Posted by: @shaqmeister

This word:

“radians”

It doesn’t mean what you think, or assume, it means.

It cannot mean what you assume it means.

Probably a more precise way to phrase this is:

It does not (cannot) reference what you think it references.

In short, it does not reference anything to do with ‘units of angular measure’. In fact it has nothing to do with angles at all.

The successes in the solution of both the Z408 (Harden, 1969) and the Z340 (Oranchak, Blake, van Eycke, 2020), separated in time by over half a century, are undoubtedly a testimony to the tenacity of their solvers and the ultimate effectiveness of cryptanalytic approaches in reaching solutions in such instances. However, with the latter success there arrives the potential for a mood of deflation when thinking about attacks on the two remaining ciphers left by the Zodiac — the Z13 and the Z32. Owing to these being unhelpfully short (13 and 32 characters, respectively) and, in the case of the 32, presenting very few instances of repeated cipher symbols, it has to be accepted that their solution falls beyond what traditional cryptanalytic approaches are able to achieve, at least on their own. If they are ever to be solved, something else will be needed alongside such methods.

The Z32 is only as long as its name, and presents 29 distinct symbols with three of these repeating once. Hardly inspiring, of itself. Yet, all is not lost to hopelessness for one important reason — that, unlike the longer 408 and 340 ciphers, prior to even seeking some plausible route to solution, we have the advantage already that we essentially know what is going to be in it. And that can — and should — make all the difference.

The Z32 arrived accompanied by a Phillips 66 map of the San Francisco Bay Area, on which had been drawn the now familiar symbol of the Zodiac, centred over Mount Diablo, alongside which comes the prompt that an adjustment in some manner (and at some point) is to be applied to account for the magnetic declination at the given location and time. Around the outside of the symbol are four numbers, placed at the quarters and proceeding clockwise in direction, that might otherwise suggest a ‘clockface’ were it not for the fact of having ‘0’ at the top instead of ‘12’.

Further to these items — the map and the code — the letter with which they were sent offered the unambiguous statement that:

“The Map coupled with this code will tell you where the bomb is set.”

Thus in the first instance, having been presented with the starting central focus point on Mount Diablo and a surrounding expanse of map in which a bomb is claimed to be located for us to find, there can be no reasonable interpretation of any kind, other than that the code will — at the very least — contain a specification of the ‘co-ordinates’ of the bomb’s location, in the form of a ‘bearing’/‘range’ pair.

The Diablo centre, marked by the siting of what amounts to a ‘compass rose’ there, of itself rules out consideration of any of the otherwise potential alternatives — longitude, latitude; grid reference; etc.

As to the fundamental intention of ‘bearing’ and ‘range’, the joint concept is an uncomplicated one. In its application, it amounts to essentially this:

Specify, in some manner, a direction from the starting point in which progress is to be made.

Specify, in addition, the extent of progress to be made along this direction — the distance to be covered.

As to the latter, this can be given as either a distance in the field (as in orienteering, say, in km) or according to the map scale.

Pertinent to the argument being made here, however, is the crucial point of note that what the specification essentially amounts to is:

a direction + a distance along that direction.

The word ‘along’, and the context of its use here, is critical to the understanding of the additional nudge towards solution that the Zodiac offered at the end of the Little List letter.

So, how is the direction of the first step defined?

Firstly, the starting point or location is considered as a centre. Then, in relation to this centre, a standard ‘base’ direction is defined to be that from which the sought direction is to be specified according to its angular distance from such in a given direction. In the present instance, the base direction is magnetic north, and the direction (bearing) we are to look for will be specified by its angular separation from magnetic north, proceeding in the clockwise direction.

Note, however. Here, the bearing — or angle — is merely the ‘means of specifying’ the ‘resultant’ direction. It is the direction which we are going to be travelling along, not the angle.

Thinking in terms of ‘directions’ as ‘lines’ helps here. Imagine the base direction as a line from the centre directed to magnetic north. Rotate this around clockwise through the given angle, and a second ‘line’ is attained and is the one we are going to be progressing along. We cannot, in any sense, go “along” the angle to attain the intended destination. What we go along is the line specified by the combination of centre, base direction and angle therefrom.

At the failure, no doubt, of the Z32 to be deciphered by this point, the Zodiac gives us an extra hint at the end of the Little List letter, almost as an afterthought.

“PS. The Mt. Diablo Code concerns Radians + # inches along the radians”

Note, first, that he uses the word “concerns” and not “contains.” The code, as analysed, will “contain” the specification of an angle. But that is not his “concern” here. Here, he is only wanting to prompt you as to the nature of what the code is conveying:

Specification of a direction (line) + distance along that direction.

 To aid clarifying this point, we only have to seek to hold all prejudgements aside and replace the word ‘radians’ with ‘x’ in the Zodiac’s statement.

“PS. The Mt. Diablo Code concerns [x] + # inches along the [x]”

In doing so, and given the discussion above, it has to be clear that for ‘x’ we can only have some word used with the intention of “directed lines” or “radial lines” or even simply “radii.” Certainly, we cannot replace ‘x’ with “angles” and reach anything meaningful.

If nothing else, it is evident that the Zodiac was extremely keen to not miss out the “#” of inches that give the range. From how it is written, it would appear that he may actually have forgotten to include this initially, squeezing it back in only after having noted its omission, then adding an arrow underneath for emphasis. No such ‘#’ was deemed necessary to precede “Radians,” however since, despite the idiom of giving such in the plural, there will only be necessity to consider one of these.

In Conclusion

All the consideration over the past five decades, as to what precise bearing might be contained within the Z32 in units of radian measure, has been nothing other than the pursuit of a massive distraction. In contrast, having here explored the only reasonable interpretation of the intent of the statement given in the Little List ‘hint’, it has to be concluded that the Zodiac — for whatever reason — is using the word ‘radian’ to convey ‘directed radial line’ and not making reference to angles in any unit. To discover the actual units of angular measure intended, we therefore have to look elsewhere.

It may be that he merely intended ‘radii’, and got the word wrong. This would be a straightforward interpretation of error, but even this is not necessary.

Here, I will omit going into any of the details, but research has turned up how there was, in the context of land surveying in California in the 1970s, published use of the word ‘radian’ to specify a directed radial line. Whether, or not, this hints at the Zodiac having had exposure to, or experience in, land surveying must, however, be left to another discussion altogether.

What this discussion, and the conclusion here, achieves as a whole, however, is crucially important.

Once we have rid ourself of the spectre of ‘angles in radians’, we can now go back to the Diablo map and look at what we were actually given — some apparent intention to instruct us that the units for the bearing in the Z32 will be whatever you want to call a 12th part of the full 360.

And with that, real progress can finally begin to be made.