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Copyright © 2007 jsd

## 1  Secchi Disk

Figure 1 shows the pattern used on a Secchi disk. Such disks have been used since 1865 to measure the transparency of water (reference 1).

Figure 1: Secchi Disk on Gray Background

The disk has two black quadrants and two white quadrants.

A secondary usage of Secchi patterns (and variants thereof) is as general-purpose fids i.e. fiducial points i.e. reference marks on apparatus and on drawings of apparatus. They have been used as such for many decades. (Lately people have come to associate them with crash-test vehicles and crash-test dummies, but that is certainly not their only use, primary use, or original use.)

The pattern’s usefulness as a fid depends in part on the following geometric fact:

The boundary between one region and another has zero width.

In this case, we are talking about the boundary lines between a light sector and a dark sector. The Secchi symbol has two such zero-width lines (or four, depending on how you count), and where they meet defines a zero-size point.

As a consequence, if you zoom in on the center of the fid, what you see is the same, regardless of scale: you see two white quadrants and two black quadrants. This scale-invariance would not be observed it you used a dot or a cross; if you zoom in too closely on a dot all you see is the interior of the dot, and you get lost in the interior. Conversely if you make the dot too small, it could be invisible at lesser magnifications.

If you want to experiment with scaling, here are Secchi patterns in various easily-scalable formats:

## 2  Endless Boundaries

Another interesting geometric property is that the boundaries constructed in this way are necessarily endless.

Here’s what we mean by that: Imagine that the degree of brightness represents height, so that black=0, gray=½, and white=1. Then there is a half-height step up going from black to gray, and another half-height step up going from gray to white. There is a full-height step up going from black to white.

In figure 1, you can imagine that the actual boundaries can be expressed as the sum of two contributions, namely an endless inward-pointing half-strength boundary surrounding each white sector, plus an endless outward-pointing half-strength boundary surrounding each black sector. Along the black/white boundaries, these two contributions add up to make a full-strength step.

This illustrates an important theorem from differential topology, namely “the boundary of a boundary is zero”. This theorem applies in higher dimensions; for example the boundary of a three-dimensional region is a shell, and such a shell is necessarily closed, with no ends or edges.

## 3  Unicode

I find it surprising and disappointing that Unicode apparently does not have a representation for the Secchi pattern … even though it has quite a number of circular entities with various halves and various quadrants colored in, as shown in the following table.

 ● 9679 25CF BLACK CIRCLE ◐ 9680 25D0 CIRCLE WITH LEFT HALF BLACK ◑ 9681 25D1 CIRCLE WITH RIGHT HALF BLACK ◒ 9682 25D2 CIRCLE WITH LOWER HALF BLACK ◓ 9683 25D3 CIRCLE WITH UPPER HALF BLACK ◔ 9684 25D4 CIRCLE WITH UPPER RIGHT QUADRANT BLACK ◕ 9685 25D5 CIRCLE WITH ALL BUT UPPER LEFT QUADRANT BLACK ◖ 9686 25D6 LEFT HALF BLACK CIRCLE ◗ 9687 25D7 RIGHT HALF BLACK CIRCLE

If somebody knows of a Unicode Secchi pattern, please let me know. If it really doesn’t yet exist, would somebody please add it, along with the other missing quadrant-patterns?

## 4  References

1.
Wikipedia article,“Secchi disk” http://en.wikipedia.org/wiki/Secchi_disk
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