio words besides red/green/blue? I don’t want color — I want direction. Top/middle/bottom? — No, I want the three directions to feel like full and equal partners.

44

I think that LSLGuy was leading up to just assigning numbers to the sextants, similarly to how the quadrants are numbered in Cartesian coordinates. But I don’t think that has the sort of symmetry you’re looking for.

As to why it’s most common to use two binary pairs for direction, it might be because we’re talking about a two-dimensional plane. More unit vectors are redundant.

45

In up/down, left/right, north/south, or east/west we use items that are thought of as opposites. And we use two pairs of them. But in no sense is up the opposite of right or east the opposite of south. They’re simply unrelated.

North/south and east/west seem like a coherent set of four items. But they’re not; or at least aren’t etymologically. They’re two unrelated pairs that are so often used in proximity that we’ve gotten used to thinking of them as a foursome.

So by analogy what we’re looking for are 3 sets of opposite pair concepts. Not two sets of triplets.

So for a deliberately terrible example we could use labels of male/female, coniferous/deciduous, and transparent/opaque. Each of those pairs is a binary concept. Also each pair has no influence on the other two pairs. They’re linguistically “orthogonal” in a hand-wavy way. Saying that something is opaque tells us nothing of its male/female-ness; etc.
Skipping over my criteria 1-3, we’d use the labels in order like male, coniferous, transparent, female, deciduous, opaque.

There’s no sense in which the ordinary quadrant words stand for their ideas better than any other random noise humans could agree to make.

I think ultimately you’re sorta stuck right there. Any word that already carries any connotation of directionality will be damaged goods, carrying inappropriate quadrant-centric baggage with it.

So you can just make up three new words, preferable with some mnemonic natural sequence, then perhaps add an “anti-” on the front for their opposites. What are the names of the first 3 letters in some written but now dead alphabet?

46

Except that, here, we don’t want our terms to be orthogonal. If you go one step coniferous and then one step opaque, you don’t end up one step male. And the only context I know of with that particular sort of orthogonality, beyond directions in 60º increments, is colors (like for quarks), which you’ve already rejected.

47

The problem is that you don’t need three coordinates to uniquely identify a point in a hexagonal grid system. You just need two, same as for a square grid system.

This image should explain what I mean: http://devmag.org.za/blog/wp-content/uploads/2013/08/screen_122.png

Each hex is uniquely identified with only two numbers, the same as would be the case for a square grid.

If we really need six unique words for hexagonal coordinates, I propose six flavors: up, down, strange, charm, top, and bottom.

ETA: Curse you Chronos!

48

Yes, you can specify a planar position, even on a hexagon grid, with two coordinates — and that is clearly better than using three if the only goal is to minimize number of coordinates. In practice, there are elegant approaches to processing on hexagon grids in which three coordinates are (redundantly) employed. I don’t think I need to point to papers in the literature of hexagon-grid processing, nor more generally to support for the thesis that Economy is not always a virtue; nor is Flexibility through redundancy always a vice. :slight_smile:

But all this is besides my point. I’m just looking for six (or three*) words to use as labels. Something somehow more intuitive than, say, “up, down, strange, charm, top, and bottom.” (I could as well use “Mon, Tue, Wed, Thu, Fri, Sat.”)

    • Three directions with an anti- prefix.
49

How about clock named directions? That’s obviously 12 directions rather than 6, but you could restrict yourself to even numbers. Bandits at 8 o’clock!

50

Although it’s tangential to the purpose of my subthread here, I’ll give examples of this. Start by imposing x, y, z axes in the plane at 120° angles to each other. Yes, the z axis is redundant, but look at what is gained:
[ul]
[li] Unlike rectangular transforms, in which F[sub]x[/sub] and F[sub]y[/sub] might be basic filters equivalent after rotation, symmetric systems on the hexagon grid will have three rotated forms of basic filters: F[sub]x[/sub], F[sub]y[/sub], F[sub]z[/sub].[/li][li] Symmetric separable transforms on the plane can be built from a 1-D transform F by convolving three versions of F: F[sup]*[/sup] = F[sub]x[/sub]•F[sub]y[/sub]•F[sub]z[/sub].[/li](This is akin to a 3-D transform but operates on the 2-D (hex-gridded) plane!)
[li] Distance measurement is awkward with the axes at 120° angles unless you use all three axes. Here are the six unit vectors in 3- and 2-coordinate versions:[/li]



                (+1, 0, 0)      --> (+1,  0)
                (0, 0, -1)      --> (+1, +1)
                (0, +1, 0)      --> ( 0, +1)
                (-1, 0, 0)      --> (-1,  0)
                (0, 0, +1)      --> (-1, -1)
                (0, -1, 0)      --> ( 0, -1)

(Isn’t the 2-coordinate form displeasing?)
[/ul]