It was whilst musing about the Pythagorean sect that the author had an idea that is so simple it would be incredible if it were not already known.
Pythagoras and his disciples needed to eat. Therefore, once they had spent time - possibly years - developing a mathematical method, these had to keep it secret.
But how would the Pythagorean triangle be used secretly on a building site,
perhaps in front of an audience?
Suppose you want to create a square building. It is an obvious idea to
make up a loop of string with a knot for each corner. The intervening
lengths of string are all equal in length.
However, surely Pythagoras would incorporate a secret, removable knot P.
This would be positioned one third of the length of a side away from
one of the permanent knots.
Pythagoras would conceal the removable knot in his hand (although for the purpose of demonstration he is showing it to you here). A helper (not shown) might head towards Polaris, the Pole Star, and fix knot A.
If one side is aligned north-south in this way, an adjacent side will be
aligned east-west, with a southerly aspect.
The helper does not need to know mathematics. He is simply told to fix
knots A, B, C and D in that order - and keep the rope taut. As he bustles
about, he captivates the audience. Nobody notices Pythagoras holding the
secret knot.
The helper proceeds to knot B. This is the Pythagorean moment
when a right-angle is created at A.
With knots A and B secure, Pythagoras himself forms the third point of
the triangle.
The helper then proceeds to knot C and fixes it without pushing the string
out of true. This consolidates the right-angle at A, and liberates Pythagoras
from his chore.
This is the point where Pythagoras makes his play. He secretly unties the
temporary knot.
A moment of high drama here, as Pythagoras drops the rope.
Yes! He actually drops the rope. Nice work if you can get it.
Pythagoras has removed the temporary knot, and concealed it in his pocket.
There is now no clue as to how the trick was done.
The helper now tensions the rope, and fixes knot D. Three right-angles
appear simultaneously at B, C and D. The helper graciously acknowledges
the cheers of the adoring crowd.
Meanwhile, Pythagoras slips quietly away to see the site manager about the cash.
The idea came to the author when he considered the Pythagorean “Triple” 3 4 5.
The author realised that it is an arithmetical progression, where the middle number 4 is also the average. Thus, the perimeter of a 3, 4, 5 triangle is 3 times 4.
Also, the perimeter of a square of side 3 is 4 times 3.
“Heureka”! The perimeter of a 3, 4, 5 triangle is that of the square on one of its sides - so a length of string can be converted from a triangle to a square.
Other Pythagorean triples are not so fortunate. The 5, 12, 13 and
the 7, 24, 25 triples are not arithmetical in their progression. On the
other hand, multiples of 3, 4, 5 such as 6, 8 10 and 9, 12, 15 are
perfectly suitable.
We have seen how the fourth side is divided up between two of the other
sides, so that the remaining side must be the shortest.
We now turn our attention to the question of whether rectangles other than squares may be created with little effort.
Buildings from the “Classical Period” of Greece (450 B.C. to 330 B.C.) were often constructed according to the “Golden Section” of root-5 plus 1, all divided by two.
Pythagoras himself rejected irrational numbers such as this (1.618033989), so they will have worked with whole numbers (integers).
The Golden Section appeared in the writings of Euclid (“The Elements”). Euclid was born in 325 B.C. - at about the close of the Clasical Period.
Integer approximations to the Golden Section came from Leonardo Fibonacci, in his book “Liber Abaci” 1500 years later - so Pythagoras cannot have used that book.
We may assume that knowledge was discovered, and then went missing again.
The scalability of Pythagorean triangles was mentioned above. One can
make a 3, 4, 5 triangle in yards and measure it in feet. It measures
9, 12, 15. Similarly, a 3-metre, 4-metre 5-metre triangle is a
300 400 500 centimetre triangle.
Here we have a 3, 4, 5 triangle scaled by 24. This creates a 72, 96, 120 triangle. The perimeter is 288 units.
This also allows two sides of 55 and two of 89 to be constructed.
We need a pair of Pythagori (is that the correct plural?)
to hold the corners, whilst the helper fixes A, B and C. Then the
Pythagoras Twins can drop the rope, and the helper can secure point D.
What emerges is a rectangle, with four perfect right-angles and sides
55 and 89.
The resulting aspect ratio is 1.6181818. This is just 91 parts per million longer than a building that had the ratio of the Golden Section. The error would be quite undiscernable.
The Golden Section (“Golden Ratio”) is said to bestow great elegance upon objects so
constructed.
Pythagoras, with his mathematics, ushered in the Classical Period of
ancient Greece, but his exact methods are today shrouded in the mists
of time.