Monday, September 27, 2010

Circle, Square, and Diamond

The scholars Koskenniemi and Parpola consider as variants of a single sign the following symbols in the Indus script:  KP355(a) SMALL CIRCLE, KP355(b) SMALL SQUARE, KP355(c) DIAMOND, and KP355(d) CIRCLE (full-sized).  The fact that these authors use the same identifying number for all four symbols reveals the fact that all these are a single sign with four variations in their view. 

The RECTANGLE appears in the top seal (above);
the VEE IN DIAMOND in the seal impression in the central illustration;
and in the bottom illustration, the Egyptian egg is visible,
a tilted oval, a determinative indicating a goddess.
While I find it reasonable to consider the small and large circles variants of one sign, it seems odd to include such distinct geometrical forms as the square and diamond shapes in the same category.  The CIRCLE, as seen in the discussion of the two-stroke sign II9, also has pointed oval variants, but while these occasionally become slightly angular, perhaps to be confused occasionally with a diamond, they remain visually distinct from squares.
The square and diamond are both angular, so perhaps it is only a matter of how one tips an object as to whether it is one or the other.  On one seal (H-6), there is a slightly tilted square, and perhaps one might be inclined to conclude from this that there is no real distinction between squares and diamonds.  On the other hand, there is a rectangular sign also, so perhaps the square should be considered the small variant of the larger rectangle sign, just as the small circle is the small variant of the larger oval or circle.  But lumping rounded signs together with angular signs seems counterintuitive to me.
What do the other list-makers say?  Iravatham Mahadevan seems to agree with Koskenniemi and Parpola in considering the four symbols previously mentioned to be variants of a single sign.  That is, the small and larger circles, the square, and the diamond are again variants of a single symbol.  In a way, Mahadevan goes further in one article, including as additional variants the circle and diamond with a small “V” shape attached inside, the whole series designated Mh261 through Mh373 (2009: 5).  Since I lack access to his original list of signs, I can give no more detail.
However, not only does Wells not agree with Mahadevan’s larger grouping, he lists and enumerates the original four in this group separately as well:  the rounded version of the CIRCLE is W382; the more pointed OVAL W348; SQUARE W526; and DIAMOND W392 (Wells 1998: appendix).  For good measure, RECTANGLE is also separate, W522.  Fairservis takes the middle ground, grouping the circles together but not the others: F-15 (SMALL CIRCLE) and F-1 (CIRCLE) both mean “sun, day” for him.  But then he has K-14 (SMALL SQUARE) which he defines as “one-quarter.” Two variants of the DIAMOND are N-4 and N-5, one just slightly rounded and hence really an OVAL, while the other is quite angular and thus truly a diamond.  These, he says, are classifiers for types of settlements. 
One might assume that we could determine which signs were variants, initially by classing the signs together based on graphic similarity, then by verifying this grouping by means of positional regularities in inscriptions.  Most of the signs in the Indus script are not particularly frequent, though, a feature of the particular symbols under consideration here.  Wells provides the information that a rounded circle occurs only once, at Kalibangan.  I see the small and round variant twice in the same inscription at Kalibangan (K-11), as well as on H-479 from Harappa, and in two inscriptions from Mohenjo daro, M-157 and M-724, making five occurrences (my “d” variant of II9).  The larger but less rounded version appears seven times at Mohenjo-daro, six times at Harappa, and once each at Kalibangan, Chanhujo-daro, and Desalpur (16 times total, according to Wells).  I find one large, round occurrence at Desalpur, Dlp-2 (my “a” version); four plump and somewhat pointed ovals, three at Harappa and one at Chanhujo daro, C-22, H-842, H-841, H-668 (my “b” variant); five thin and quite pointed ovals, two at Mohenjo daro, two at Harappa, and one at Kalibangan (my “c” variant).  All of this means I missed one somewhere. 
Angular and rounded signs appear in the same inscription from Harappa.

As for the angular “variants,” SMALL SQUARE (IV4) is supposedly another singleton, like the SMALL CIRCLE, this time found only once at Harappa, according to Wells (H-682).  Once again, I see more than that: the square in the broken seal, L-80, is smaller than the post next to it, and there appears to be a badly executed one in M-412 as well.  Its right side is a bit dented.  A slightly tilted SQUARE begins the inscription on H-6, smaller than the oval-topped DUBYA to its right.  Another, more evenly positioned, begins the inscription on H-389, also smaller than the CHEVRON to its right.  Yet another begins the inscription on H-682, smaller than the TRIPLE-BELTED AITCH to its right, as well.  However, we should note that the SQUARE on L-80 is not in initial position, since a tiny piece of the preceding sign remains to its left, though not enough to identify what it was.
On one of the copper tablets, the inscription has been highlighted in white (M-1529), making it appear that there are two small squares.  However, on the duplicates without highlighting (1528-1532), these elements are more clearly grids.  Thus, I do not count these as instances of the SMALL SQUARE.  Similarly, the squarish signs on H-199 though H-202 could be SQUARES.  But close inspection reveals hints that these, too, are grids.  I do not count them as instances of this sign, either.

HOV diamond on modern (rectangular) traffic sign.

I tend to think the SMALL SQUARE is the smaller variant of RECTANGLE, IV5, which Wells considers a singleton appearing on M-331.  It is the same height as neighboring signs in this inscription.  Other signs show this type of variation also, appearing small on some seals, enlarging to fill the available space on others.  I see another RECTANGLE on H-517, beside a MAN HOLDING DOUBLE POSTS.  It, too, is the same size as its neighbor sign.  There are three RECTANGLES in a row on H-97 on the right end of the bottom row, all the same size as the LAMBDA to their right.  One more instance occurs in second position beside the PALM SQUIRREL on H-419, just before the break.  This RECTANGLE is again essentially the same size as the preceding sign, or at least not square and not substantially smaller.  Two questionable possibilities include on the seal impression M-1383, where it is in initial position and very thin, and Kd-17 on a pot shard from Kot Diji.  The latter is broken, making identification uncertain.  The sign might also be an instance of the MALLET, VEST, or BATTERY, signs I have not yet discussed.
The DIAMOND, IV9, occurs 10 times at Mohenjo-daro, once each at Harappa and Chanhujo-daro.  This makes a total of 12 occurrences, of which we should note that some are significantly smaller than other signs in the same inscription, including those on M-667, M-742, and M-1002, while others are the same size as the other signs (a characteristic I posited as separating the SQUARE and RECTANGLE as variants of a single sign).  These statistics are somewhat contradicted by the Koskenniemi and Parpola concordance, but all of the symbols appear far fewer than 100 times, by anyone’s count.  Statistics are only meaningful where there are sufficiently large amounts of data.  This is an elementary fact of statistics known as the Law of Large Numbers.
Still, if each site in the Indus Valley produced a single, distinct symbol out of the set (CIRCLE, SQUARE, DIAMOND), we might interpret this as evidence of regional variation of a single symbol, despite the relative paucity of data.  Chanhujo daro only has a DIAMOND, no SQUARE or CIRCLE.  Desalpur has a CIRCLE and no SQUARE or DIAMOND.  Lothal has only a SQUARE and no CIRCLE or DIAMOND.  So far so good.  But these three sites do not yield many seals or tablets.  The sites where most artifacts appear are Mohenjo daro and Harappa.  It so happens that at each of these two sites we find that the CIRCLE, the SQUARE, and the DIAMOND all appear.  Even the rare RECTANGLE occurs at both Mohenjo daro and Harappa.  Due to these overlaps, it is clear that geography does not explain the variation. 
Mimbres fish showing tilted grid filler (diamond shapes as artifact).

If each of these symbols always appeared in the same location within inscriptions, say, always at the beginning or always at the end, this would support the hypothesis that all are variants of a single sign, even in the absence of large numbers of occurrences (i.e., even if the distribution is not statistically significant).  This is not the case either, though, and positional frequencies are particularly unrevealing for singletons (Korvink 2007:61).  The DIAMOND occurs in initial position five times out of 12 occurrences.  But it also occurs as the second, third, fourth, and next to last or eighth of nine signs.  The CIRCLE also appears in multiple positions from initial to final.  The less common SQUARE is most often initial, but it also appears medially.  The RECTANGLE, rarest of all, occurs at the right end of two lines and at the left end of another.  That’s not much to go on, but it is enough to say that it is not standardized in a single position.
My own perusal of the Corpus suggests that, in general, signs can often be found in smaller versions at one or the other end of an inscription, especially longer ones, as the “scribe” runs out of room, or where symbols must be fitted around part of the icon, such as above the horn of the “unicorn.”  This appears to explain the small and oval variants of the CIRCLE.  Where space is cramped, signs are either thinner from side to side, or smaller over all.  This same explanation, it seems to me, would be the best explanation for the existence of both a SQUARE and a RECTANGLE.  The SQUARE is the smaller variant of the RECTANGLE.  The DIAMOND and the CIRCLE also have their smaller variants.  Thus, this phenomenon is not unexpected for the RECTANGLE/SQUARE.
We might also reason that if circles, squares, and diamonds were indeed equivalent in this script, then we would expect to see circular, square, and diamond-shaped variants of each of a variety of symbols.  For example, there is a CIRCLED VEE (in both round and oval variants), so there should also be a VEE IN DIAMOND.  And so there is.  We should also expect a VEE IN SQUARE.  This is considerably less common than the other two, but it does appear (e.g., K-85 and B-18, possibly K-122 and M-1428).  So far so good for our hypothesis. 
There is another symbol that seems to have four little vee shapes creating an outlined “X” shape, the CIRCLED FAT EX.  This, too, comes in both circular and oval forms, and there is the expected FAT EX IN DIAMOND.  Unfortunately for our thesis, there is no *FAT EX IN SQUARE, unless it appears in the third volume of the Corpus, which I have yet to see.  There is a CIRCLED TRIDENT as well as a TRIDENT IN DIAMOND.  Again, though, there is no *TRIDENT IN SQUARE.  Is this pattern telling us something?
But there is more.  While there is a six-point asterisk in a circle, the CARTWHEEL, there is no *ASTERISK IN DIAMOND, nor an *ASTERISK IN SQUARE.  This symbol only has a round version, and it is not the only one (e.g., there are only rounded versions of the CIRCLED DOT and DONUT or circled circle).  There are also signs that only appear in rhomboid form.  While there is a CIRCLED VEE and a VEE IN DIAMOND, there are several variations on a sign with VEE AND TRI-FORK IN DIAMOND.  But there is no comparable circular variant to any of these (no *CIRCLED VEE AND TRI-FORK).  Nor is there a square or rectangular variant of any of them. 
There is a DIAMOND WITH (3) SHISH KEBABS on it, but no circle or square is decked with a kebab (i.e., a vertical line crossed by three to six short horizontals).  Squares and rectangles partake of fewer variations to create new signs, but even they have their unique symbols.  There is a DOUBLE-BELTED RECTANGLE, but no *DOUBLE-BELTED DIAMOND or *DOUBLE-BELTED CIRCLE.  There is the MALLET, a square with a post on top.  But there is no diamond with a post on top.  Instead, there is a diamond with a post below it and several shorter ones on top, the HAIRY DIAMOND LOLLIPOP, a rather different sort of symbol.  The dotted circle appears with a post above and another below as the SKEWERED DONUT, but again, that is rather different.  It does not seem that these basic geometric shapes – CIRCLE, DIAMOND, SQUARE – are fully equivalent to one another.  A circle is not always the same as a diamond or square.
Again, there might be some social reason for one “scribe” to prefer rounded “variants” and another to prefer angular “variants,” perhaps based on social class, ethnicity, religion, etc.  If this were so, then we would not expect both rounded signs and angular signs to occur in the same inscription.  That is, even if the basic geometric shapes are variants in only some cases, we should be able to describe the context in which each occurs, whether we know the precise reason for it or not.  But the Corpus reveals a number of examples where both rounded and angular signs are, in fact, found in the same inscription.  It is therefore preferable, for the time being, to assume that the different shapes represent different concepts.
In addition to such internal evidence, one may examine other ancient scripts for external parallels to determine whether it is likely that particular symbols are variants of one another.  In Egyptian hieroglyphs, there are a number of circular glyphs.  These include, from smallest to largest N33 (grain of sand), D12 ( pupil of an eye), S21 (a circle in a circle, a ring), N5 (a dotted circle, the sun), N9 (a circle bisected horizontally, the moon), Aa1 (a striped circle, perhaps a placenta), O50 (a dotted circle, a threshing floor), O48 (two short vertical strokes in a circle representing a building at Hierakonpolis), O49 (a circle containing a “fat ex” depicting an encircled crossroads), X6 (a small square resting on the inside bottom of the circle depicting a type of bread loaf) (Gardiner 1976: 546-548)..  In hieratic writing, the fine distinctions among many of these circular glyphs often fail to appear.  Several are simply dark circles, so that the reader must rely on context to distinguish them.  Even so, their continued existence on stone monuments in fully detailed form and their distinct uses reveal a series of distinct signs. 
Not only are size and internal details significant in Egyptian, so are details of shape.  Ovals are not the same as circles and these are not to be confused with quadrilaterals.  Among the ovals, two have pointed ends as among the Indus symbols (specifically D21 and V38, the first a horizontal oval representing a mouth, the second a vertical oval representing a bandage).  Among quadrilateral forms, squares and rectangles, there are again a number of distinct glyphs.  The smallest is a square, Q3, depicting a mat.  A horizontal rectangle, O39, is a stone slab or brick.  Other rectangular symbols are distinguished by internal or external markings (N37, a garden pool of water, N38 the pool showing the sides, N39 the pool showing the water, O6 a rectangular enclosure representing a mansion, temple, or tomb, O36 a wall, S32, a fringed piece of cloth).  There is no diamond in Gardiner’s list, although one does appear in the older dictionary published by E.A.W. Budge, which includes symbols from the whole range of periods, not just Middle Egyptian (1978, Vol. II, p. 947, in the name of Auhep, sanctuary of the god Ahi; possibly a variant of Gardiner’s Aa5, a kind of stacked double chevron, part of the steering gear of ships).
If we rely on an Egyptian parallel, then, we would conclude that size may be significant and meaningful in Harappan symbols.  Small signs may not indicate the same thing as larger ones, even where other details are the same.  Details (whether internal or external) should also be considered significant until there is unequivocal evidence that they are not meaningful.  Thus, we would be quite hesitant to group together the simple DIAMOND and the one with internal detail, VEE IN DIAMOND.  We would not group together the SMALL SQUARE and the RECTANGLE.  We would not even group together the CIRCLE which is rounded and the OVAL which is pointed at both ends.
The earliest script found in southern Iraq is termed proto-cuneiform.  In this symbol system, circles also come in more than one size and this distinction has meaning.  Both types of circle most frequently function as numerical symbols, indicating measures or quantities.  A small circle can represent either 10 measures of grain or 10 head of cattle, while the larger circle indicates 3,600 measures of grain but does not figure in measure of cattle (Schmandt-Besserat 1996: 119).  Context may or may not make the distinction clear to a modern reader, but presumably the ancient scribes knew what they were counting or measuring in all cases.
Small and large circles represent these same two numbers, 10 and 3600 respectively, in the proto-Elamite sexagesimal enumeration system, which was used to count discrete inanimate objects (Englund 2001).  The smaller circle also appears in other proto-Elamite numerical systems.  It has the value of 10 in the decimal system used to count discrete animate objects, and 10 as well in both bisexagesimal systems, one of which enumerates grain, the other rations (larger and smaller measures differ in each enumeration system). 
In the capacity system used for barley, the small circle has a slightly different role.  Ten barley measures indicated with the smaller circle are the equivalent of a single measure indicated by the larger circle.  Neither circle, in simple and unmodified form, appears among the capacities for emmer wheat, so these are not abstract numerals.  In proto-Elamite area measures, six of the smallest area (each represented by a horizontally positioned wedge) make up the second measure (represented by a similar wedge with an impressed circle inside), of which three make up the measure represented by the small circle.  Ten small circles make up a single measure represented by a large circle.  Thus, while a given symbol may have more than one specific meaning, in context its distinct uses become clear.  In each case, size is significant, while shape often is, though not invariably.  Thus, the proto-cuneiform scripts used in Sumer and Elam suggest that size and shape may prove significant among Indus symbols as well.
However, the evidence from the Near East is somewhat equivocal on this matter.  Among non-numerical symbols, proto-cuneiform makes use of a symbol termed LAGAB.  The variant LAGAB~a is a circle which came to mean “block (of wood), slab (of stone), trunk (of a tree).”  However, early on, this symbol was often used to represent a head of small cattle, especially a type of sheep.  Internal markings might indicate something about the specifics of age or gender.  A pointed oval, vertically positioned is KI@n, which came to mean “earth, place, area, ground.”  A diamond shape is HI, which came to mean “to mix; mixed.”  Variants of the latter include HI@g~a with internal striping and HI@g~b, which is a horizontal oval, pointed at the ends, also with internal striping.  The variant of the first symbol, LAGAB~b is a square.  Thus, the circle and square are variants of one another, to a certain extent, though the square variant contains other symbols less often.  And the diamond and oval shapes are variants of each other, although once again the oval variant seems less common.  And this oval variant is invariably horizontally positioned, in contrast to KI, which is normally oval but vertically positioned.  So, it would seem that some distinctions are more important than others and context helps to make the difference.
In the earliest Chinese writing, found on so-called oracle bones, and in the somewhat later system called Old Seal writing, there is a circular symbol once again.  This is now pronounced wei2, “a round, a circumference, an enclosure; to contain” (Wieger 1965: 188).  In later calligraphic writing, this circle becomes a square.  As a radical (part of a compound sign or character), the circle most often holds another symbol inside it.  We may thus consider it the Chinese version of the large circle.  For example, with the pictograph of a pig inside, it becomes hun4, “sty.”  A circle of the same size as the enclosure, but with a dot or short horizontal line inside, is ri4, “sun, day” (Wieger 1965: 311).   
These and other circular characters found in early Chinese become squares in later calligraphy.  Thus, while a square and a circle are equivalent in Chinese, each symbol has its own place.  A circle and a square do not normally appear together in an inscription or passage, although one might see Old Seal style in a “chop” or signature seal, or in other special circumstances alongside normal calligraphic characters.  There is no diamond in this symbol system.  Since the Indus script does not seem to change its character over time, the Chinese model suggests that the contemporary shapes of angular and rounded signs of varying sizes may represent different ideas and should not be conflated.
In conclusion, I tend to think a CIRCLE may be either a real circle or an oval, but not a square or a diamond (II9).  A SQUARE may be either a real square or a RECTANGLE, but not a circle or a diamond (IV4 AND IV5).  Finally, a DIAMOND may be small or full sized, but it is not a circle, not an oval, not a square, and not a rectangle.  It is, quite simply, a diamond.  And while it is the basis of the some of the most common Indus signs, it is surprisingly rare elsewhere.  Except for the one instance in the older dictionary, it does not seem to occur in Egyptian.  Nor does it seem to appear in Old Chinese.  As an individual symbol, it does not appear in the rock art of North America.  It only shows up in multiples, where it is essentially an artifact of either strings of “XXXXX,” or else a grid that is angled (Newcomb 1996: 188, Pl. 138, no. 19-B, string of X’s; 196, Pl. 147, no. 23-G, chain of X’s; Heizer and Baumhoff 1984: 176, fig. 113b, tilted grid; 196, fig. 133e).
It is present in Luwian hieroglyphs, where it represents the syllable mi.  I mentioned earlier the appearance of the diamond in proto-cuneiform, as HI.  A diamond also appears in the Old European markings of the Danube Valley, often with a central dot (OE 218).  In Hopi art, in the American Southwest, a diamond shape can represent the world (Kabotie 1982: 35).  It also occurs in proto-Elamite (M218).


Budge, E.A.Wallis. 1978. An Egyptian Hieroglyphic Dictionary in Two Volumes. Vol. II. New York: Dover. (orig. 1920)

Hesselt van Dinter, Maarten. 2006. Tribal Tattoo Designs from the Americas. Mundurucu Publishers.

Kabotie, Fred. 1982. Designs from the Ancient Mimbrenos with a Hopi Interpretation. Flagstaff, AZ: Northland Press.

Old European:

1 comment:

  1. The diamond/lozenge/rhomb sometimes represents jewell, lightning bolt or vajra/vajre (or viraj?) or phallic/lingam [also compare Ingwaz rune], or vulva, eye, or earth. The Mayan glyph Lamat [your Lambda, Semitic lamed] has a diamond-square abit like a "star".

    It is said that the "circle, square and triangle" are the main base of most occultic symbols (&/or geometry?)