Unravelling Dholavira's Geometry - Michel Danino

UNRAVELLING DHOLAVIRA’S GEOMETRY

Michel Danino*

Paper contributed to Recent Researches in Archaeology, History and Culture (Festschrift to Prof. K.V. Raman), P. Chenna Reddy, ed., Agam Kala

Prakashan, Delhi, 2010, pp. 179–193.

Dholavira, a major site of the Indus-Sarasvati civilization in Kachchh district of Gujarat, was discovered in the 1960s by Jagat Pati Joshi and excavated in the 1990s under the direction of R.S. Bisht. It is characterized by monumental architecture, massive fortifications, the use of dressed stones, elaborate water management — and meticulous town-planning of a unique kind. A look at its map (Fig. 1) reveals a distinct geometry involving multiple enclosures (“citadel,” middle and lower towns) laid out in what appears to be specific proportions.

In several papers,1 R.S. Bisht described Dholavira’s special features, including its proportions. Some of his observations (reformulated here following my system of notation spelt out in Table 1) were:

The city’s length (east-west axis) and width (north-south) are precisely in a ratio of 5 : 4 (Lt / Wt = 5/4);

the castle’s proportions follow the city’s ratio of 5 : 4 (Lci / Wci = 5/4); the castle’s northwest corner is located on the city’s main NE-SW diagonal; the citadel (castle and bailey together) has a ratio of 1 : 2; the middle town’s length and breadth are in a ratio of 7 : 6 (Lm / Wm =

7/6); the ceremonial ground’s proportions are 6 : 1 (Lg / Wg = 6).

* Independent researcher, author of several books and papers on Indian culture and protohistory, including the Indus-Sarasvati civilization, its legacy, and aspects of the Aryan issue. Email: [email protected].

Unravelling Dholavira’s Geometry / p. 2

Abbreviation Meaning measurement (in metres) Mnemonic clue

Lt Length of city 771.1 Subscript “t” stands for “town”

Wt Width of city 616.85

Lm Length of middle town 340.5 Subscript “m” stands for “middle” town

Wm Width of middle town 290.45

Lg

Length of ceremonial ground 283 Subscript “g” stands for “ground”

Wg Width of ceremonial ground 47.5

Lci Inner length of castle 114 Subscript “ci” stands for “castle inner”

Wci Inner width of castle 92

Lco Outer length of castle 151 Subscript “co” stands for “castle outer”

Wco Outer width of castle 118

Table 1


Clearly, those ratios cannot be attributed to mere chance.2 But if they were consciously determined, that raises some weighty questions: (1) Since one proportion may be the logical consequence of another, what were the initial principles adopted by Dholavira’s architects? In mathematical terms, what are the basic assumptions, or axioms, required to work out the totality of Dholavira’s geometry? (2) Are similar proportions found in other Harappan cities? (3) Can we figure out some of the units used in planning and measuring out the city? (4) Can we suggest reasons for those units and for the planners’ initial ratios? (5) Are some of those proportions found in later classical Indian town-planning or architecture?

This paper does not claim to answer these questions in full; it only attempts to work out a few firm elements of this largely virgin field.

Before we proceed, we should mention Bisht’s estimate of an error margin on measurements of 0.2 to 0.5%; they “may be slightly higher than the actual, not less.”3 We should add that whatever ratios the Harappans may have decided on, irregularities on the terrain and alterations over the millennia certainly introduced small deviations from the “ideal” proportions; if, therefore, we find specific ratios to be within 1% or 2% of actual measurements (either provided above by Bisht or read on Dholavira’s map), it will be legitimate to consider them as potential deliberate choices.

Let us now adopt the following additional definitions (Table 2 below, to be studied in conjunction with Fig. 2):

Abbreviation Meaning

F The castle’s outer north-western corner, a major “focus” of the city

A Distance from the city’s northern wall to F

B Distance from the city’s southern wall to F

C Distance from the city’s western wall to F

D Distance from the city’s eastern wall to F

P Distance from the city’s western wall to the middle town’s western wall

Q Distance from the city’s northern wall to the middle town’s northern wall

Lm1 Western portion of Lm up to F

Lm2 Eastern portion of Lm from F

K Width of the castle’s western and eastern fortifications

Table 2


Basic Choices for Dholavira’s Geometry

Studying Dholavira’s geometry, I have reached certain conclusions as regards a coherent system to make sense of its fundamental principles and proportions; some of the latter may not appear “natural” to us at first sight, yet I will show that they are without a doubt conscious choices, and will suggest their possible raison d’être. They are also the choices that yield the least error margins when compared to the actual dimensions on the ground.

Let us keep in mind that the city in its mature phase (shown in Fig. 1) did not appear overnight: it evolved from earlier stages through alterations and expansions (detailed in Bisht’s papers); for instance, the dimensions of the castle, the earliest part of the city, were altered when the middle town was added (shortly before the mature phase), so as to bring them in line with the desired proportions. Therefore, more than the exact sequence implied below, it is the choices and their consequences that matter.


(1) The architects, as pointed out by Bisht, decided to have the city’s length and width in a ratio of 5 : 4 or 1.25: Lt / Wt = 5/4. (This is correct to 0.005%!)

(2) They chose the same ratio for the castle, both as regards inner and outer dimensions: Lci / Wci = Lco / Wco = 5/4 (the error margin for the first is 0.9%, for the second 2.4%).

(3) Then they had to set the location of the outer fortifications with regard to the castle. Certainly, the two rivers, north and south of the city, provided a desirable frame, especially in view of the city’s advanced water harvesting system fed by them. In addition, the castle’s north-western corner (F) was regarded as a focal point (probably because it overlooked the ceremonial ground below not far from the latter’s centre) and Dholavira’s planners decided to have F on the city’s NE-SW diagonal, a decision consistent with the ratio 5 : 4, since all points located along this diagonal would exhibit that ratio between their distances to the city’s northern and eastern boundaries. This can be expressed by D / A = 5/4 or again C / B = 5/4. (The actual ratios, measured from the published map of the city, are 1.23, within 1.6% of 1.25).

(4) F’s precise location on this diagonal now needed to be determined. This amounts to choosing a fixed ratio between A and B or C and D, or again between A and C or B and D (any one of those four choices would do). Measurements taken on the map show that A is very precisely 1.5 times C: A / C = 3/2 must have been the architects’ choice in fixing F’s location.

(5) As pointed out by Bisht, they chose 7 : 6 as the main ratio for the middle town: Lm / Wm = 7/6, with a mere 0.5% as error margin.

(6) Now they had to determine the middle town’s dimensions: I propose that its length was chosen to be four-ninths that of the city, i.e.: Lt / Lm = 9/4 (the error margin is only 0.6%).

(7) A strong indication that the proportion 9/4 was deliberately chosen is that we find it repeated between the outer castle and the middle town: Lm / Lco = 9/4, very precisely (0.2%). (This now fixes the castle’s dimensions.) In plain English, the length of the castle is to that of the middle town what the length of the middle town is to that of the city — clearly not an accident. This fundamental principle is called recursion, and we will return to it.


(8) The middle town’s dimensions now fixed, its precise position within the city had to be set. The decision seems to have involved a choice to keep the middle town’s north-west corner equidistant from the city’s northern and western boundaries: P = Q. This is precisely the case as far as the map shows (and is also pointed out by Bisht).

(9) Returning to the castle, with an error margin less than 0.7%, I found its inner length to be three-fourths of its outer length: Lci / Lco = 3/4. The width of the eastern and western fortification walls, K (half of Lco – Lci), being irregular, Bisht takes care to note that the dimensions for Lci and Wci are averages; nevertheless, this attractive ratio is unlikely to be due to chance.

(10) The architects chose the proportions indicated by Bisht for the ceremonial ground (Lg / Wg = 6, verified within 0.7%).

(11) In addition, they chose the ground’s length to be exactly five-sixths of the middle town’s length: Lg / Lm = 5/6 (precisely verified within 0.3%).

(12) Finally, the architects chose the proportions indicated by Bisht for the bailey (1 : 1).

Calculations

With these raw data, the city’s geometry is completely defined; we can now work out other underlying features of Dholavira. As these calculations use only the most basic algebra, I give them only in outline (referring by their numbers to the preceding 12 basic choices). Below are some of the possible results (with the most significant ones emphasized in bold type).

General proportions

(13) (3) and (4) give us A / Lt = 12/23; C / Lt = 8/23; D / Lt = 15/23. We therefore have A / Wt = 15/23 which is within 2.2% of 2/3, a proportion that may have tickled the architects’ hunger for perfect ratios, especially as it applies to the location of F, thus nearly two-thirds down the width of the city, also two-thirds down its length (since D / Lt has the same value).

(14) Using A + B = Wt and C + D = Lt, we get A / B = D / C = 15/8.


Middle town

(15) Using (8) in conjunction with obvious equations Lm1 + Lm2 = Lm, P + Lm1 = C, and Q + Wm + B = Wt, we get P / Lt = Q / Lt = 68/483. A fair approximation (within 1.5%) of this fraction is 1/7.

(16) Another result from the same equations is Lm1 / Lm2 = 75/86, a fraction very close (by 0.3%) to 7/8.

(17) (6) and (11) together yield Lg / Lt = 10/27 (correct to 0.9%).

Castle

(18) (7) and (9) together show that the castle’s internal length is precisely equal to one third of the middle town’s length (the error margin is a mere 0.4%): Lci / Lm = 1/3. This striking ratio must have appealed to the architects.

(19) Combining it with (6), we get Lci / Lt = 4/27. This is just 0.2% away from the figure supplied by the actual dimensions, an excellent agreement.

(20) Another consequence is Lco / Lt = 16/81, verified within 0.9%. (Bisht’s proposed Lco / Lt = 1/5 is 2.1% off, although this is an acceptable approximation.)

(21) Finally, (9) also gives us K = Lco / 8 = Lci / 6. These are highly attractive proportions; the first yields 18.88 m, the second 19 m. The actual average (half of Lco – Lci) works out to 18.5 m, close enough in view of the irregular width of the castle’s fortification walls.

Other cities

A natural question is whether some of Dholavira’s important proportions are reflected in other fortified Harappans cities and towns. Unfortunately, few of them have been fully excavated, few excavation reports are published in full, and accurate data (precise measurements or good maps) are not easily available; indeed from one publication to another we sometimes find different values for the same dimensions. The following observations are therefore a first sampling, subject to correction and intended to stress the need for a systematic study:

The ratio 1 : 2 of Dholavira’s citadel (bailey + castle, both of equal length), is common enough: we find it in the citadels of Mohenjo-daro (roughly


380 x 190 m),4 Harappa (415 x 195 m)5 and Kalibangan (120 x 240 m)6 — it is highly significant that this ratio should be common to at least four major citadels. In addition, it is found in Surkotada’s overall dimensions (130 x 65 m).7

The ratio 1 : 3 between Dholavira’s Lm and Lci is that of Mohenjo-daro’s so-called “college” (69 x 23.4 m).8

The ratio 1 : 6 of Dholavira’s ceremonial ground is reflected in Lothal’s docking yard (average dimensions 216.6 x 36.6 m).9

The ratio 3 : 2 is found in a sacrificial pit at Kalibangan (1.50 x 1 m).10

Finally, Dholavira’s prime ratio, 5 : 4, is the same as Lothal’s overall dimensions (280 x 225 m),11 an important observation in view of Lothal’s relative proximity. But we also find it further afield, for instance in Harappa’s so-called “granary” (50 x 40 m),12 and in a major building of Mohenjo-daro’s HR area (62 x 50 feet or 18.9 x 15.2 m).13

Systematic studies of Indus-Sarasvati cities would surely bring out more such examples. It is clear that ratios were an important part of Harappan aesthetics, partly, as we will propose later, for reasons of sacredness.

Dholavira’s master unit of length

Dholavira’s engineers must have used a precise unit of length to measure or calculate the various dimensions involved. This unit can be calculated without any prior assumption. To do this, we only need to recall the city’s basic proportions and find out whether most of its dimensions can be expressed in terms of whole multiples of such a unit (i.e., without remainders).

Let us name this unknown unit “D” (for Dholavira). The smallest dimension among those we have considered is the average thickness of the castle’s western and eastern fortifications, K. If K can be expressed in terms of D, we will have:

(a) K = nU, where n is an integer.

Now, according to (21) above, we have:

(b) Lci = 6K = 6 nU, and therefore Wci = 4/5 Lci = 24/5 nU.

(c) Similarly, Lco = 8K = 8 nU, and therefore Wco = 4/5 Lco = 32/5 nU.


Let us move to the middle town, using (18) and (b):

(d) Lm = 3Lci = 18 nU.

As regards the ceremonial ground, (10) and (11) above give us with (d):

(e) Lg = 5/6 Lm = 15 nU, and Wg = 1/6 Lg = 5/2 nU.

We end with the lower city, using (6) and (1) with (d):

(f) Lt = 9/4 Lm = 81/2 nU, and Wt = 4/5 Lt = 162/5 nU.

If we look at all these equations together, it is plain that if we are to get whole multiples of our unit D for all these dimensions, we need an “n” which will be the smallest common multiple of the denominators of the fractional results found in (b), (c), (e) and (f). (In plain English, we need an “n” that will cause those fractional results to disappear, leaving just integers.) Since the only denominators are 5 and 2, by choosing n = 5 x 2 = 10, we will have no fractional results. Let us therefore rewrite the above results with n = 10:

(g) K = 10 D

(h) Lci = 60 D, Wci = 48 D

(i) Lco = 80 D, Wco = 64 D

(j) Lm = 180 D

(k) Lg = 150 D, Wg = 25 D

(l) Lt = 405 D, and Wt = 324 D.

In addition, simple calculations establish that:

(m) P = Q = 57.02 D, almost exactly 57 D.

(n) Lm1 = 84 D, Lm2 = 96 D (exact results).

Finally, we have so far left apart the “bailey,” the western part of the city’s citadel, whose inner dimensions are 120 x 120 m according to Bisht, and therefore (with less than 0.3% error margin):

(o) Lb = Wb = 63 D.

It is important to understand that equations (g) to (l) are nothing but the direct consequence of the proportions we worked out earlier: the two sets of equations — the earlier proportions and these multiples of D — are


mathematical equivalents. But this new expression in terms of D brings out Dholavira’s underlying mathematical harmony much more tellingly, as we now see in Fig. 3. Moreover, the simplicity of the new expressions of the dimensions — with a remarkable number of multiples of 10 — is an intuitive confirmation that our scheme of ratios must be the one Dholavira’s planners adopted.

There is one exception, though: we have left Wm out of the above calculations. Wm is six seventh of Lm, and this ratio was clearly as deliberate as the others. Why not include it, then, in our scheme of units? Theoretically, it can be done: since Lm = 18 nU, Wm = 108/7 nU. However, including this equation in the set (a) to (f) would add an inconvenient denominator: 7. To get rid of it, the smallest possible “n” will now have to be 2 x 5 x 7 = 70. We could reformulate all dimensions with this new “n,” but there are two good reasons for not doing so: first, the new multiples (all seven times the ones listed in (g) to (o) ) will be awkward; the scheme’s simplicity and elegance will be gone; second, as I will shortly show, in our previous calculations, D is of the order of 1.9 metres, a unit suited to the measurement of dimensions ranging from the castle to the outer city; if we force Wm into this scheme, the basic unit will now be divided by 7, i.e. about 27 cm, which is hardly convenient to measure out dimensions such as the city’s length (771 m). In a complex scheme like Dholavira’s, with a variety of ratios at work, it is to be expected that some of the dimensions will not have simple expressions; it is remarkable enough that only one should be in this situation. We will therefore be content with Wm = 154.3 D (derived from Wm = 6/7 Lm, and (j) above).


Dholavira's Angula and Dhanus

We must now determine D’s value and see if we can make sense of it.

We can extract that value from any of the equations (g) to (l). It stands to reason that we will get the best result if we choose the ratio that had the least error margin in comparison to actual measurements; this is the case with the city’s outer fortifications (Lt / Wt = 5/4, remarkably accurate to 0.005%), which are also the least disturbed perimeter: the castle or the ceremonial ground underwent successive alterations, which explains the slightly higher error margins there. Now since Lt = 771.1 m = 405 D, we get D = 1.904 m.

If we could now establish a simple relationship between D and a known Harappan unit, it would provide a second confirmation that we are on the right track. It is tempting to try and correlate those dimensions with one of the three known Harappan scales: those of Mohenjo-daro (6.7056 mm), Harappa (9.34 mm) and Lothal, the last suggesting itself as a more appropriate candidate in view of the geographical proximity. Its basic unit was 1.77 mm (as 26


divisions, marked by 27 graduations, cover 46 mm on Lothal’s ivory scale14). Dividing D = 1.904 m by the Lothal unit, we get 1075.7, or, with an approximation of just 0.4%, 1080. This last number can be written 108 x 10, or D = 108 x 10 UL (where “UL” is the Lothal unit of 1.77 mm).

So expressed, D begins to make sense. Indeed, 10 times the Lothal unit, i.e. 17.7 mm, has often been compared to India’s traditional angula (related to the width of a finger), which according to the metrologist V.B. Mainkar was 17.78 mm.15 Let us define a “Lothal’s angula” as AL = 10 UL; we can now write D = 108 AL. In other words, Dholavira’s unit is equal to 108 Lothal angulas (which works out to 1.91 m).

This may appear to be mere jugglery, a simple matter of definitions. But as it happens, “108 angulas” is a traditional measure and concept in classical India. For instance, one of the systems of units described in Kautilya’s Arthashastra (usually dated a few centuries BCE) fits very well in the Dholavirian scheme: in a section on measurements of space and time,16 we are told that “108 angulas make a dhanus (meaning a bow), a measure [used] for roads and city-walls....” This is all the more striking as the second function of that particular unity, to measure out “city walls,” is exactly the case in Dholavira. Our unit “D” stands not only for Dholavira but also for a Lothal dhanus.

Ancient India devised multiple systems of units for a variety of purposes. Units of lengths provide the following examples:

From the Arthashastra to Varahamihira’s Brihat Samhita (of the sixth century A.D.), a hasta, for instance (“hand,” often translated as “cubit”), would usually be 24 angulas, but could also be 27 or more; a danda (meaning a staff), identical to the dhanus, was 4 hastas, and therefore either 96, 108 or more angulas.

The Natyashastra (2.9), in its description of three types of playhouse, states that “their lengths, fixed in terms of cubits as well as dandas, are 108, 64 or 32.”17

The Brihat Samhita (58.30) prescribes a height of 120 angulas for sculptures of Rama and Bali, and 108, 96 or 84 angulas for other deities.18

A little further (68.105), it confirms that the height of a tall man is 108 angulas, that of a medium man 96 angulas, and that of a short man 84 angulas.19 If we use Lothal’s angula (only one of several possible choices in


classical times), we get 1.91, 1.70 and 1.49 m respectively, quite consistent with “tall,” “medium” and “short.”

A classical treatise of Hindu architecture, the Manasara, states (9.9) that “the wise architect should measure with the rod of Dhanurgraha cubits,”20 that is to say, the dhanus made of 4 cubits of 27 angulas, i.e. 108 angulas.

Significantly, 108 angulas has long been one of the principal standard heights for statues of Hindu deities sculpted by traditional Sthapatis.21 According to Isabella Nardi, “Many [early texts] concentrate on the description of an image of 108 angulas in length.”22

This relationship between historical canons and Dholavira’s protohistoric unit of length is too pronounced to be accidental.

I will provide below a few more confirmations of the system of unit we have worked out, but it is now time to sum up Dholavira’s dimensions in terms of D or dhanus as in Fig. 3, and to work out their deviations from the actual measurements. For the purpose, however, we first need to determine D’s value as precisely as possible. Is it 1.904 m, as calculated above from Lt? Or is it 108 x 1.77 cm = 1.91 m, as suggested by the Lothal scale? The difference of just 6 mm might appear insignificant, yet it matters when multiplied hundreds of times to measure out the city’s length. Calculations show that if we adopt 1.91 m, all theoretical dimensions of Dholavira are on the higher side, on average by 1%; this would imply that measurements are all on the lower side, yet Bisht certifies just the opposite.23 1.904 m therefore seems to be a better candidate. Moreover, when the excavation report of Lothal mentions 26 divisions covering 46 mm, it fails to include an error margin (an imperative in metrology); as a result, the figure of 46 mm appears suspiciously rounded off: a difference of 0.2 or 0.3 mm might well have escaped notice. A dhanus of 1.904 m corresponds to a basic unit of 1904 / 108 = 1.763 mm; multiplying by 26 divisions, we get 45.84 mm, just 0.16 mm away from 46 mm and probably within the reading’s error margin. Until the Lothal scale is measured with greater accuracy, it is safer to stick to D = 1.904 m.

Let is now see in Table 3 the deviations this leads to, rounding results to the first decimal:


Length

(in terms of unit D)

Theoretical measurement

(in metres)

Actual measurement

(in metres)

Deviation (%)

Lt 405 771.1 771.1 0.0

Wt 324 616.9 616.85 0.0

Lm 180 342.7 340.5 +0.6

Wm 154.3 293.8 290.45 +1.1

Lg 150 285.6 283 +0.9

Wg 25 47.6 47.5 +0.2

Lci 60 114.2 114 +0.2

Wci 48 91.4 92 – 0.7

Lco 80 152.3 151 +0.9

Wco 64 121.9 118 +3.2

Lb, Wb 63 120 120 0.0

Table 3

With a single exception in the outer width of the castle (built through successive alterations), deviations are extremely modest: 0.4% on average without Wco, 0.6% with it. The system of ratios and units proposed certainly permits a consistent reading of Dholavira’s dimensions. (It is true that the theoretical dimensions remain mostly on the higher side, but now much less so; perhaps some erosion in the fortifications can account for this.)

More on Harappan Units and Ratios

If our system is the one Dholavira’s engineers actually used, we should expect to find its ratios as well as units reflected elsewhere in the Indus-Sarasvati civilization (not necessarily in every dimension, since we know that Harappans used at least three units of lengths). I have shown above that Dholavira’s ratios are indeed reflected elsewhere. As regards units, we have numerous hints that D was a crucial one in Harappan town-planning. (We can now round it off to 1.9 m, since none of the following dimensions are very precise.)

We mentioned above that Mohenjo-daro’s citadel measures roughly 380 x 190 m. That amounts to 200 x 100 D, rather neat figures.


Kalibangan’s citadel is 120 x 240 m (120 m is also the inner dimension of Dholavira’s “bailey”) which is equivalent to 63 x 126 D.

Harappa’s “granary” measures 50 x 40 m, the smaller side being precisely 21 D. (There could be a connection with 63 D found in Kalibangan’s citadel and Dholavira’s bailey, since 63 = 3 x 21.)

We also mentioned the case of a major building in Mohenjo-daro’s HR area: its dimensions, 18.9 x 15.2 m, can be expressed neatly as 10 x 8 D.

Kalibangan’s streets have widths in arithmetic progression: 1.8, 3.6, 5.4 and 7.2 m.24 While those are not multiples of D, I am tempted to suggest that 1.8 m, the basic street width, is close to 1,000 Lothal units or 100 AL (Lothal angulas), the difference being only 3 cm or 1.7%. In that case, the streets can all be expressed in terms of hundreds of Lothal angulas.

Banawali’s bigger streets measure 5.4 m,25 close to 300 AL.

A compelling case comes from Dholavira itself, which must have been an impressive lake-city with reservoirs covering over 10 hectares. One of the largest, just south of the castle, was cut in the rock and measured 95 m x 11.42 m (the width being measured at the top: it is a bit narrower at the bottom).26 These dimensions are exactly 50 x 6 D.

Lothal’s best-known fire altar presents us with an interesting case. It measures 2.65 x 2.74 m.27 The first dimension is almost exactly 150 AL. This may point to a different kind of unit: Baudhayana’s Shulbasutra (1.3) mentions two types of foot (pada): a “small foot” of 10 angulas, and a “big foot” of 15 angulas. The Lothal ivory scale happens to have longer dividing lines every 15 graduations, which (when multiplied by 10) are suggestive of a “big foot.” Let us define a Lothal foot as 15 Lothal angulas: FL = 15 AL

(i.e. 26.55 cm); we can now say that the smaller side of the fire altar is precisely 10 FL, while the bigger side would be nearly 10 FL + 5 AL, i.e. 101/3 FL. I also noted (on the detailed sketches published in the excavation report) that the widest walls in Lothal’s houses are exactly 3 FL wide.

Again at Dholavira, a “ceremonial pathway” descending from the castle to the ceremonial ground has a width of 9 m.28 This same width is found at Mohenjo-daro. Apart from being 5 times the basic street width of Kalibangan and Banawali, it is close to 4¾ D (0.3%), but an equally close expression is 34 FL. Only a careful study of Dholavira’s streets (most of


whose dimensions are unpublished) would determine which of the two units was used to measure them out.

In his 1984 paper,29 V.B. Mainkar explored many other such suggestive correspondences, with brick sizes in particular.

Finally, a remark on the three known basic Harappan units of length: Mohenjo-daro’s (6.706 mm), Harappa’s (9.34 mm) and Lothal’s (1.77 mm). Mainkar30 noted that 10 Mohenjo-daro units (67.06 cm) + 15 Lothal units (26.55 cm) = 10 Harappa units (93.4 cm, correct to a mere 0.2%*). This can be written 10 UM + 15 UL = 10 UH, certainly a conscious connection, since we have already discerned the use of multiples of 10, and also FL = 150 UL. The same equation can be written UH = UM + 3/2 UL.

It would take a thorough study of the dimensions of Harappan bricks, buildings, streets, reservoirs and fortifications to build a coherent and complete picture of Harappan metrology as regards units of length, but we seem to have got hold of a few central principles and units.

Auspicious Ratios and Fractions

However, there remains to explore the reason, if any, behind Dholavira’s chief ratios such as 1 : 2, 3 : 2, 5 : 4 or 7 : 6. Technically, it would have been much simpler to have the castle, the middle and the lower town square; why did the town-planners choose the harder way? I suggest that such ratios were used because they were perceived to result in auspicious or sacred proportions. In fact, it is possible to outline the chain of transmission of some of these ratios.

This is the case with 5 : 4, Dholavira’s key ratio. (I mentioned its triple presence at the start of this paper, and could have added the existence of 5 salients on the northern side of the middle town, against 4 salients on its eastern and western sides.) If we study the description of the trapezium-shaped mahavedi, the main Vedic sacrificial ground, given in the Shatapatha Brahmana,31 we find that its western side is 30 steps long, while the eastern side is 24 steps: 30 : 24 is the same as 5 : 4. In the later Shulbasutras of Baudhayana (4.3) those dimensions are expressed as 30 prakramas (a unit roughly equal to 54 cm) and 24

* The deviation drops to 0.1% if we use a value of 1.763 mm for Lothal’s unit, as suggested by Dholavira’s length (771.1 m divided by 405 units, divided by 108 and by 10). This is one more indication that I was not wrong to suggest that 1.77 mm seems to be on the higher side.


prakramas. It is interesting to note that in Baudhayana’s system of units (1.3), some units are in the same ratio: the pada (= 15 angulas) vis-à-vis the pradesha (= 12 angulas), and the purusa (5 aratnis) vis-à-vis the vyayama (4 aratnis).

But the most striking evocation of Dholavira’s proportions comes from various traditions of Vastu Shilpa. Varahamihira, for instance, writes in his Brihat Samhita (53.4 & 5): “The length of a king’s palace is greater than the breadth by a quarter.... The length of the house of a commander-in-chief exceeds the width by a sixth....”32 These two proportions (1 + 1/4 and 1 + 1/6) are better expressed as 5/4 and 7/6 — very precisely Dholavira’s basic ratios! This is too much of a coincidence. Varahamihira adds that the best of five possible widths for the king’s palace is 108 cubits; of course this cannot apply to a whole city like Dholavira, but it is intriguing that its width Wt, expressed as 324 D, should be three times 108, a number held as sacred in classical Hinduism, initially because of astronomical considerations.33 We find, therefore, number 108 doubly embedded in Dholavira’s town-planning: as one third of the city’s width (in terms of D), and as D = 108 AL.

It is clear that specific proportions regarded as auspicious in Harappan times were carefully preserved and later integrated in a systematic approach to architecture.

Significantly, Varahamihira’s formulations are in terms not of direct, final ratios, but of the addition to the width of a fraction of itself. This addition of a fraction (without which we would have a perfect square) seems to be intended to express a process of expansion, of auspicious increase symbolizing prosperity. Indeed, a whole branch of Shilpashastra, which spells out the classical canons of Hindu architecture, deals with fractions and with ways to create series of them.34 As one example among hundreds, the Manasara (35.18–20) explains how “the length of the mansion [to be built] should be ascertained by commencing with its breadth, or increasing it by one-fourth, one-half, three-fourth, or making it twice, or greater than twice by one-fourth, one-half or three-fourths, or making it three times.”35 Apart from the initial square, this gives the ratios 5 : 4, 3 : 2, 7 : 4, 1 : 2, 9 : 4, 5 : 2, 11 : 4, 1 : 3. Of these, five (the first, second, fourth, fifth and last) are found in Dholavira, which in addition used an increment of one-sixth (as we can see in the ceremonial ground and the middle town).


In the above Manasara example, 9 : 4 is significant: it is the ratio recurring between the lengths of the city and the middle town, and between the lengths of the middle town and of the castle. Now we can see, thanks to the Manasara, that 9 : 4 is intimately connected to 5 : 4: 5/4 is one plus one-fourth, while 9/4 is two plus one-fourth. The classical tradition of fractional increments must without doubt be traced to Harappan architecture.

So is it with the principle of recursion (or repetition of a motif), illustrated by the recurring ratio 9 : 4. We find this principle at work in many Hindu temples, where shikharas of increasing height build up towards the towering last one, or in symbolic representations of the universe such as the Srichakra.36 Finally, let us recall that the ratio 1 : 2, found in several Harappan citadels, including Dholavira’s, is recommended in various texts for villages and towns, but found in practice mostly in Hindu temples.

Dholavira: a Case for Continuity

There are important conclusions to be drawn from the remarkable precision of Dholavira’s planning and execution. We have seen a careful application of ratios and increments intended to create proportions perceived as auspicious. We have reconstituted the system of units used by the town-planners, based on a “master unit” equal to 108 times 10 gradations on the Lothal scale. We have constantly noted the use of multiples of 10, 100 and 1,000, confirming the partial use of a decimal system, which series of Harappan weights had already established. Such considerations are bound to expand our understanding of how much mathematics Harappan engineers were familiar with. No doubt, it might have taken no more than ropes and a few sticks to measure out Dholavira’s fortifications on the ground, but the remarkable precision achieved, despite the terrain’s irregularities, defies explanation at the moment; and the concepts guiding those ropes and sticks around 4,600 years ago were sophisticated by any standard.

I have shown that most of Dholavira’s key principles reappear later in a sacred context (in fact, the whole of Shilpashastra and Vastushastra is regarded as sacred). This allows us to see the city in the same light as historical cities from Sisupalgarh to Vijayanagar, that is, as “the apparent intent ... to interweave, by means of geometry, the microcosm and the macrocosm,” writes astrophysicist J. McKim Malville while commenting on Dholavira’s features.37 However alien to our “modern” mind, this creation of a sacred geography was the primary


concern of the ancient town-planner and architect, from Indus-Sarasvati cities to classical ones.

Indeed, the many points of contact between Indus-Sarasvati and classical architectural units, canons and usages testify to a strong continuity from protohistoric times; that, in itself, is nothing to be surprised at: despite the absurd dogma of a nearly complete cultural break between “pre-Aryan” Indus-Sarasvati civilization and “Aryan” Ganges civilization, there has been a steadily mounting body of evidence of Harappan techniques, crafts, ornaments, art forms, customs, rituals and religious beliefs being transmitted virtually unchanged from one to the other.38

A case in point is that of the early historic city of Kampilya (the capital of Drupada’s southern Panchala kingdom in the Mahabharata), recently identified with a complex of ruins called Drupad Kila, a few kilometres away from the modern village of Kampil (in western Uttar Pradesh). In the 1990s, an Italian team led by Indologist Gian Giuseppe Filippi measured a rectangle of 780 x 660 m, with the longer side oriented east to west; it noted, not without surprise, that this was almost identical to the layout of Dholavira (which was being excavated at the same time): 771 x 617 m, with the longer side in the same direction.39 Kampilya may indeed provide one more link in this long chain.

Be that as it may, the continuity in architectural traditions which Dholavira bears witness to will certainly strengthen archaeologist Jim Shaffer’s thesis of a strong connection between Harappan urbanism and the urbanism of historical times, which he finds connected by “a unique cultural tradition traceable for millennia.”40

* * *

Acknowledgements

I am grateful Prof. B.B. Lal and Dr. S.P. Gupta for their encouragements; also to Prof. Subhash Kak, who has researched underlying principles of Hindu architecture, for his useful comments on an earlier draft, in particular regarding the concept of recursion. Deep appreciation is also due to Dholavira’s team of excavators, without whose careful measurements no precise analysis of Dholavira’s geometric scheme would have been possible.


References & Notes 1 Bisht, R.S., 1997. “Dholavira Excavations: 1990–94” in Facets of Indian Civilization — Essays in

Honour of Prof. B.B. Lal, ed. J.P. Joshi (New Delhi: Aryan Books International), vol. I, pp. 107–120. ___, 1999. “Dholavira and Banawali: Two different paradigms of the Harappan Urbis forma” in

Puratattva (New Delhi), No. 29 (1998–99), pp. 14–37. ___, 2000. “Urban Planning at Dholavira: A Harappan City” in J. McKim Malville & Lalit M.

Gujral, Ancient Cities, Sacred Skies: Cosmic Geometries and City Planning in Ancient India (New Delhi: Indira Gandhi National Centre for the Arts & Aryan Books International), pp. 11–23.

The last two papers delve more particularly on Dholavira’s geometry. 2 R.S. Bisht proposes some more ratios, but they do not form a consistent system with those

mentioned here, and most have high error margins; I have therefore not discussed them here. 3 Bisht, R.S., 2000, p. 18. 4 Lal, B.B., 1997. The Earliest Civilization of South Asia. (New Delhi: Aryan Books International), p.

106. 5 Ibid., p. 112. 6 Ibid., p. 122. 7 Ibid., p. 135. 8 Ibid,. p. 108. 9 My calculations of those average dimensions are based on the dimensions of the four sides

provided by Rao, S.R., 1979, Lothal, a Harappan Port Town (1955–62) (New Delhi: Archaeological Survey of India), vol. 1, p. 123.

10 Lal, B.B., 1998. India 1947–1997: New Light on the Indus Civilization (New Delhi: Aryan Books International), p. 96.

11 Lal, B.B., 1997, p. 129. 12 Kenoyer, Jonathan Mark. 1998. Ancient Cities of the Indus Valley Civilization (Karachi & Islamabad:

Oxford University Press & American Institute of Pakistan Studies), p. 64. 13 Madhukar K. Dhavalikar & Shubhangana Atre, 1989. “The Fire Cult and Virgin Sacrifice: Some

Harappan Rituals,” in Old Problems and New Perspectives in the Archaeology of South Asia, ed. Jonathan Mark Kenoyer (Wisconsin: University of Wisconsin,), pp. 195–197.

14 Rao, S.R., 1979, vol. 2, p. 626. (S.R. Rao commits the surprisingly common mistake of dividing the span of 46 mm by the 27 graduations; but it is by the 26 divisions formed by the graduations that 46 mm must be divided.)

15 V.B. Mainkar, 1984. “Metrology in the Indus Civilization” in Frontiers of the Indus Civilization, eds. B.B. Lal & S.P. Gupta (New Delhi: Books and Books), p. 147. Note that the angula certainly had varying values; it is enough for us here that the “Lothal angula” was of the same order.

16 Arthashastra 2.20.19. See The Kautilya Arthasastra, tr. Kangle, R.P., 1986 (New Delhi: Motilal Banarsidass), part II, p. 139.

17 Natyashastra, 2.19. See Ghosh, Manomohan, 1967 (Calcutta: Granthalaya, 2nd ed.), vol. 1, p. 19. 18 Varahamihira’s Brhat Samhita, trans. Ramakrishna Bhat, M., 1981 (New Delhi: Motilal

Banarsidass), vol. 1, p. 556. 19 Ibid., p. 642. 20 Acharya, Prasanna Kumar, 1934. Architecture of Manasara (republished New Delhi: Munshiram

Manoharlal, 1994), vol. IV in Manasara Series, p. 63.


21 See for instance: Ganapathi Sthapati, V., 2002. Indian Sculpture and Iconography: Forms &

Measurements (Pondicherry: Sri Aurobindo Society & Ahmedabad: Mapin Publishing), part III. 22 Nardi, Isabella, 2006. “On Measuring Images: a Critical Analysis of the Theory of Talamana,” in

Sahrdaya: Studies in Indian and South East Asian Art in Honour of Dr. R. Nagaswamy, eds. Bettina Baumer, R.N. Misra, Chirapat Prapandvidya & Devendra Handa (Chennai: Tamil Arts Academy), p. 260.

23 See note 3. 24 Lal, B.B., 1997, p. 127. 25 Ibid., p. 127. 26 Bisht, R.S., 1999, p. 28. 27 As measured on the sketch in Rao, S.R., 1979, p. 97. 28 Ibid., p. 31. 29 Mainkar, V.B., 1984. Unfortunately, some of his calculations had rather high error margins;

moreover, he worked out a wrong value for the Lothal unit (see note 14 above). 30 Ibid., p. 146 (but Mainkar’s deviation is 0.8% because of his wrong value for the Lothal unit). 31 Shatapatha Brahmana, I.1.2.23, quoted in The Sulbasutras, Sen, S.N. & Bag, A.K., eds., 1983 (New

Delhi: Indian National Science Academy), p. 170. 32 Varahamihira’s Brhat Samhita, op. cit., vol. 1, pp. 451–52. 33 Kak, Subhash, 2000. The Astronomical Code of the Rgveda (New Delhi: Munshiram Manoharlal), pp.

101 & 124. 34 Pillai, Govind Krishna, 2004. Hindu Architecture (Silpa-Sastra) (Delhi: Bharatiya Kala Prakashan),

chapters 5 & 8. 35 Acharya, Prasanna Kumar, 1934, p. 374. 36 Kak, Subhash, 2006. “Art and Cosmology in India” (Patanjali lecture given at Center for Indic

Studies, University of Massachusetts, Dartmouth). 37 Malville, J. McKim, & Gujral, Lalit M., 2000. Ancient Cities, Sacred Skies: Cosmic Geometries and City

Planning in Ancient India, op. cit., p. 3. 38 See for instance:

Kenoyer, Jonathan Mark, 1998. Ancient Cities of the Indus Valley Civilization (op. cit.);

Lal, B.B., 2002. The Sarasvati Flows On: The Continuity of Indian Culture. New Delhi: Aryan Books International; Danino, Michel, 2003. “The Harappan Heritage and the Aryan Problem” in Man and Environment, (Pune), vol. XXVIII, No. 1, pp. 21–32.

39 Filippi, Gian Giuseppe, & Marcolongo, Bruno, eds, 1999. Kampilya : Quest for a Mahabharata City (New Delhi: D.K. Printworld), pp. 19 & 39. Note that Kampilya’s main ratio is 1.18 (13/11), which is 5% off 5/4. This “deviation” could be the result of a deliberate choice for Kampilya or could have other causes; we must suspend our judgement until the city is excavated.

40 Shaffer, Jim G., 1993. “Reurbanization: The Eastern Punjab and Beyond” in Urban Form and Meaning in South Asia; the Shaping of Cities from Prehistoric to Precolonial Times, eds. H. Spodek and D.M. Srinivasan (Washington D.C.: National Gallery of Art & University Press of New England), p. 54.

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Unravelling Dholavira's Geometry - Michel Danino