Magic Square: Worlds in Space-Time

For milleniums, all these multi-pattern squares have served as boardgames and temple decoration all around the world.

Tic-Tac-Toe or three-in-a-row is one of the most famous square used as a boardgame.

One needs to make three in a row orthogonally or diagonally and block their opponent from completing three in a row. Considering winning combinations, there are 255,168 possible games and 362,880 ways of placing your pieces and your opponent's pieces on the board. Regarding the boards previously showed, all strategical rules are all the same: blocking your opponent from making a row (a chain), and making your own. A similar process of blocking a cell from making a chain of proteins.

This square was found in Lascaux and dated back to 15,000-12,000 BC. A rare abstractive figure for that time. Actually, once can notice the 9 squares (3x3) of the magic square.

Symetrical figures are created when numbers are linked orderely.

Lo Shu (it means book of the river) is part of the legacy of the most ancient Chinese mathematical and divinatory tradition (over 5000 years old). The odd and even numbers alternate in the periphery of the Lo Shu pattern, the 4 even numbers are at the four corners, and the 5 odd numbers(outnumbering the even numbers by one) form a cross in the center of the square. The sums in each of the 3 rows, in each of the 3 columns, and in both diagonals, are all 15 (fifteen is the number of days in each of the 24 cycles of the Chinese solar year). Since 5 is in the center cell, the sum of any two other cells that are directly through the 5 from each other is 10.

The magic square based on a binary system

According to Wim van Binsbergen : In the most archaic Sumerian writing (ca. 3000 BCE) the agricultural field was simply represented by a rectangle divided by vertical lines: the image of a field divided by irrigation ditches...which ultimately led to the standard character.

In Chinese (Hân Yîng Cídian 1988; cf. Wieger 1965: 316f) the character for field is (t'ien):

This representation of ‘field’ is already attested in the most archaic Chinese writing on seals and oracle bones (2nd mill. BCE), as:

In Ancient Egyptian hieroglyphic, again, the oblong grid: has the cognate meaning of ‘district’, "administered land area" (Faulkner 1962: 54, 178 and passim) — which was rendered in Greek as nomós, and is generally¸considered to represent a (manually) irrigated field.

This board was often found associated with a game called the "Alquerque". The word alquerque might come from the Arabic word "El-Quirkat" meaning "the mill".

Alquerque is known to date back at least as far as 1400BC, since boards have been found cut into the roofing slabs of the temple at Kurna in Egypt. A game called Quirkat is mentioned in an Arabic work of the 10th Century AD.

Alquerque in the wall of a medieval mill in "Arroyo de los Molinos", Andalucia, Spain.

From the "Catedral de Ourense" (Orense), in Galicia, Spain.

Nine Men's Morris or Merels board

Supposedly, the first carved board ever discovered was found 3500 years ago at the temple of Kurna, Egypt. However, in 2006, archeologists have found a Men's Morris engraved stone dated back to the last Ice Age. The last Ice Age started about 50,000 years ago and ended around 8,000 BC.

This engraved rock is thought to be an early games board (Nen Men's Morris). Photo Creswell Heritage Trust. Creswell Crags, England is a Scheduled Ancient Monument and Site of Special Scientific Interest and could also become a World Heritage Site.

Even, Earth and Underworld

The science of trinity...

According to Bengt Hemtun, " This is the Gateway to Underworld made of clay to manifest the concept..."

"To that we can add this golden plate from a Stonehenge grave. The groves show the three steps to Underworld."

Children's Scribbling and The Invention of Writing

"Only one thing is certain - that written language of children develops in this fashion, shifting from drawings of things to drawings of words. The entire secret of teaching written language is to prepare and organize this natural transition appropriately...Make believe play, drawing and writing can be viewed as different moments in an essentially unified program of development of written language...The discontinuities and jumps from one mode of activity to the other are too great for the relationship to seem evident." Lev Vygotsky, "The Prehistory of Writing," an essay, c. 1930

Basic Scribbles

From the book Analyzing Children’s Art, 1969, written by Rhoda Kellog.

According to Catherine Baily in The Development Of Children’s Drawings and Paintings :

I have observed that every child uses many of these scribbles in his/her early drawings and that each child also has their favorites.

In general, the first outlined shape that children make is circular, and Thomas and Silk quoted Arnheim (1956) writing that “the first enclosed form made by children—the “primordial circle”—appears to be able to stand for almost any object from the child’s environment” (35). It is with this beginning of closed shapes, with the circle that can stand for anything, that most children then move on to more complex shapes and then eventually representational objects.

In the full-fledged Diagram stage, children begin to draw six different Diagrams, those being the rectangle, the oval, the triangle, the Greek cross, the diagonal cross, and the odd shape. Developmentally, “the Diagrams indicate an increasing ability to make a controlled use of lines and to employ memory” (Kellogg 45).
Kellogg and O’Dell note that “almost from the moment they are able to draw shapes in outline form, children begin to combine these forms into designs” (43). With just these two examples of children moving from outlined shapes to designs it is clear that this progression is not only a speedy one, but that it also occurs naturally without any outside guidance.

Maybe scribbling practices and entoptic experiences encouraged the formation of a different eye-span, which became specialized in tracking forms within forms, and consequently led to the invention of writing.

1. Free examination
2. Estimate material circumstances
3. Give the ages of the people
4. Surmise what the family had been doing befoe the arrival of the unexpected visitor
5. Remember the clothes worn by the people
6. Remember positions of people and objects in the room
7. Estimate how long the visitor had been away from the family

In the 1950s, Alfred L. Yarbus* did important eye tracking research and his 1967 book, Eye Movements and
Vision, is one of the most quoted eye tracking publications ever. For example he showed the task given to a subject has a very large influence on the subject's eye movements. He also wrote about the relation between
fixations and interest:

"All the records (…) show conclusively that the character of the eye movements is either completely independent of or only very slightly dependent on the material of the picture and how it was made, provided
that it is flat or nearly flat." The cyclical pattern in the examination of pictures "is dependent not only on
what is shown on the picture, but also on the problem facing the observer and the information that he hopes to gain from the picture."

*Alfred L. Yarbus was a Russian psychologist who studied eye movements in the 1950s and 1960s.

According to Hoffman, current consensus is that visual attention is always slightly (100 to 250 ms) ahead of the eye. But as soon as attention moves to a new position, the eyes will want to follow.

We still cannot infer specific cognitive processes directly from a fixation on a particular object in a scene. For
instance, a fixation on a face in a picture may indicate recognition, liking, dislike, puzzlement etc. Therefore eye tracking is often coupled with other methodologies, such as introspective verbal protocols.

Rock Carving and the Nature of Things

Celestial maps, seasons, moon cycle, time...


Source: Scandinavian Society for Prehistoric Art

At the time of these celestial images the equinox were around a half turn apart from today's spring equinox at the ecliptic. That means Ramadan and end of moon season was in our Pegasus that was known as the Stag/ deer / reindeer. The image of the falling Stag gives the picture another dimension and later on end of "first half-year" was animated as a fallen man. At the same time solstices were in both end of Watergate/ Milky Way. (B. Hemtun)

From Skaane we know that the footprints has to do with the season. In Egypt an empty boat symbolised start of season. The footprint with the snake has the toe beneath and symbolises water and fertility running down. That helps us understand the artefact from Isturitz that is maybe 10,000 years elder and the cave was used from around 30,000 BC onward. We see the footprint under ground and the spiral symbolises the flow and the other signs are other times. (B. Hemtun)

The nature of things
"Entoptic or celestial map? or both. The nature of things is decoded through the use of drugs (see the other world), and the process of celestial mapping is just another representation of the same world."

Drugs and the Origin of Art

The theory of entoptic phenomena is simple : phosphenes (or entoptics) are generated in the neural system, and anyone can see them. These visions are enhanced by the taking of hallucinogenic drugs; these drugs may have been taken as part of a shamanistic ritual and the images "seen" were then drawn by the visionary.


Heinrich Klüver who, in 1926, noticed that hallucinations seemed to occur in two stages, the first being related to four types of geometric: the grid, described variously as lattice, filigree, honey comb, grating, fretwork and chessboard; cobwebs; tunnel, also associated with cone, vessel, funnel, alley; and spirals. The second stage was that of iconic images which Klüver interpreted as being drawn from memory. An interesting point is that there seemed to be constants of theme in the more elaborate iconic images, the most common being religious symbols and images, followed by images of small animals and human beings (Siegal, 1977).


"The long association during the Upper Palaeolithic of entoptic with often remarkably 'realistic' depictions, combinations of entoptic and iconic elements predicted by the neuropsychological model, the occasional combination of different iconic images... the location of much of the art in remote galleries and inaccessible diverticules..." and that these are "better explained by the more extreme varieties of altered consciousness than by infantile perception." (Lewis-Williams, 1991)

Another concern is that of sequence in time. Leroi-Gourhan (1968) divides Upper Palaeolithic art up into a chronology of styles. The 'signs' fit into this chronology and change over the progression. This change is not problematic for the iconic images since in Lewis-Williams and Dowson's theory they are largely culturally determined derivatives from the fundamental forms, but according to their theory the entoptics have their origin in the nervous system itself - they are fundamental universals and should not change as a function of time and style. Leroi-Gourhan said "At Lascaux I really believed they had come very close to an alphabet." (recalled by Brigitte Delluc and Gilles Delluc of the Musee de l'Homme, Paris, in Lewis-Williams and Dowson, 1988), and it is likely that early sign-alphabets would be indecipherable to all but those members of
the group that developed them, the ones who knew what they stood for.We've had a nervous system capable of hallucinating for millions of years, our hands have been free from the task of locomotion for millions of years, yet art doesn't appear until possibly 300,000 BP on the evidence of an engraved bone (Lewis-Williams and Dowson, 1988) and doesn't really blossom until the Upper Palaeolithic.

One of the characteristic properties of intoxication by hallucinogenic substances such as LSD and Mescaline is a tendency to spontaneously perceive regular geometrical forms. Heinrich Klüver (1966) has categorized certain types of patterns, or form constants, commonly observed under mescaline intoxication, which subjects described as lattice, fretwork, filigree, honeycomb, and chessboard patterns, as suggested in figure A, as well as cobwebs, tunnel and funnel patterns as suggested in figure B. These same form constants are also commonly observed under a wide variety of conditions of psychological stress, or threshold consciousness, including falling asleep, waking up, insulin hypoglycemia, the delerium of fever, epilepsy, psychotic episodes, advanced syphilis, sensory deprivation, photostimulation, electrical stimulation, crystal gazing, migraine headaches, dizziness and a variety of drug intoxications (Siegel 1976). The diversity of different conditions which provoke the same kinds of patterns suggests that these form constants reflect some fundamental property of the visual system.

Siegel (1976) observes that the visual patterns seen under LSD intoxication are strikingly similar to the primordial or archetypal forms such as the mandala, the mystic symbol of the universe employed in Hinduism and Buddhism as an aid to meditation. Moreover, as many anthropologists have noted, the hallucinogen-inspired art of many primitive peoples often contains similar geometrical patterns of form, color, and movement (Lewis-Williams & Dowson 1988). Actually, the same kinds of form constants also appear commonly in art which is not hallucinogen-inspired, most especially in the patterns of non- representational or decorative art and ornamental design. For although there is a great degree of variation in the arts of different cultures, there is also much that is common among them.




The world's oldest example of abstract art, dating back more than 70,000 years, has been found in a cave in South Africa. Supposely, "modern" human behaviour started only around 40,000 years ago. We can notice complex geometric patterns; a double-wave pattern.




Sources:
http://www.wynja.com/arch/entoptic.html
http://www.nap.edu/readingroom/books/bio73h/kluver.html
http://sharp.bu.edu/~slehar/webstuff/dirhr/section.html
http://news.bbc.co.uk/2/hi/science/nature/1753326.stm