Note:As at July 2018, due to circumstances outside our control, the container has had to be repositioned by 6 or 7 metres but this should not cause a problem. Altering the puzzle was considered unnecessary...
The area where you will find the cache can be busy with passers-by during the day. Please try to be discrete when seeking and replacing the container but please put it back in the obvious place. On the plus side, there is a convenient bench to sit on while writing your log.
Babylon was an important city in Mesopotamia, the remains of which are situated 85 km south of Baghdad in Iraq. Babylon is understood to have been founded in 1894 BCE (BC) and is thought to have been the largest city in the world during its prime with a population of some 200,000 inhabitants. Among the significant things for which Babylon is known, is the importance of its mathematicians, and it is recognised, from clay tablet records, that they understood the concept of a mathematical constant in the relationship between the circumference of any circle and its radius. That constant is today represented by the Greek letter Pi; and the relationship is that Pi = the circumference (C) of a circle divided by its diameter, and the diameter of a circle is equal to 2 x its radius (r), the radius being a straight line joining the centre of the circle to any point on its circumference. So, Pi= C / 2r.
The Babylonians, however, did not name the constant Pi.
In fact it was not until the 18th century CE (AD) that this representation of the constant came into use. Prior to that, the symbol was used to represent the “periphery” (meaning perimeter or circumference) of the circle, being the Greek equivalent to the first letter of the word. An Englishman, William Jones, a mathematics teacher, is credited with being the first author to use the Greek letter for this constant in his 1706 work, A New Introduction to Mathematics, a book based on his teaching notes intended for beginners. Elsewhere, Jones gives some credit to an English astronomer and mathematician by the name of John Machin, who himself developed a quickly converging series that he used to compute Pi to 100 decimal places. It was Euler, however, who brought the symbol into popular usage as a result of frequent references in his later 18th century works on mathematics.
The best Babylonian approximation of the value of the constant (out by about 0.5%), is thought to have derived from an observation that a regular hexagon inscribed within a circle could be used to estimate the length of the circle’s circumference as 6 times the length of its radius, based on the fact that the 6 sections of its perimeter are sides of equilateral triangles the other sides of which are the circle’s radii.
This is our Six Slices of Pi:
What is the arithmetic fraction, and its decimal representation to 9 decimal places, thought to have been associated with the Babylonian approximation of Pi?
In the 3rd century BCE (BC), one of the greatest scientific minds of antiquity, Archimedes of Syracuse, extended the Babylonian approach. The approach he used is known as “exhaustion” whereby the circle is filled with polygons having an ever increasing number of sides until an ever better approximation of the circumference is reached. He also adopted the idea of both inscribing and circumscribing the circle with these polygons so as to calculate lower and upper limits to the approximation.
Using regular polygons with 96 sides, what fractions did Archimedes obtain for these lower and upper limits of approximation of Pi? And what are the decimal equivalents of each, to 9 decimal places?
It would be remiss not to mention the Chinese mathematician Lui Hui, who lived in the 3rd century CE (AD), as he invented an algorithm for the calculation of Pi (known as Lui Hui’s algorithm – surprise, surprise!). With this iterative algorithm it was theoretically possible to calculate Pi to any required accuracy based on bisecting regular polygons. Doing this kind of calculation was not easy using “counting rods”, which were used in calculating by the ancient Chinese for more than two thousand years. To get around this problem, Lui Hui discovered a short cut to his algorithm which gave a value for Pi accurate to 4 decimal places. What is the fraction associated with this “quick method”? And what does that fraction generate as a value for Pi to 9 decimal places?
Zu Chongzhi , another Chinese mathematician and astronomer who lived in the 5th century CE (AD), calculated an approximation of Pi that remained the most accurate available for near on a millenium, being accurate to 6 decimal places and to within 0.000009% of the value of Pi. This was based on a fraction he called Milü (meaning “detailed ratio” in Chinese). He obtained the result by approximating a circle with a 24,576-sided polygon. This was truly amazing considering the counting device he used for recording results was a pile of wooden sticks (the counting rods) laid out in certain patterns.
What was the ratio (fraction) he obtained and its decimal equivalent to 9 places?
Take these 5 sets of data you have gathered and calculated; add to them the true value of Pi to 9 decimal places; you now have another Six Slices of Pi.
Use these to determine the co-ordinates of the cache, as follows:
N 51° 03.ABC W 000° DE.FGH where:
A is the 3rd Post-Decimal Point Digit (PDPD) in the Babylonian approximation
B is the 6th PDPD of Archimedes upper limit approximation
C is the sum of all the PDPDs of Lui Hui’s approximation divided by 2
D is the 4th less the 8th PDPD of Archimedes lower limit approximation
E is the 7th PDPD of Zu Chongzhi’s approximation
F is the 5th PDPD of Pi Day’s date and time reference (true value of π to 9 digits)
G is the 7th PDPD of Archimedes upper limit approximation
H is the 8th PDPD of Zu Chongzhi’s approximation
You can validate your puzzle solution with certitude.