Peculiarities Concerning Π

In addition to my previous blog concerning the “Derivation of Π”, there are numerous other ways to arrive at Π’s value, some of which are very strange indeed. As well, many other peculiarities have been discovered concerning this mystical number. I will describe some of these here, but be forewarned, some of these facts concerning Π are truly weird, and will make you wonder more about this amazing number. You can call them purely coincidental, or believe that something more exists, but mathematical fact cannot be denied. Here are a whole slew of unusual facts about the number Π:

Randomness of Π

Π has been calculated by computers to a value consisting of trillions of decimal places. Numerous peculiar number sequences can be randomly found in the decimal values given the fact that there are just so many numbers in the decimal places to choose from.

Say we were to divide Π’s decimal places into groups of ten, starting immediately after the decimal place (3.-----). What is the probability that ALL ten digits (0-9) would be found in such a group of ten? The probability has actually been calculated as 1 in 40,000 chance of finding all ten digits in any given group of ten numbers after the decimal place of Π. For example, it could happen at the 3,678^th group, the 9,572^th group, or the 29,468^th group, but how soon does it really occur?

Despite this chance of 1 in 40,000, ALL ten digits are found in the 7^th group after the decimal place, as seen below!!

3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825……

Coincidental? Read on.

Π and Probability

Although we had previously detailed that Π is defined as the ratio of a circle’s perimeter to its diameter, it also seems to “pop up” in many other areas which are not in the least related to circles or geometry. Here are some examples:

Numerous integers have what are known as “unique prime divisors”, meaning that they are able to be divided only by prime numbers, and only once from each of these prime numbers. 105 is an example of an integer with the prime divisors: 3, 5, and 7. Notice that each prime divisor is only used once, and thus 105 is an integer with “unique” prime divisors. But what is the probability of randomly selecting an integer with unique prime divisors? Let’s just say that it involves Π: 6/Π² is the probability in fact!!

Another example of Π’s involvement in probability is really fascinating:

Buffon’s Needle

In 1777, Georges Louis Leclerc, Comte de Buffon published his needle problem. The problem is as follows:

Take a piece of paper with parallel lines, with equal distance d between them. Next, take a needle of length l, in which l < d. Subsequently toss the needle onto the paper numerous times. What is the probability that the needle will touch one of the parallel lines? Buffon calculated the probability as:

Probability = 2l/Πd

Astounding how Π pops up here. Of course, if the length of the needle l is equal to the distance d between the parallel lines, then the formula reduces to:

Probability = 2/Π

Another mathematician named Mario Lazzarini actually tried this little experiment with 3408 tosses of the needle, counted the number of times it intersected the parallel lines to calculate the probability, and subsequently solved the above equation for Π to get the following:

Π = 3.1415929

An amazingly accurate calculation of Π.

Unusual Calculations of Π

Numerous calculations of Π or ratios involving Π are truly strange indeed. Here are several:

Π/2 = (2 x 2)/(1 x 3) x (4 x 4)/(3 x 5) x (6 x 6)/(5 x 7) x (8 x 8)/(7 x 9) x …(2n x 2n)/(2n – 1)(2n + 1)

Another calculation:

Π²/6 = 1/(2)² + 1/(3)² + 1/(4)² + 1/(5)² + …..

There are numerous such convergences to Π or ratios of Π, which I will not present here.

Π – A Number of God?

Is Π a number of God, or the Devil? Well, Π appears in the Old Testament in a relatively strange manner, yet it also has unusual mathematical relationships to the “Number of the Beast”-666. Read the following and decide for yourself!!

Π in The Bible

A description of what is believed to be Π is stated in the Bible in two different places, although these passages are identical with the exception of one word. The following appears in both 1 Kings 7:23, and also in 2 Chronicles 4:2:

“And he made the molten sea of ten cubits from brim to brim, round in compass, and the height thereof was five cubits; and a line of thirty cubits did compass it round about.”

This passage details measurements of a huge ritualistic fountain (“the molten sea”) in King Solomon’s temple. It describes the circumference of the fountain (“a line of thirty cubits did compass it round about”), as well as the diameter of the fountain (“ten cubits from brim to brim”).

Therefore, it appears that this passage has crudely described the value of Π as 3 (Π = circumference/diameter = 30 cubits/10 cubits = 3). Not bad for something which was written around ~1000 B.C.!! HOWEVER, that is not the end of the story in the least!!

Notice above that I stated that the passages in 1 Kings 7:23 and 2 Chronicles 4:2 were identical except for one word, although I only gave you one passage. Well, that is because they are identical in the English translation of The Bible. Let us go back to the original Hebrew in which the passage was written.

In both of these passages, the word “a line” was given in English. However, in Hebrew, two different words for “a line” are used in the 1 Kings passages as compared to the 2 Chronicles passage. Both of these Hebrew words are best translated to mean “a line measure” in English.

In 1 Kings 7:23, the original Hebrew for “a line” is written like this:

קוה

In 2 Chronicles 4:2, the original Hebrew for “a line” is written like this:

קו

A wise biblical scholar named Elijah of Vilna corresponded these Hebrew letters to their numerical sequence in the Hebrew alphabet, and came up with the following (recall that Hebrew is read right-to-left):

קוה =100 + 6 + 5 = 111

קו =100 + 6 = 106

Take the ratio of these two numbers:

111/106 = 1.0472

Multiply this by the originally calculated Biblical version of Π:

1.0472 * 3 = 3.1416

Is this just a really, really strange coincidence, or a deeper hidden truth within the Bible? I’ll let you decide for yourself.

Π and 666

Π has also been found to be related to the number 666, using a little mathematical wizardry of course.

First, some unusual peculiarities concerning the number 666:

Written in Roman numerals, 666 is DCLXVI. Notice that all Roman numerals are used only once, except M (1000).

666 is also equal to the sum of the squares of the first seven prime numbers:

666 = 2² + 3² + 5² + 7² + 11² + 13² + 17²

Other unusual calculations of the number 666:

666 = 1⁶ – 2⁶ + 3⁶

666 = 6 + 6 + 6 + 6³ + 6³ + 6³

666 = (6 + 6 + 6)² + (6 + 6 + 6)² + 6 + 6 + 6

666 = 1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 5³ + 4³ + 3³ + 2³ + 1³

Now to some mathematical relationships between Π and 666:

Take the first 9 digits of Π, starting with the 3 before the decimal point, and divide them up into groups of the three:

314 159 265

A Pythagorean relationship (a² + b² = c²) can be formed from the 2^nd two sets of numbers plus introducing the number 212 into the mix:

159² + 212² = 265²

Take 666 and dividing by this new number 212:

666/212 = 3.1415 = Π

A little bit of a stretch, but nevertheless it is odd.

Another unusual relationship:

The sum of the first 144 digits of Π is equal to 666. Recall that 144 is the square of 12, which can also be written as follows:

144 = 12² = (6 + 6) x (6 + 6).

All of these examples highlight the peculiarities concerning the number Π. There are many more which are quite fascinating, but I have presented the most unusual. The mysteries of Π are numerous, and we are only just beginning to understand this constant and its role in mathematics, linking geometric circles, probability, Biblical representations of Π, and the number 666. Does Π ever end? Not from what we can tell currently, and there appears to be no end in sight. Taking all of this into account, I kid when I say that Armageddon may arrive at the moment we finally reach the end of this strange constant.