Pi: Flirting with Infinity

Pi is one of my favourite values (A few of my others are 0 and i.) It's absolutely beautiful - from the Pi character itself to what it represents to the very numbers. I've memorized about twenty digits of Pi - my boyfriend knows 30 - and there's this incredible rhythm to it. It really does feel more like a meditative chant than a recitation. Sometimes I recite it in my head when I need to clear my thoughts.

Pi is the ratio of a circle's circumference to its diameter. That's it. It's breathtaking in its simplicity. And yet it is one of the most important values in mathematics - it crops up in the most unexpected of places. It is transcendental, which implies, of course, that it is infinite. Pi has been calculated to about five trillion digits, repeatedly analyzed for repetitions and codes (you can search for special numbers in the first 200 million digits of Pi here) and has eluded all attempts by man to find some sort of pattern. It's a bit of a majestic tease.

So, this ode to Pi isn't without cause. The other day I decided to research into something I had been aware of for some time - the Bible's definition of Pi. That's right, the Bible has a definition of pi - not explicitly, of course, but easily calculated. The verse we're concerned with is 1 Kings 7:23 (NIV):
He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a line of thirty cubits to measure around it.
Take the ratio of the circumference to the diameter: 30/10 cubits = 3. 3. The Bible says Pi is equal to exactly 3!

The Bible, Mr. Arie & Pi

This is a problem! A religious text, said to be directly given to man from an omnipotent god, claims that pi is exactly 3. The implications of this are profound - one simple verse casts doubt on the correctness of the holy scriptures. For if this one verse is wrong - and it is, at least taken at face value - than what of the other verses in the bible? Are some - or most? - of them wrong too?

The first page I landed on relating to the subject was this one written by a Mr. Arie Uittenbogaard (I'll call him Arie).

The first page is good. He intelligently dismisses some very weak explanations for the difference between the biblical pi and the mathematical pi. This is looking promising! I am always ready to hear intelligent arguments, even if they are in the scripture's favor.

Page two - Arie's personal argument. The first couple paragraphs are devoted to the idea that a cubit as a unit of measure wasn't precise - it could vary depending on who was measuring. You know, I can accept that, since I don't know for sure. It's a good thought, worth looking into. (more on this after) There's much more on the page, though, so obviously Arie has more reasons why this verse isn't a problem. I keep reading:
(emphasis mine)
But suppose that a measuring line dropped from heaven and the vessel maker now had a standard length of exactly 10 meter [...] and the vessel maker made the vessel precisely 10 meters in diameter. That means that the circumference of that vessel was precisely 10 times pi = 31,41592... meters. Or in words: Thirty-one meters, plus forty-one centimeters, plus five millimeters, plus nine micrometers (that's pretty much the limit of accuracy used in modern engineering), plus a few picometers, plus the width of a few molecules, plus the width of some atoms, plus an electron more or less, plus a couple of the smallest units of length possible in the universe, called Planck-Wheeler lengths.
Smaller detail than that is not possible, but pi goes on! Pi goes on and on to describe smaller and smaller detail, but all of it is fictional and untrue.

Pi lies. Pi lies by nature of its transcendence, and has no relationship to reality. In fact, by its very nature, the more accurate we represent pi in numbers, the less accurate it represents nature!

Sacrilege! I nearly screamed. Heresy! Lies, damn lies!

Pi is nature. Pi is reality. It is only by the doings of humanity that it is assigned a numerical value. It is only because of our artificial constraints that pi is not finite.

We'll be getting into some number theory here, but it must be done to show just how wrong, obscene and idiotic such a statement is. The "artificial constraints" we place on pi has to do with our base-ten number system. If you already understand bases feel free to skip ahead to "Implication".

On Bases

For this section I am going to have to be careful with my terms. I will spell out a conceptual value (such as ten). The word "decimal" will refer to the base-ten number system, and the term "fractional value" will refer to a partial, noninteger value (for example the ".65" in 4.65 is a fractional value).

There are more number systems than the one we use every day. In fact, there are an infinite number of number systems! Of course only some of them are practical. The ones we humans commonly use are binary, decimal and hexadecimal. You have probably heard of these terms. But what, exactly, does it mean?

The base of a number system defines how many digits it uses. The binary system uses two digits (0 and 1), the decimal system uses ten digits (0,1,2,3,4,5,6,7,8,9) and hexadecimal uses sixteen digits (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F). Each "place" of a number can count up to the number of digits it has (in decimal, you can count from zero to nine, in hex you can count from zero to fifteen) before "rolling over" into the next place value. Using the value thirty-four as an example:



There are plenty of comparison charts available on the web. :)

Here's another way of looking at it, a bit more mathematical than the above. Each place "x" in a number represents the value of the digit it represents multiplied by baseX and then the values are added to get the value of the number. He's an illustration using the three base systems above:

Decimal
The value thirty-four in decimal is represented by 34.
Place 0 equals four (4 x 100 = 4 x 1 = 4)
Place 1 equals thirty (3 x 101 = 3 x 10 = 30)
Add thirty and four and you get thirty-four!

Binary
The value thirty-four in binary is represented by 100010.
The zeroith place equals zero (Decimal: 0 x 20 = 0 x 1 = 0).
Place 1 equals two (Decimal: 1 x 21 = 2 x 1 = 2)
Place 2 equals zero (Decimal: 0 x 22 = 0 x 4 = 0)
Place 3 equals zero (Decimal: 0 x 23 = 0 x 8 = 0)
Place 4 equals zero (Decimal: 0 x 24 = 0 x 16 = 0)
Place 5 equals thirty-two (Decimal: 1 x 25 = 1 x 32 = 32)
Add thirty-two and two and you get thirty-four.

Hex
The value thirty-four in hex is represented by 22
Place 0 equals two (Decimal: 2 x 160 = 2 x 1 = 2)
Place 1 equals thirty-two (Decimal: 2 x 161 = 2 x 16 = 32)
Thirty-two plus two gives thirty-four.

There. We now hopefully understand how whole numbers work. But there are more than just whole numbers - in fact, there is an infinite number of real numbers that aren't simple integers. We must be able to represent fractional values! In decimal, we have a system where each digit in place X to the right of a decimal point represents (Important to note is that we start counting the place number X at one instead of zero.):
Digit value * 1/ 10X
And we then sum the fractions. So the decimal value 0.6437 is equal to:
(6 / 101) + (4 / 102) + (3 / 103) + (7 / 104)
= (6 / 10) + (4 / 100) + (3 / 1000) + (7 / 10000)
= 0.6 + 0.04 + 0.003 + 0.0007

In binary, the fractional values are described by a sum of fractions of the form
digit value * 1/2place
and, indeed, any base's fractional values are represented by a sum of fractions
digit value * 1/baseplace
Returning to binary, the decimal value of the fraction 0.011 is:

(0 / 21) + (1 / 22) + (1 / 23)
= (0 / 2) + (1 / 4) + (1 / 8)
= 1/4 + 1/8
= 3/8

Implication

We now know how fractional values are represented in various base systems. But here's the rub. With our decimal system - with any number system - there are values we cannot represent! There are values that cannot be represented as a sum of fractions - and especially not as a sum of a specific type of fractions. In decimal, for example, we cannot precisely represent the value obtained by dividing one by three. The fraction 1/3 in decimal is 0.33333... repeating for infinity.

For example, there are valid decimal fractions that cannot be represented in binary. Since all fractional values in binary have to be represented by a sum of fractions described by
1 x 1/2place
there are plenty of perfectly normal decimal values that binary cannot describe. (This is why computers have a hard time with computations involving floating-point numbers e.g. fractional numbers. They cannot precisely or accurately represent many decimal values in the finite amount of memory they have (maximum of 64 digits of binary, usually) so the answers they come up with [because they convert from decimal input to binary for the computation back to decimal for the output] can sometimes be wrong.)
We can get really, really close to many values - close enough for government work, as they say - but never exactly.

For example, the value obtained by dividing one by ten (1/10) is perfectly reasonable in decimal. It's simply 0.1 . However, an attempt to convert this to binary results in the infinitely repeating 0.0001100110011... (worked out example here)

You may begin to see where this is leading...
Yes, our chosen numeral system of base ten cannot represent pi finitely, but that does not mean that pi is "fictional and untrue." It simply means that our artificial construct of a number system is not not capable of expressing a specific value. Pi exists and it is natural and true. It is our attempts to wrangle the infinitness of numbers into a finite form that is unnatural and untrue.

Where Arie went wrong

And that's it, mostly. Arie is trying to prove the scriptures correct by proving mathematics is wrong without having even a basic understanding of mathematics. He further proves his ignorance with another "gem":
Pi is defined as the ratio between C and D of a circle. In 1761 Johann H. Lambert proved that pi is not rational. That means that whatever pi may be, it's certainly not a ratio .

Pi is a paradox and the proof that items that cover an unequal number of dimensions (diameter is 1D; circumference is 2D) are not always compatible.
Here, Arie is relying on the definition of a rational number. The definition is as follows:
...any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero
Arie overlooks a very significant detail: a rational number can be expressed as a ratio of two integers. Any number that cannot is irrational. It is impossible to obtain pi by dividing one integer by another - but, we can obtain pi by dividing any integer by a fractional number, or a fractional number by an integer, or a fractional number by another fractional number. A ratio is simply a unitless quotient. "Circumference:diameter" is the same as "circumference/diameter". And it will always give you pi. And never three.

So, what about the Bible?

There are really two possibilities, and both have their implications for the validity of the Bible.

As Arie himself argued first before making a fool of himself, a cubit may not have been a standardized unit of measure. However, we do have to remember that since pi is a ratio, any unit could be used to obtain it. If I measured a circumference to be 10 snakes long I will find the diameter to be about 3.18 snakes long, as long as I use the same snake every time. A cubit as a unit of measure is roughly equal to a man's forearm - so maybe two different forearms were used instead of an equivalent measuring rod, although this says some interesting things about the integrity of the engineers contracted to craft the holy vessel.

The second possibility is a case of bad precision: about 30 divided by about 10 is about three. This could even potentially be due to restrictions from the language. As Arie states (uncited, of course, and I am unable to find any other information on the topic, so take it with a grain of salt. Interestingly, he also seemingly misses the significance to his argument)
[...]even if the Hebrews had decimal notation, which they didn't.
If the Hebrews had no way of representing fractional numbers and simply rounded up or down to the nearest integer... that explains how they get 3 for pi. But it also means that there may be several other values in the Bible that are lacking precision too. Would it have a great impact on the validity of the scriptures? Don't ask me, I'm a computer scientist, not a theologian! ;)

And now for something [not] completely different

And now, with all that nasty business behind us, I leave you with my favourite ode to Pi. Parts of the lyrics literally send chills down my spine:



When ink and pen in hands of men
Inscribe your form, bipedal "P"
They draw an altar on which
God has slaughtered all stability
No eyes could ever soak in all the places you anoint
And yet to see you all at once we only need the point
Flirting with infinity, your geometric progeny
That fit inside you oh so tight
With triangles that feel so right

3.14159265358979323846264338327950288419716939937510582097494459

Your ever-constant homily says flaw is discipline
The patron saint of imperfection frees us from our sin
And if our transcendental lift shall find a final floor
Then Man will know the death of God where wonder was before

2 things about

Pi: Flirting with Infinity
  1. This one, along with your argue with a creationist, really show some of the problems ppl have with relating biblical interpretations to secular thoughts. I just tend not to bring it up in conversation since neither side will find themselves satisfied with the other's interpretation.

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  2. I agree!
    I really don't mind when people arrive at spiritual conclusions when given scientific arguments, but I really dislike when they use faulty logic to validate their beliefs. It's really all or nothing - if you want to try to use science to justify parts of your belief you must apply its standards to all your belief, and if you want to use scientific arguments to debunk something that contradicts your belief, you must do it properly.

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