Tuesday, September 7, 2010

Is a Circle a Square or a Diamond?

In the Koskenniemi and Parpola concordance, inscriptions are shown in columns that center on the shared symbol (1982). Signs designated 355 in their list include those I term CIRCLE, SQUARE, and DIAMOND.  The CIRCLE appears twice, in two sizes, which I will term SMALL CIRCLE for the small version and simply CIRCLE for the larger one.  To distinguish the various shapes given the same KP number, we can give them small letters thus: KP355a SMALL CIRCLE, KP355b SQUARE, KP355c DIAMOND, KP355d CIRCLE.

Anti-Evil Eye circle
Wells shows two versions also, but distinguishes them by shape, not size.  W382 is round and thus a proper CIRCLE while W348 is pointed at top and bottom and thus more accurately termed an OVAL.  Fairservis distinguishes two versions in addition, making his distinction by both size and shape.  F-1 is a tall and pointed oval, representing the sun, meaning "to set; to rise, do, obtain."  F-15 is a smaller and more rounded version associated with partnership which Fairservis suggest is always found with animals.  Mahadevan gives the sign the number Mh373 and groups CIRCLE, DIAMOND, and SQUARE together as do Koskenniemi and Parpola.

According to the Koskenniemi and Parpola concordance, the SMALL CIRCLE appears in first position 8 times, all alone in the second line, and before the following symbols: MAN HOLDING POST; QUADRUPLE DOTTED EX; BACK CEE; SPACESHIP; MALLET; 3 TOED FOOT; and POT. Two such circles appear alone and initially in combination 4 times, before BATTERY (and again this way in line 2 of the same inscription); and before BOAT & BENT OAR.

In medial position, the SMALL CIRCLE appears 15 times, before BATTERY; DOWN JAY followed by another SMALL CIRCLE (twice); PRAWN; SINGLE POST followed by another SMALL CIRCLE (twice); SINGLE POST followed by an abraded area; DOUBLE POSTS; BACK CEES (6 times); and before CUP. Doubled in medial position, it appears 3 times, preceding POT twice, and once before VEE IN DIAMOND. In these same inscriptions it follows POT; SPEAR (twice); STRIPED TRIANGLE; RAKE; MAN HOLDING CUP; FISH; BELTED FISH; QUINT-FORK; TRIPLE POSTS; BACK CEE; BI-QUOTES; and also appears after an abrasion. Duplicated in medial position twice, it follows CUPPED SPOON; and DOWN JAY.

In final position 8 times, SMALL CIRCLE follows TRIDENT; COMB; SKEWERED CHEVRON; QUINTUPLE QUOTES; DOWN JAY; and POT (three times). The duplicated SMALL CIRCLE only appears in final position when its duplication comprises the whole of the inscription.


The larger CIRCLE is the same size as other symbols and appears less often in the KP concordance.  In initial position, CIRCLE appears six times, preceding TRIFORK (twice); SQUARE CORN HOLDER; CHEVRON; HAIRY HUNCHBACK; and BATTERY. In medial position, it occurs six times, preceding RAKE; DOUBLE BACK CEES; AITCH; and an abrasion. In these inscriptions, it follows BACK CEE; MAN HOLDING STICK; BI-QUOTES; POT; and an abrasion.


SMALL SQUARE appears seven times. It follows MAN HOLDING STICK; POT; MAN HOLDING QUOTE; and DOUBLE POSTS. It stands in first position three times, preceding SINGLE POST (twice); CHEVRON; BACK CEE; BATTERY; TRIPLE BELTED AITCH. It does not occur in final position.


DIAMOND is more common than the preceding, occurring 22 times. It follows DOUBLE POSTS (four times); BENT HORN; MAN HOLDING STICK (five times); STRIPED MALLET (four times); and SINGLE QUOTE. It is in initial position seven times, preceding MAN; BUD; SHISH KEBAB TOPPED DUBYA; LOOP TOPPED DUBYA; BI-QUOTES; SINGLE POST (five times); DOUBLE POSTS; DOUBLE POSTS & DIAGONAL; CHEVRON (four times); EX UNDER TABLE; MALLET; AITCH (twice); and POT. Like SMALL SQUARE, it does not occur in final position.


Of these proposed variants, only the SMALL CIRCLE is found in final position. It appears in all three positions, initial, medial, and final. The others – CIRCLE, SQUARE, and DIAMOND – appear initially and medially only. SMALL CIRCLE is also the only one of these four that occurs in duplicated form. If we look at the signs that these “variants” precede, we see the SMALL CIRCLE, SQUARE, and DIAMOND before the SINGLE POST, but CIRCLE does not occur before that symbol. SMALL CIRCLE and DIAMOND occur before DOUBLE POSTS, but that is not true of SQUARE or CIRCLE. SMALL CIRCLE and CIRCLE both appear before DOUBLE BACK CEES, but SQUARE only precedes the single CEE, and DIAMOND is not seen before either sign. SQUARE, DIAMOND and CIRCLE precede the CHEVRON, but SMALL CIRCLE does not. DIAMOND and CIRCLE precede AITCH, but SMALL CIRCLE and SQUARE do not. The symbols in this group, therefore, do not unequivocally appear in precisely the same positions, although some of their co-occurrences overlap. Since none occurs particularly frequently, this could be just a result of chance. But the hypothesis that they are variants of a single symbol is not strongly confirmed by position.

One might hypothesize reasons for the variations among these possible variants and test these against occurrences. Variation might be based on geography, for example. Scribes in the southern area of the Harappan zone might prefer, say, the rounded variants, SMALL CIRCLE and CIRCLE (as well as Wells' oval version not separately classified by Koskenniemi and Parpola), whereas their northern neighbors might prefer the angular variants, DIAMOND and SQUARE. Or, perhaps the DIAMOND might be found mainly in the east, the SQUARE in the west.

Since Wells provides data on which seals contain each symbol, showing provenance, one can easily see which symbols occur where and how often. However, the SMALL CIRCLE does not appear in this list as a separate symbol.  I examined the photographs of the circles noted.  It appeared to me that the one on Dlp-2 is the largest and roundest, although it has a noticeable point at top and bottom.  Four were distinctly more slender and more pointed at top and bottom: C-22, H-668, H-841, and H-842.  Quite thin and pointed were five others: K-11, H-664, H-764 (small), M-131, and M-121.  Both round and small, there were these five: H-479, M-724, M-157, and K-40 (two circles on this seal).   My judgment of these shapes does not tally with either that of Wells or of the other scholars.  While I hardly expect anyone to take my word for what is a circle and what is an oval, I do think this goes to show there is indeed a bit of variation going on here.

Relying on Wells' statistics, we can say that the DIAMOND occurs 10 times at Mohenjo daro, once at Harappa, and once at Chanhujo daro (W392). The SQUARE appears only once, at Harappa (W526) (a bit of embarrassed throat clearing here as I point somewhat nervously at what seems to be another small SQUARE on L-80, perhaps a badly executed one on M-412, maybe even a couple on M-1529 if those aren't grids with their internal markings washed out). Thus, two “variants” occur at Kalibangan (SMALL CIRCLE and CIRCLE). A different pair of variants appear at Mohenjo daro (CIRCLE and DIAMOND). Three “variants” are found at Harappa (CIRCLE, DIAMOND, and SQUARE). The hypothesis of strictly regional variants is not confirmed.

We may wish to back up a moment and determine whether or not the CIRCLE and OVAL are really the same thing, as I've been assuming.  Let us hypothesize that the CIRCLE alternates with the thin, pointed OVAL (no KP version shown) for reasons of economy of space. That is to say, W382 and W348 the round circle and the pointy oval both = CIRCLE.  We would look to see whether circular forms are found more often in short inscriptions or where there are no signs of crowding. Where there are more symbols in an inscription and these appear cramped, perhaps KP's SMALL CIRCLE and Wells' OVAL versions appear, in order to take up less space.

In favor of such a proposal, we can point to M-777, in which two very small and thin ovals appear on either side of a DOWN JAY, in an inscription with four other symbols (Shah and Parpola 1991). This adds up to a relatively lengthy seven symbols, unless Wells is correct that CIRCLE/OVAL – DOWN JAY – CIRCLE/OVAL together comprise a single symbol. Here, space seems quite limited, the symbols cramped. In M-724, a small and thin oval appears underneath BI-QUOTES along with three other larger symbols. The size, shape, and placement of the oval beneath the quotes seem to represent an effort to fit everything neatly in a small, cramped space.

In M-786, a small and thin oval is placed over the head of the bovine icon, before an inscription containing four larger symbols. Again, size, shape, and placement of the oval seem intended to consume the least space possible. In M-1173, the sign on the far right before the break is partly abraded, but this appears to be a very pointed oval after four other symbols, each of which is made tall and thin. In M-1457, a small but rounded oval, with only slightly pointed ends, appears at the far left in the top row of an inscription containing three more signs in the top row and four more in a second, lower row. In this copper tablet, the initial sign is the smallest one, although all of those in the top row are smaller than those in the second row. The “scribe” seems to have made an effort to use up as little space as possible in that top row, perhaps anticipating running out of room for the second row.

Another copper tablet, M-1498, has a small, thin, and quite pointed oval at the far left, with about five more signs on the right. Four verticals joined by a horizontal at the bottom, the HAIRPICK, has five shorter strokes above it, not particularly neatly lined up (QUINTUPLE QUOTES?). Making the left-most sign oval in shape, long, and thin seem to be ways of saving space once more. Stacking two symbols may be another space-saving maneuver.

In contrast, on M-1644, a stoneware bangle, both the overlapping circles on the left and the single CIRCLE on the right are very round. Here there is plenty of room for the brief inscription on this large object and the “scribe” appears to have taken advantage of this. A pot impression, M-1375, also has a fairly round and full-sized CIRCLE, preceding five verticals and a “V” shape with a central post (DUBYA). The bar seal M-1332 contains just four symbols of approximately equal height and girth. The circular sign has been chipped here, making it impossible to identify precisely, but its round shape is clear enough. In each of these cases, where there is no problem of running out of room, full-sized, circular symbols appear rather than thin ovals or the smaller circle.

However, on seal M-1092, there are only two symbols and plenty of room. Even so, the “circular” symbol on the left (broken off at the top) has a definite point at the bottom. The oval on M-1205, side F, is by itself and has even more room. It is still a pointed oval, not a fully rounded circle. On M-1630 and M-1631, two stoneware bangles, the inscription has plenty of room again, and while the initial circle is round, it is quite a bit smaller than the other signs. It is hard figure out what was going on in the "scribe's" mind here. 

Copper tablet M-1540 contains three “circular” signs, along with three or four others. On the far right, a tall and wide circle contains a trident. Beside it on the left are two overlapping “circles” with three backslashes inside. Past a sign resembling an upside down “y” and another “y,” right-side up this time, there is a third “circle,” perhaps with a central dot. All three of these rounded signs are pointed at the top and bottom and were apparently drawn with two strokes, one on the right and one on the left. Thus, the evidence remains slightly equivocal on the hypothesis of OVAL and SMALL CIRCLE as variants of CIRCLE. It appears to me that “scribes” generally must have drawn circles with two lines, going from the top to the bottom. Such a practice would tend to create somewhat pointed figures even where there was room to make them nice and round. 

Thus, we may tentatively consider size to be mainly a function of space, insofar as explaining SMALL CIRCLE and CIRCLE as variants. We may also tentatively state that the oval versus circular shapes are either a result of a writing style using two strokes (and thus inadvertent), or a space-saving measure (and thus intended). Again, the limited number of occurrences of each “variant” does not allow for a statistical test, but close analysis can suggest lumping together at least the circles and ovals of various sizes.

One might then hypothesize further that the pointed oval could easily become a DIAMOND if it were easier to carve straight-sided figures than curved shapes on seals. In favor of such a hypothesis, several proto-Elamite symbols appear similar to Harappan symbols except that the proto-Elamite ones are angular while the Harappan ones are curved (e.g., proto-Elamite M260 resembles the Harappan FISH). If ease of carving were the case, perhaps older artifacts would evince mostly curved figures (circles and ovals), while those made in later periods would be more likely to contain straight-sided figures (squares and diamonds). Alternatively, hard objects like seals and stoneware bangles might bear mostly straight-sided figures, while pots would have mostly curved symbols since they could be marked while damp and malleable. Again, we are faced with the fact that there are insufficient numbers of each of the basic signs on all artifact types, so we cannot statistically distinguish all these possible “variants.” Nor is there information about comparative age of various inscriptions in either the Corpus or Wells. The hypothesis is not easily tested, then, at least not directly.

But we might try a logical test.  For example, if we assume that a circle, a square, and a diamond are essentially the same sign as basic symbols, separated only by time or age, we might also assume that the same rule would also apply to more complex symbols. In several instances, one of these basic geometric shape contains another symbol as a form of ligature.  For example, there is a VEE IN DIAMOND symbol (W390; 105 occurrences).  We would expect to see a VEE IN CIRCLE variant (W341; 127 occurrences). Both of these receive the same identifying number of KP385, indicating Koskenniemi and Parpola’s view that they are variants of a single symbol, just as the basic circle and diamond are classed as one sign. And we would expect to see a VEE IN SQUARE as well, if CIRCLE, SQUARE, and DIAMOND are truly variants.  But there is no KP VEE IN SQUARE or in Wells' list.  Mahadevan mentions one, giving it the same number as the VEE IN DIAMOND, Mh267.  On a broken seal from Banawali there is something that may be a VEE IN SQUARE (B-18), but then it may be another of the tiled VEE IN DIAMOND signs (e.g., L-29, M-72, M-142, M-1152).  One other possibility is the exceedingly dim and difficult-to-read tablet M-1428.  But since this is so very hard to make out, one cannot really prove the case with this single example either.

Again, there is an oval containing four “v” shapes (FAT EX IN CIRCLE, W223, 11 occurrences), as well as a diamond-shaped “variant” (FAT EX IN DIAMOND, W391, 14 occurrences). Once more, however, there is no *FAT EX IN SQUARE. As a third example, let us consider the sign resembling a spoked wheel (CARTWHEEL). If squares and diamonds, alongside circles and ovals, are all variants of a single symbol, one would expect not only a spoked circle, but also a spoked diamond and a spoked square (or, in other terms, a six-legged asterisk inside a circle, a diamond, and a square). Wells shows oval and circular variants as versions of a single sign (W342, 125 occurrences), just as there is only one KP number (378). Once more though, there is no *ASTERISK IN SQUARE. This time there is no *ASTERISK IN DIAMOND either.

There are more examples along these lines.  The SQUARE is found with a vertical line perched on top, making a MALLET.  Is there a DIAMOND MALLET?  No.  Is there a CIRCLE MALLET?  No.  If we flip the image upside-down and have the geometric shape on top and the line coming down from it, then we can find something similar for the CIRCLE.  That is, if we put another circle inside our first circle, making a DONUT and then put the line beneath it, then we have the DONUT LOLLIPOP.  But that would make us expect a diamond inside a diamond as another type of lollipop.  This we do not find.  Instead there is a diamond without anything inside but with five prongs coming out the top, the HAIRY DIAMOND LOLLIPOP.  Besides that, the square does not participate as a lollipop, but its close relative, the rectangle does.  This is the HAMMER.

There is the ever-popular trident inside a circle and it is found inside a diamond, invarious positions.  But we don't see it inside the square.  Instead, it sits on top of something that may or may not be a square.  It may only be the battery -- an open, three-sided relative that didn't quite make it to square-dom.  And so it goes.  Where we find three little circles stacked in row, we fail to find three little diamonds or three little squares.  There is a dot in a circle but no dot in a diamond or dot in a square.  There is a hammer in a diamond, but no hammer in a square or hammer in a circle.  Quite simple, while there are occasional overlaps, the shapes are not completely interchangeable. 

Our tentative hypothesis of square, circle, and diamond being variants of each either is not finding much positive evidence, then, although there is (somewhat equivocal) evidence in favor of circles and ovals of large and small sizes being equivalent. Let us try one more hypothesis. If a “scribe” preferred a diamond-shaped variant for one symbol, it stands to reason that he would also prefer diamond-shaped variants for other symbols in the same inscription, at least where these angular types were an option. It seems reasonable to make such an assumption, even though we cannot fathom the reason for the preference.

Unfortunately for this last hypothesis, we can easily find examples where diamond shapes and circular shapes appear in the same inscription. Following our earlier suggestion, we will eliminate some possible variables, first, by examining only the data from a single site, Mohenjo-daro. In M-726, the inscription begins with a pointed, oval CARTWHEEL and BI-QUOTES, after which there is another pointed, oval sign, TRIDENT IN CIRCLE. So far so good. But after four more symbols, the DIAMOND appears near the end. Perhaps we could save our hypothesis of personal preference by suggesting that the seal carver was getting tired as he approached the end of this inscription. Could that be why he resorted to an easy, straight-sided figure after laboriously carving two rounded ones?

However, M-714 has the VEE IN DIAMOND in second position, followed by SINGLE QUOTE and then TRIDENT IN CIRCLE (unless it is the simpler PACMAN). Similarly, in M-724, the first sign is VEE IN DIAMOND, followed by the common end-of-prefix sign, BI-QUOTES, over a SMALL CIRCLE. These show the reverse of the order predicted by our feeble “weary seal carver” hypothesis. He surely would not have started off tired and become rested as he went along.

If rounded shapes and angular shapes are really equivalent in their basic form, one is hard pressed to develop a hypothesis to explain the seal carver having the option to make one sign either straight-sided or curved, but being forced to make other, similar signs always curved or always angular. Since specific symbols must be rounded and other specific symbols must be angular, depending on internal or external “diacritics,” it is clear that curved and angular forms were distinguished at some level.  There are other cases, as well. The same VEE IN DIAMOND and TRIDENT IN CIRCLE also appear together in M-806, separated once more by a single sign. VEE IN DIAMOND also appears at the beginning of M-894, CIRCLE appearing at the end.  Here again, we cannot assume that a weary seal-carver switched from the easy method (diamond shape) to the difficult (rounded shape).

Worst of all for this hypothesis, M-855 contains both VEE IN DIAMOND and VEE IN CIRCLE (not to mention DOT IN CIRCLE as well, another symbol that has no diamond-shaped or square variant). Both rounded signs and diamond-shaped ones co-occur on M-656, M-852, M-889, and M-894 in addition.

Remember, we had decided originally just to consider the examples from Mohenjo daro. So perhaps all this is a peculiarity of this one site?  Alas, no.  Similar data comes from seals from other sites as well. VEE IN DIAMOND and TRIDENT IN CIRCLE appear in a seal from Harappa, H-135A and the list goes on.  Thus, it is not simply a matter of one site including both types in a single inscription, either. Thus, it does not seem likely that ease of carving or personal preference determined the shape of the “variant.” In fact, we are hard pressed to come up with any more hypotheses to explain the circles, squares, and diamonds as variants of a single symbol. Hence, we conclude that the circular and oval shapes are the only true variants, while the square and the diamond are both independent.

Parpola proposes the meaning “egg” for the circle/oval (2009: 66). He admits, “The square-shaped graphic variant does not agree so well with the ‘egg’ hypothesis, but is not an insuperable difficulty, as the square shape of the Chinese sign for ‘open mouth’...shows” (2009:67).

Let us now consider other writing systems, following Parpola’s lead, and take Chinese first. It is true that circular or oval shapes formerly appeared in Chinese, both in oracle bone writing and in Old Seal writing, as noted in the previous section. The modern character for kou3 represents the mouth, written as a small square, just as Parpola points out (Wieger 1965: 180). Earlier, however, this was not a square, but a “U”-shape with a horizontal across near the top. Occasionally, instead, it took a “Y” shape with that same horizontal. Another character, wei2, was originally a large circle, meaning “a round, a circumference, an inclosure, to contain” (Wieger 1965: 188). It is now a rectangle, typically somewhat larger than the “mouth” symbol. As a radical, the enclosure often contains other symbols, where the smaller mouth generally appears beside others in compound characters. Other examples would be simple to show as well, all with the same pattern: the sun, the field, the eye, etc. In each case, a rounded symbol originated in oracle bone writing and continued in Old Seals, but is now invariably square in calligraphy and printing. Thus, these two basic geometric shapes are only equivalent diachronically in Chinese. They do not occur synchronically in the same text.

In ancient Egyptian, there are a number of circular signs: D12 (pupil of the eye), N5 (sun), N9 (moon), N10 (moon), N15 (star), N33 (grain of sand), (48 (building), O50 (threshing floor), S21 (ring), X6 (loaf of bread), Aa1 (placenta).  There are ovals: D21 (mouth), H8 (egg), O47 (building at Hieraconpolis), V38 (bandage), Z8 (determinative round).  There are also rectangular glyphs, most of which represent buildings (O39, N37, Q3, etc.). Each of these, as well as other similar shapes, is distinct from the others. Shape, size, orientation, and the presence of internal or external elements are all significant in Egyptian.

In early Sumerian proto-cuneiform, there are round symbols (circles or, less often, ovals), diamonds, and rectangles, among other shapes (Schmandt-Besserat 1992: 72-78). A circle within a circle represents a lamb, a crossed circle indicating a grown sheep. A diamond (as well as a diamond within a diamond) represents something sweet, perhaps honey. A square with an internal square – as well as some extra short horizontals on the edges – denotes a type of mat or rug, while a rectangular sign with an internal tic-tac-toe marking means “wool, fleece.” In the earliest script of southern Iraq then, both shape and size were significant just as in Egyptian, as well as internal markings.

Somewhat later, the circle becomes LAGAB, "black, slab (of stone), trunk (of tree)."  The oval with its pointed ends, on the other hand, becomes KI, "earth, place, area, ground."  In proto-Elamite the only type of circle was made by impressing the round end of the stylus into the clay (M351).  There is another shape similar to a diamond but elongated like some particularly horrible pills I once had to take for a long time -- M296.  Perhaps this shape was the local angular equivalent of an oval.

Taken together, these internal and external lines of evidence do not support the hypothesis that the circle, diamond, and square are variants.

In later Luwian hieroglyphs, a circle or oval came to mean FEMINA/MATER, or "woman/mother."  Certainly in this script one could not substitute a square to mean the same thing.

Circles appear in North American rock art (Newcomb 1996: 154, Pl. 107 no. 14; 155 Pl. 108 no. 3 and no. 7; 156 Pl. 109 no. 7; and Heizer and Baumhoff 1984: 101 fig. 38c; 157 fig. 94 a).  The latter authors note that circles appear as a motif 443 times, quite common motif.  This is not counting the times they appear in clusters or chains, also common motifs.  However, in the Southwest as in China, the time period makes a difference.  During an early period the style of rock art includes mostly curvilinear forms, so this is called the Curvilinear Style.  The later period is characterized by the Rectilinear Style.  In this later period, squares and diamonds largely replace the motifs with circles and ovals.  Although I have little to base this on, I suspect that the difference has to do with weaving.  Australian art largely lacks the rectilinear motifs.  The earlier North American art lacks rectilinear motifs.  The European Paleolithic art also seem to lack rectilinear motifs to a remarkable degree.  I have not observed them in the art of the South African San either.  None of these people were weaving, as far as I am aware.  However, once weaving is introduced, this medium forces rectilinear patterns into existence because of its nature.  And this, I suspect, brings them into other areas such as rock art.  It's only a hypothesis.  But that may explain why the diamond, for example, is relatively rare compared to the circle.

As for the circle, it appear rock art of Australia mutiple times at Mount Yengo Rock shelter, New South Wales (Flood 1997: 200).  Circles also appear at Sundown Point in northwest Tasmania (1997: 233).  Interestingly, circles and lines were among the first motifs to appear in the drawings of my preschool children.  And when a certain little tyke of two years made a great mass of circles in purple on a sheet of typing paper one day and I asked what it was, she boldly announced, "A kangaroo!"  I proudly posted the "kangaroo" on the refrigerator.  The next day I pointed it out to her and reminded her that I had put her kangaroo on the fridge.  She frowned at it and said, "That's not kangaroo!  That's Papa's belt!"

So what do kangaroos and belts have in common?  They're evidently both round somehow.  And so are squares and diamonds!

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