You maybe did not expect the above. The original patent did, after all, stipulate that a single unit of 𝜏 would mark the zig-zag boom cross-over locations. And also that a single, co-linear wire should connect one zig-zag apex to it's opposite fellows. Clearly I violate that original concept.
I voilate it (very slightly) as follows. It is not a pefectly straight line which runs apex-to-apex passing through the boom on its way. That line is actually bent. At boom cross-over, there is a tiny change in vector by some small fraction of a degree. Thus the saw teeth themselves are not perfect isosceles triangles. Very nearly, but not quite. Were you able to tell? I'm betting not.
Picture an imaginary string drawn taut between opposing apexes of opposite saw teeth. That line will pass directly through the otherwise lonely 𝜏-point situated between element cross-over 𝜏-points. And there the string will form right angles with the boom. The string will also divide the opposing saw teeth each into a pair of mirrored unequal right triangles. It has to be thus, since 𝜏-divisions are unequal by definition. And, for this being the case, neither are the boom cross-overs quite perfectly straight.
Now lastly picture the angle formed between that taut string and the forward edge of the saw tooth which it bisects. I am inclined to call this measure the saw tooth's Leading Half-Angle. I further like to suppose that it even significant. Reference this link: XML
All in all, a slight deviation from the 1966 patent. A trivial one, but still thought I'd mention.
How that transpired is like so. I was purposely lazy. I chose to employ double the number of half-size divisions for 𝜏. I did it because saw tooth elments are an add-on feature to my pre-existing calculator which was set up originally for log-periodic dipole arrays. So the math for those was already in place. And so I made (somewhat contorted) use of that rather than trouble myself at special geometry for something entirely new.
My excuse (if I need one) shall be this. The original zig-zag patent did not have a boom. The venerated L. B. Cebik, W4RNL (SK), demonstrated the advantages of lower ohmic losses by adding a boom. For being uncertain, quite how W4RNL arranged his own models, I came up with this on my own. Perhaps I mirror his own choice unknowing? The real truth is that I was too lazy to work out a fresh new algorithm for calculating the intersection of straight lines pivoted toward one another from boom cross-over locations at 𝜏-divisions. It would have taken just only some trig ... so that's pretty lazy, I'm sure you'll agree. I did make a start in that direction, but then...
After a bit of toying around in the CAD platform Rhino 6, I saw as how the differnce between what I
ought to do, and what could be more easily done, then talked myself out of the former. Which in programming
circles, is no vice at all*.
*
"We will encourage you to develop the three great virtues of a programmer: laziness, impatience, and hubris."
-- Larry Wall, inventor of Perl
If you're familiar with LPDAs, then it may have struck you that I build arrays in reverse of established precident. I build from front to back, not back to front. My thoughts at the time hinged (pun intended) upon the apex of the angle α as being, ahem ... pivotal.
In the NEC and ANT file formats, wire #1 is generally the source. And an LPA has its source at the front. So, for consistency's sake, I set up my calculator to ennumerate wires from front to back. I was eager to move ahead with the coding, so didn't agonize over that much. Thus did it occur that the whole program branches out from a central array which orders wire numbers from feed wire in front to stub at the back. The feed wire must always be there, after all, while the stub is entirely optional. This being the case, it now would feel awkward should enumerate the divisions of 𝜏 in reverse of all else.
The electrons themselves, so they inform me, don't care at all. They happily flow in reverse of that direction toward which they were originally assigned. Which, once the fact was discovered, left stubborn academics having to re-arrange their minds to think in terms of hole flow instead. I rest my case.