Primordial Pi

It is one of the first constants we learn in school, and perhaps, one of the most mysterious. It forms the basis of modern trigonometry, and is involved in relations as deep as the infamous Euler’s Identity. It seems strange, even almost impossible, that mathematicians might have ever functioned without Pi. But, as is so often the case, with exception to the ancient greek, ancient mathematicians found a way.

Mesopotamian and Babylonian Pi:

Strictly speaking, it was not that ancient mathematicians were truly working without pi, rather that they dealt with it indirectly and were never fully aware of its intricacies.

One of the most striking examples of such indirect use is the mathematics of Mesopotamia. The Mesopotamians quite aware of the importance of constants in geometry. In fact, rather than define important relationships between the sides, areas, and volumes of fundamental geometric shapes using formulae, as we know them today, the Mesopotamians actually defined these geometric relationships purely in terms of ratio coefficients.

For instance, given an equilateral triangle, such as the one below:

Equilateral

The mesopotamians held that:

\overline{AD}=\overline{BC}\cdot\frac{7}{8}

Thus, were a Mesopotamian to attempt to find the height of an equilateral triangle given its base (which they considered to be its defining feature), he or she would simply multiply the length of the base of this triangle by \frac {7}{8}.

The Mesopotamians had many such coefficients, including two for their circles. Interestingly, the Mesopotamians did not view the radius of a circle as its defining feature, but instead defined their circles in terms of their circumferences. Thus, the Mesopotamians did not have a constant for converting the radius of a circle to its circumference, but instead, they had a constant for converting the circumference of a circle to its diameter. This coefficient was as follows:

Diameter=\frac{1}{3}\cdot Circumference

That’s right! In other words, the Mesopotamians held that:

\Pi=3

The mesopotamians also had a coefficient for finding the area of a given circle in terms of its circumference, and this too implied a value of 3 for Pi.

It is not known explicitly where or how such an amusingly simple approximation for Pi originated, but, it is not unreasonable to assume that the Mesopotamians simply decided that the value three was close enough.

Notably, the Babylonians did discover a more accurate approximation for Pi (namely, \pi=3.125) by inscribing a regular hexagon into a circle, but it seems that the babylonians did not realize the worth of this better approximation, as it was never utilized (as far as we know) for circular geometry.

Indian Vedic Pi

On the opposite side of the spectrum, in terms of simplicity, is the Vedic conception of Pi. The Vedic contribution to mathematics comes in the form of the Sulbasutras, a series of texts produced in India during the Vedic period to aid with the construction of religious altars. Consequently, the Sulbasutras were less concerned with mathematical exactness or proof, and more concerned with applicability and general reliability. Thus, as one might expect from something more akin to to an engineering textbook than a mathematics textbook, the Sulbasutras give no proofs for any of the relationships they present. Moreover, while some of the relationships expressed are, remarkably, quite correct, a great deal of them are only approximations, and, further adding to the chaos, the Sulbasutras make no distinction between what is approximation, and what is not.

Pi, naturally, is among these approximation-type relationships, and is one of the ugliest. As it happens, there is not any single unifying approximation for Pi present within the Sulbasutras. Instead, the Sulbasutras host a great variety of approximations to Pi, with several texts containing more than one. Among the list of these approximations are:

\pi=\begin{cases}2.99\\3.00\\3.004\\3.029\\3.047\\3.088\\3.1141\\3.125\\3.16049\\3.2022\end{cases}

It is important to realize that most, if not all, of these approximations are implied. That is, the Sulbasutras do not directly state these as being values of Pi (nor do they acknowledge the existence of Pi in the first place). Instead, the Sulbasutras present various equations for various calculations involving circumferences, diameters, areas, and volumes that, in order to be mathematically correct, would require Pi to have one of these above values. This lends some explanation as to why Pi has no consistent value in the Vedic texts. The writers clearly were not thinking in terms of a unified Pi, but rather were thinking more in terms of the operations that use Pi. It is only natural that they would use and imply entirely separate approximations in problems that, in their perspective, were as well entirely separate and unrelated.

Ancient Egyptian Pi

Perhaps the most elegant ancient approximation of Pi, outside those developed in and around ancient Greece or China, is that of the Egyptians. As with the Vedic texts, and with the Mesopotamians, the Ancient Egyptians’ value of Pi was, once again, implied. However, unlike the Mesopotamians, this implied value was, relatively, quite accurate. As for whether or not Egypt used this implied value as a ubiquitous standard is up for debate. Unfortunately, very few Egyptian mathematical manuscripts have survived (two, to be exact), and the implied approximation of Pi appears in only one problem in only one of these manuscripts, as well as a few other problems based on the method that it presents. Thus, it is impossible to say whether or not this implied value was standard, as there is not a diverse problem set against which to test.

So, what is the approximation in question? Put briefly, this implied value for Pi is:

\pi=\frac{256}{81}\approx 3.16

The problem from which this implication comes is as follows:

Example of a round field of diameter 9. What is the area? Take away 1/9 of the diameter; the remainder is 8. Multiply 8 times 8; it makes 64. Therefore, the area is 64.

— Problem 50 of the Rhind Papyrus (qtd. in Katz 8)

Written in modern mathematical terms, this problem states that:

A={\left (D - D/9 \right)}^{2}

Or, in other words, that:

Area = { \left ( \frac {8} {9} Diameter \right)}^{2}

Comparing this to the standard equation for the area of a circle:

Area = \pi { \left ( \frac { Diameter }{2} \right)}^{2}

We can find the implied value of Pi with some algebra:

A=\pi{\left(\frac{D}{2}\right)}^{2}={\left(\frac{8}{9}D\right)}^{2}\\ \pi \cdot {D}^{2}\cdot \frac{{1}^{2}}{{2}^{2}}=\frac {{8}^{2}}{{9}^{2}}{D}^{2}\\ \pi \cdot \frac{{1}^{2}}{{2}^{2}}=\frac{{8}^{2}}{{9}^{2}}\\ \pi=\frac{{8}^{2}}{{9}^{2}}\cdot \frac{{2}^{2}}{{1}^{2}}=\frac{64}{81}\cdot \frac{4}{1}=\frac{256}{81}

And thus, we have arrived at our implied approximation for Pi.

Like the approximations for Pi present in other ancient cultures, how the Egyptians came up with this implied value is not explicitly known. However, unlike the approximations for Pi in other cultures, the Egyptians left a clue. In problem 48 of the same Rhrind Papyrus, a small octagon-like shape is inscribed within a square. The problem is not explicitly given. Instead, the only things present are the calculations:

8 \cdot 8=64

and:

9\cdot 9=81

along with the aforementioned geometric inscription:

problem48

(Image courtesy of The Number Warrior)

 

If we assume that the area of the square is the larger of these two calculations (9\cdot9), and if we assume that the octagon was meant to be constructed symmetrically with non-diagonal sides of length 3, then the following Picture arises:

Octagon1

If we separate the larger square into nine subs-quares, (and thus, the octagon into five squares and four half-squares), it is easy to see that the octagon occupies seven of these nine new squares:

Octagon2

 

In other words, the octagon occupies \frac{7}{9} of the area of the square. Put mathematically, this is:

{A}_{Octagon}=\frac{7}{9}\cdot{A}_{Square}=\frac{7}{9}\cdot{9}^{2}=63

At first, this apparently random octagon seems quite irrelevant… But consider, for a moment, the possibility that the octagon is an approximation to a circle inscribed into the same square:

Octagon3

If this possibility holds true, then this octagon becomes an approximation for the area of a circle with diameter 9. In other words:

{A}_{Circle}=\frac{7}{9}\cdot{A}_{Square}=\frac{7}{9}{D}_{Circle}^2

If we compare, as we did before, to the standard equation for the area of a circle in terms of its diameter, we may once again solve for Pi:

{A}_{Circle}=\frac{7}{9}{D}_{Circle}^{2}=\pi{\left(\frac{{D}_{Circle}}{2}\right)}^{2}\\ \frac{7}{9}{D}_{Circle}^{2}=\frac{1}{4}\pi{D}_{Circle}^{2}\\ \frac{7}{9}=\frac{1}{4}\pi\\ \pi=\frac{28}{9}=\frac{252}{81}

As you can see, this implied value value for Pi (\pi=\frac{252}{81}) is exceptionally close to the originall implied value, \pi=\frac{256}{81}. Still, this slight difference is troubling. Fortunately, it, too, has a theoretical explanation. To understand this explanation, it is important to remember that the Egyptians were, most likely, not even aware of these implied values of Pi. From the perspective of the Egyptian scribe who produced the above octagon inscription, the important ratio derived was that between the square of the diameter of a circle, and its area (this ratio being \frac{7}{9}). It is quite possible that this scribe had originally set out to find a ratio by which to multiply the diameter of a circle before it is squared. In this case, the ratio that the scribe would have truly wanted, is \sqrt{\frac{7}{9}}, not \frac{7}{9}. Unfortunately, as the scribe would have presumably known, \frac{7}{9} does not have a rational square root. This scribe might have noticed, however, that the fraction \frac{64}{81} (which is quite close to \frac{7}{9}=\frac{63}{81}) does have a rational square root, and simply chosen to round to this value. Taking the square root of \frac{64}{81}, we have  \sqrt{\frac{64}{81}}=\frac{8}{9}, which is constant originally used in Problem 50 of the Rhind Papyrus, as described above.

There is, notably, some criticism surrounding this explanation. Skeptics, in particular, note the rather misshapen nature of the octagon inscription, positing that it was, perhaps, originally intended to be a circle, or that the octagon was intentionally misshapen such that its area actually does come out to be 64. Others argue that the less misshapen interpretation of the octagon is correct, but that the discovery of \frac{64}{81} as a square alternative to \frac{63}{81} might have been made through more geometric means. For instance, the scribe might have noticed, as shown below, that by removing 17 of the total of 18 1×1 squares in area lying outside the octagon, a perfect square is produced. The scribe, seeing an opportunity for simplicity, might have simply decided to ignore the final 18th square, and use the sides of the original and resultant squares in his or her final ratio. Counting the length of these sides does, indeed, reveal that this ratio is \frac{8}{9}.

Octagon4

 

Offline Work Cited:

Katz, Victor J. A History of Mathematics: Brief Edition. Boston: Pearson/Addison-Wesley, 2003. Print.

 

 

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