Earth resistance testing testing with Euclid and Da Vinci

It may seem unlikely that there could be a connection between an ancient Greek mathematician, a renaissance genius and the mundane subject of earth resistance testing but, as we’ll see shortly, there most certainly is! And the connection itself is just about as unexpected and unlikely as its existence. Hopefully your interest has by now been piqued, so it’s time for – if you’ll forgive the expression – a down-to-earth explanation.

Engineers and technicians who have been involved with earth resistance testing will recognise the diagram below, which illustrates the basic principle of earth resistance using the fall-of-potential method.

Without going into the theory, the way the test works is that a test electrode (ROD 2 in the diagram) is driven into the ground some distance from the electrode that’s being tested (ROD 1). A measured current is then passed between the two electrodes. A third electrode (ROD 3) is then driven into the ground at various points along the straight line between ROD 1 and ROD 2. The voltage at ROD 3 is measured for each point and, because the current flowing in the circuit is known, the resistance at each of the points can be easily calculated using Ohm’s law.

Assuming that there are no complicating factors, the result of plotting resistance against the position of ROD 3 will be a graph like that shown in the bottom half of the diagram. Notice that this graph has a near-horizontal section where moving the electrode hardly changes the resistance. The resistance measured at this point is the resistance of the electrode under test.

This is a highly simplified explanation, of course, but if you want more detail simply download the application guide “Getting Down to Earth” from the Megger website. But what does all of this have to do with Euclid and Da Vinci?

Well, in the real world, there’s often no time to take a whole series of readings with ROD 3 at various locations in order to measure the earth resistance of an electrode. It would be much quicker and more convenient to take a reading with ROD 3 in just one location but, of course, it would be essential to know that this location corresponded with the flat section of the curve.

Mathematical analysis shows that the location that best meets this criterion is 61.8% of the distance between the electrode under test and ROD 2. This is shown in the diagram as the 62% line, but that’s just rounding up. Position your test electrode at the 61.8% mark and, in simple cases at least, you’ll only need to make one measurement to determine the earth resistance of the electrode under test.

But look at that number – 61.8%. Or, to express it another way, 0.618. Does that look familiar? No? Well how about 1.618? If you’re still puzzled, 1.618 – which is actually the first few digits of an irrational number that, like π, goes on forever – is the Golden Ratio. Known since the time of the ancient Greeks, this ratio crops up in all sorts of unlikely and apparently unrelated places.

Euclid is believed to be the first to have defined the Golden Ratio, and he used it extensively in his great mathematical book, Elements. Many find evidence of the Golden Ratio in the composition of great works of art by Leonardo Da Vinci, not least in his Last Supper and the Mona Lisa. It also describes some aspects of disposition of leaves on many types of plant. Researchers have even reported that the Golden Ratio has connections with DNA in the human genome.

So why is there a connection between the Golden Ratio and earth resistance testing? At first sight, this connection appears surprising and even unlikely, but on thinking about it, the Golden Ratio crops up in so many contexts that perhaps it’s not really surprising at all that earth resistance testing is on the list. But when it comes to giving a definitive answer as to why this connection exists, no one seems to know. Of course, if you have the answer, don’t hesitate to get in touch – we’d love to be the first to publish it!