Subject |
: |
Mind switch? |

Author |
: |
Vincent Keers |

Date |
: |
09-30-2000 on 08:02 a.m. |

Dear Wendel, I'm sorry, but I began this new tread because the previous Tessellation tread took so long loading. Hope you don't mind. First I think it's essential to know where we draw the line in order to see the tessellation. I took the liberty to change the perspective on the Belouch design. Your interpretation on the Alhambra design is correct in your perception. Only I saw flowers with stars. So I did not see your tessellation which is in my perception: a leaf and a star; the star beïng the result of the setting of the leafs. The two together forming the next tessellation. So it's a tessellation within a tessellation. Pffjuh! So if a tessellation is the product of one design creating another design and we draw the line-stop! - the Alhambra isn't a tessellation. But...if we extract the whole flower with all it's leaves the result is the star and it's a tessellation. Here's another mind-switching rugdesign. This Bokhara is a tessellation of the star producing in it's setting a "Mother Goddess with two animals at her side". In the perseption of that cult it's a correct tessellation. But only because they see the Mother Goddess with the breasts, feet up in the air and the two animals at her side. So a tessellation is the product of cultural programming, and therefore it's a mind-switching(left-right) experience for us. Best regards, Vincent |

Subject |
: |
RE: Tessellation II |

Author |
: |
Steve Price |

Date |
: |
09-30-2000 on 09:26 a.m. |

sprice@hsc.vcu.edu
Dear Vincent, You're right, of course, about the fact that seeing a
tesselation is a product of the cultural baggage we tow around with us.
More importantly, I think, is that we too often forget that seeing
almost any image is culturally conditioned. Part is hard-wired into
our brains, but part isn't. The anthropologist (at least I think it was an
anthropologist, it really doesn't matter, though) who found that Turkmen
couldn't see the images in a newspaper picture is a good example of this.
Regards, Steve Price |

Subject |
: |
RE:Tessellation II |

Author |
: |
Vincent Keers |

Date |
: |
09-30-2000 on 10:49 a.m. |

Dear Steve, Ying-Yang is a tessellation within it's own private universe, the circle. And because the straight line is allways curved, the perfect tessellation doesn't excist, without the border wich is set by the artist, hoping the viewer will pick up the message. If we layout Escher's drawing around the globe, it can't be done without shrinking the design until we reach a point w're lost in a dark hole. So Escher has it's limmits (the paper he worked on) too, because shrinking is out of the question. Where this all leads to, I don't know. Oh, yes.....communication. It's the artist that sets the border. Best regards, Vincent |

Subject |
: |
RE:Tessellation II |

Author |
: |
Wendel+Swan |

Date |
: |
09-30-2000 on 11:15 a.m. |

wdswan@erols.com Dear Vincent and Steve, Perhaps the discussions about tessellation have gone on a bit too long for most of the readers, but I feel some references and clarifications are in order. Vincent posted this assumption: "if a tessellation is the product of one design creating another design ... " and then spoke of "perception." To Vincent's statement that " ... a tessellation is the product of cultural programming ..." Steve agreed that "seeing a tesselation is a product of the cultural baggage we tow around with us." A tessellation is not "one design creating another design" and cultural bias has, so far as I can tell, nothing whatever to do with whether a tessellation exists, nor does perception. A tessellation is more than the repetition of a shape; it is the repetition of a shape so as to fill a plane. While perception is may be irrelevant, definition is. In the other thread on this topic, John Howe cited us to the Math Forum, on which site a number of references can be found relating to symmetry and tessellations. http://forum.swarthmore.edu/geometry/rugs/symmetry/grids.html Carol Bier's definition at that site is: "A tessellation is a pattern formed by the repetition of a single unit or shape that, when repeated, fills the plane with no gaps and no overlaps." One may find through a search for sites on tessellation a wide variety of papers that discuss in excruciating detail the mathematical bases for tessellations in ways that I can't begin to understand. Nowhere will you find references to cultural biases or perceptions. Look elsewhere on the Math Forum at: http://forum.swarthmore.edu/sum95/suzanne/whattess.html Here is another discussion of the definition of tessellation: "A dictionary will tell you that the word "tessellate" means to form or arrange small squares in a checkered or mosaic pattern. The word "tessellate" is derived from the Ionic version of the Greek word "tesseres," which in English means "four." The first tilings were made from square tiles. A regular polygon has 3 or 4 or 5 or more sides and angles, all equal. A regular tessellation means a tessellation made up of congruent regular polygons. (Remember: Regular means that the sides of the polygon are all the same length. Congruent means that the polygons that you put together are all the same size and shape.)" I admire and appreciate the most recent Lesghi star drawings that Vincent posted, but they don't really address the topic of tessellation. I am clearly not a mathematician, but symmetry and tessellation are topics of mathematics, which is perhaps the most universal and culturally unbiased "language" that I know of. Escher's drawings are endlessly fascinating in that he bridges a gap between mathematics and representational art. Some of his drawings are true tessellations and in other instances only sections (sometimes more than one) are true tessellations. Whether one sees a winged horse or a Mother Goddess or a star is irrelevant to the process of creating a tessellation, however interesting that realistic image may be in the resultant artistic creation. Best regards, Wendel |

Subject |
: |
RE:Tessalation? |

Author |
: |
Yon Bard |

Date |
: |
09-30-2000 on 06:20 p.m. |

Wendel, applying your definition literally, one concludes that any pattern that can be conceived as made up of identical rectangles is a tessalation. For example, if you take the usual Turkmen pattern of primary and secondary guls, then the rectangle consisting of a primary gul and the quarters of the surrounding four secondary guls forms a tessalation; the same applies if you reverse the roles of the primaries and scondaries. The same can be said of the boxes that make up the aina-gul pattern plus half the width of the surrounding border strips (or the complete adjoining borders on, say, the bottom and right side of each box). Would you accept these as examples of tessalations? Regards, Yon |

Subject |
: |
RE:Mind switch? |

Author |
: |
Vincent Keers |

Date |
: |
09-30-2000 on 07:40 p.m. |

Dear Wendel' I'm sorry, but I have to react. If people are losing interest; I'm not posting on Turkotek for the amusement of others. I hope people are participating, although the reaction is poor. You got to the point. Mathematics doesn't concern itself with the design or cultural aspects. So whatever is inside the triangle or square doesn't matter as long as it only appears one time. In Eschers frogs and lizards he created originaly "quarter" frogs and lizards in a square. A system very basic in the rug designs. By shifting and mirroring and turning the original square we get the total image of whole frogs and lizards and in this way it's a tessellation. Why look for tessellation if we follow the dictionary? Why look for tessellation if it's only resricted to one design? There are 17 ways of tessellation. And those 17 tessellations take us inside the rugs design. That's what is of intrest. I do not think designs are only copied or invented by different cultures, I think the tessellating aspect has allways been in the dark. I did not "invent" the pictures of the Belouch or the Stars. The Star/Goddess design is on my website for two years now. The Tau, Swastika, Cross (oops there I go again) design, is the product of tessellation over a period of 10,000 years. I posted it, but all I get is. "Two intersecting lines"; "It's a very old, well known design"; "It's known all over the world" But I'll stop now. Afraid people will lose intrest. Thank you for this very beautiful, never ending subject, it did trigger the correct handle (POINT) (Which in tessellation produces a _______ line) Best regards, Vincent |

Subject |
: |
RE:Mind switch? |

Author |
: |
Vincent Keers |

Date |
: |
09-30-2000 on 08:36 p.m. |

Anyone interested, Great site: http://www-sphys.unil.ch/escher You can download your own Escher sketch software. University of Lausanne, Switzerland. Takes a while to load, but it's worth it. It's fun. Best regards, Vincent |

Subject |
: |
RE:Mind switch? |

Author |
: |
Patrick+Weiler |

Date |
: |
09-30-2000 on 08:56 p.m. |

jpweil00@gte.net Vincent, The pattern you introduced from the field of your Baluch rug shows up frequently in a popular type of Baluch bagface. It has a design of what Balpinar/Hirsch call "interlocked offset octagons" in their book Carpets of the Vakiflar Museum in Istanbul. It is not a tesselation because there is a diamond shaped space between the corners of the octagons. However, this resultant diamond space becomes the center of an octagon identical to the major octagons by using the corners of the major octagons as corners too. Because the colors are the same, it presents an optical illusion by forcing your mind to jump from grid to offset grid trying to "see" which grid is the dominant one. They show a picture of a rug, plate 5, 15th century, Eastern Anatolia, which is identical to the Baluch pattern from four centuries later. They say that Akkoyunlu and Karakoyunlu Turkmen and Timurids occupied this territory of Eastern Anatolia. The design shows up in many art forms of the era and they say that "Interlocked octagons are also woven on the contour-zili mafrashes of some of the Turkmen tribal groups" including Moschkova, 1977, plate LXXII Figure 23. Boucher shows this pattern in plate 45 of his book Baluchi Woven Treasures. I suspect that the weavers found as much fascination in these designs as we do today. Patrick Weiler |

Subject |
: |
RE:Tessellation II |

Author |
: |
Wendel+Swan |

Date |
: |
09-30-2000 on 09:41 p.m. |

wdswan@erols.com Dear Yon and all, I believe that you would find, by reviewing the various sites on symmetry, that a tessellation is created by a perimeter (i. e., a shape) and not by the contents of that perimeter. So if you have a field full of rectangles all of the same dimensions, you have a tessellation. And it doesn't matter what is within those rectangles. The tessellation rug in this salon contains, in the diamonds of the "S" shape, some devices that are roughly boxes with crosses on them and the triangles have smaller devices. All of this filler is irrelevant. The "S" shape creates the tessellation. In your example of the primary and secondary guls, the rectangles (consisting of a primary gul and the quarters of the surrounding four secondary guls) do indeed form a tessellation. In the case of primary guls in which lines pass through the centers, one could have the secondary gul at the center of the rectangle. The repetition of that rectangle over the field would fill the plane and you would have the field of a Turkmen carpet. On that basis, you could take any rug with a repeating design, draw a rectangle around the repeating element, translate that rectangle all over the plane and say that you have a tessellation. But in all cases it would be a tessellation of rectangles, not of the contents. Take, for example, a Senneh rug filled with rows of identical repeating botehs. Drawing rectangles around the botehs will not convert the botehs into a tessellating shape; it will only create a tessellation of rectangles that happen to contain botehs. The fact that the device within the rectangle (or whatever tessellating shape you chose) is symmetrical makes no difference. A Bakhtiyari garden carpet could be an example of a tessellation of rectangles (containing asymmetrical devices) in which the contents are not repeated and do not correspond to the contents of another rectangle. Turkmen guls are created through the application of other principles of symmetry and they may be symmetrical in two parts or four parts. Reflection and/or rotation will result in the creation of the whole gul from one part. I believe that Escher has, in some instances, used the term tessellation in conjunction with artistic works in which the contents of the shape align with corresponding contents of other rectangles, but that is where art may differ from mathematics. Perhaps it is helpful to think of a tessellation (as least as we may confine it to discussions about carpet designs) as a grid. A tessellation is a type of grid that may have very irregular lines but each segment of the grid is identical in shape to all the other segments. Escher's winged horse or his lizards are good examples of tessellations created by highly irregular lines. Best, Wendel |

Subject |
: |
RE:Tessellation II |

Author |
: |
Wendel+Swan |

Date |
: |
09-30-2000 on 10:17 p.m. |

Dear Vincent, Perhaps I am misreading or misinterpreting what you have written, but allow me to supplement your comment that: "There are 17 ways of tessellation. And those 17 tessellations take us inside the rugs design." You are perhaps referring to basic symmetrical repetition to create a two-dimensional field pattern. There are seventeen systems that mathematicians classify as symmetry groups. Those systems consist of various combinations of translations, reflections, glide reflections, and rotations. Using one of these 17 systems makes all field patterns, including tessellations. Sometimes the same result can be achieved by using any one of several alternatives among the seventeen. Keep on posting. Best regards, Wendel |

Subject |
: |
RE:Tessellation II |

Author |
: |
R. John Howe |

Date |
: |
10-01-2000 on 07:09 a.m. |

Dear folks - Just to follow what Yon, Wendel and Vincent have said above, it is in fact the outside "silhouette" of a figure not its internal mechanisms that determine whether a design is a tesselation. So one has to identify that perimeter accurately before applying the test of "no gaps, no overlaps." This is what I meant when I said early on in the first thread that a great deal hangs of the "level" at which a design is read. It is better to say, that one must choose the "silhouette" to be tested. Second, Wendel mentions that it's probably best to think of a tesselation as a "grid" and if you go to the Carol Bier link I cited above you will see that she in fact begins her discussion of tesselation by providing a variety of grids. Third, Yon has asked and Wendel has agreed that many Turkman patters with major and minor guls can be read as tesselations if one does an unexpected thing. That is to treat lines drawn vertically through the center of the major guls as the sides of the rectangular silhouette to be considered and another set of lines drawn horizontally through others of them as the top and bottom of this same silhouette. The rectangle defined in this way has the minor gul at its center. This species of tesselation is one of the more interesting instances because although a grid of squares might be seen as the most ordinary sort of tesselation, the character of this square is rather unexpected and violates our usual way of reading Turkmen pieces in which, I think, we tend to treat the major gul as primary. This is also an instance that shows that perception not just mathematics is truly implicated in the "reading" of tesselations. There is in fact a debate about how Turkman designs with major and minor guls should be read. Jim Allen, for one, has insisted that the Turkmen themselves likely constructed and read these guls in terms of the tesselated squares we have identified here. He has argued that the "true" major gul has the minor gul at its center. Some of us, myself included, have smiled about this interpretation. But this discussion should in part make us more open to such possibilities, since our pursuit of the varieties of tesselation have led directly to them. Interesting stuff indeed, R. John Howe |

Subject |
: |
RE:Tessellation II |

Author |
: |
Vincent+Keers |

Date |
: |
10-01-2000 on 07:45 a.m. |

Dear Wendel, I think we're having a Babylonic discussion. You're looking at the product/result of a tessellation and look back in time. I look at the beginning of the tessellation as the artist had to make it and how it's possible in making the tessellation as perfect as it can be. Maybe the picture will contribute. If I want to make a tessellation of a simple form, so everything fits in perfectly and has exactly the same dimension, and in between distance/space, so it's the perfect repetition, (Pfffjuh) of one design, I'll divide in four parts: In order to get C, I make B1:The product of A. In order to get it perfect I pick up the centre of B. Now it's a square and easy to make. I do not think the mosaïc artist made a realistic design, he knew what it would look like finished. But in order to get it perfect he too had to divide it in four, mirrored etc. As you see in the "Soumack Bird" design in the salon, the design has originaly been divided in squares that had triangles within them. Put all the squares into the field and change the colours of the triangles. I do not think the artist made an, out of the blue, S shape, for the S's had to fit in perfectly. I hope you can agree with this. The only problem is: With a simple, universal designs it isn't so hard. With more complex, culture related designs, we can'tt "translate". Best regards, Vincent |

Subject |
: |
RE: Another tessellation rug |

Author |
: |
Wendel Swan |

Date |
: |
10-02-2000 on 10:22 a.m. |

Following is another interesting tessellation rug that appeared in Herrmann VII. While quite rare, this pattern is not unique and can be seen in a few other examples, mostly Eastern Anatolian. The tessellating shape is the zigzag that crosses the field, including the "C's" at the peaks. Wendel |

Subject |
: |
RE:Mind switch? |

Author |
: |
Vincent+Keers |

Date |
: |
10-03-2000 on 06:07 p.m. |

Dear all, Tessellation it is. Not from the triangle. I took out a part, of which the tessellation has been made. Kurdish "Arrow" design kilim. Best regards, Vincent |

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