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1. 1. geometric • three dimensional • representational
2. 2. written by mark jansen appendix by f.e. barline, jr. designed and illustrated by fred fortune
3. 3. As the popularity of thread design spreads throughout the world, I am continually impressed with the endless artistic variations possible with the use of colored sewing thread. Some of these variations are presented in this book. More important, however, are the basic theory and techniques explained here in simple language and with which the most spectacular thread design masterpieces are created. I wish the reader a stimulating and enjoyable excursion into this challenging medium. Mark Jansen Campbell, California March, 1974 introduction The use of colored thread as a medium in art has become the source of a whole new wave of craft innovations. This booklet deals with the most dramatic of these recent applications, known as Mandala® geometric thread design. "Mandala" is an ancient word from the Far East, used in the context of transcendental meditation. It is vague in meaning, but refers to a visual pattern which draws a viewer's attention into its center. The original mandalas were great paintings of objects arranged in circles, the effect of this symmetry being to minimize the influence of unwanted distraction. This same function is inherent in the patterns shown throughout these pages and is why the name "Mandala" was chosen to identify this new art form. The unique feature of Mandala* geometric thread design lies in the fact that each pattern is a set of points, called vertices, located very carefully on a closed conic curve such as a circle or an ellipse. The vertices are then connected with straight lines according to a strict numerical progression, resulting in an impressive finished thread design. The formation of curved lines from straight ones can be understood by considering the idea of a tangent. In the case of a circle, any straight line which touches the circle at exactly one point is called a tangent. A tangent can be drawn at any point along any arbitrary curve as long as the curve is smooth (doesn't have an end point or sharp corner) at that point. Simply stated, this means that any curve may be thought of as the intersection of infinitely many tangents. From this geometrical idea, it is quite natural to see how curved patterns can be defined with suitably arranged straight lines. Let us look at two fairly simple illustrations of this proposal. In the first drawing below, you see a right angle whose legs have been equally subdivided. When straight lines are drawn between the resulting points as shown, a hyperbolic curve is formed. The third drawing shows a circle with equidistant points located around its circumference. If a set of equally long line segments is used to connect these points, a small circle appears, defined by tangents! Note that it is the placement of the outside points, called vertices or generating points, which determines the contour of the resulting curve. Notice in the circular example that the generating points could have been connected with a different set of lines. The number of points skipped between connected points could have been four, or six, as well as five. Consider the effect of superimposing different straight line combinations drawn from the same set of generating points, each defining a different inner circle. The separate combinations then become circular Mandala™ layers.
4. 4. one two three sum In general, for N generating points in a circle, the number of ways to connect all the points with lines of equal length is equal to N/2 if N is even or (N - 1)/2 if N is odd. This means that in the example, there are (47 - 1)/2 = 23 possible layers. circular, elliptical, and starburst designs As shown previously, it is the arrangement of generating points which determines the shape of each design. In this section, we will show how these points are accurately located for each of the three basic patterns shown and how the resulting points are threaded. Designing A Circular Template Needed for laying out a precise template are a sharp pencil, paper, a ruler, some string, a hammer, nail, a flat board, and a compass. With these tools anyone can geometrically construct the set of generating points for circular, elliptical and starburst geometric thread designs. Let's look at the simplest case and consider the steps for drawing a circular template: 1, Cut a piece of paper square to the desired size of background. Locate the center of the square by drawing diagonal lines connecting opposite corners. The two diagonals intersect at the center point, which will eventually be the center of the circular design. 2, Draw a circle using a small nail driven into the center point and a piece of string set to the radius of the desired circle. 3, Subdivide the circle into a prime number of equal parts. A prime number has no divisors other than one and itself, like 11, 41, 67, and 101. With a prime number of generating points, each layer in the design can be made with one continuous strand of thread, lied off neatly only where the ends meet. See Appendix A for more on prime numbers. As outlined above, constructing a circular template is fairly simple. You should take care, however, in completing each step. The accuracy, for instance, of finding the center point and drawing a circle will determine how well the finished design is centered.
6. 6. Successive layers are done in identical fashion to the one just described except that a different number is used in the counting procedure. The next larger circle would be formed by connecting every 24th pin, and so on, up lo the desired number of layers. If the design you wish to make doesn't have 53 divisions like the one above, then you might want to know how to figure out the number of points to count over in forming the different layers. For those willing to accept some abstract symbology, the table below will be of some help in threading a Mandala® pattern with N generating points, assuming N is a prime number; Layer to be formed 1st (smallest) 2nd 3rd Number of pins to be counied (N - 1)/2 (N - 3)/2 (N - 5)/2 ...etc. Apply this simple procedure to the threading of each layer, taking care to maintain tension in the thread. It is very important that the threads lie in tight, stable, straight lines. Designing An Elliptical Template In geometry, the circle is known as the simplest of a series of curves known as "conic sections." Simply stated, a circle is that set of points which are all the same distance from the center point. What most people don't know is that a circle is just a special case of a more general closed curve, the ellipse. The ellipse may be thought of as the path of a point, the sum of whose distances from two fixed points, called "focal points," is constant. The above relation defines a very profound geometrical symmetry. The path of a planet around its sun is known to be an ellipse. In fact, any mass continually orbiting any source of gravity travels in an elliptical path with the source of gravity located at a focal point- See figure above. Consider what happens as the distance between the focal points diminishes. As the points come together, the ellipse fattens, and if the points become one and the same, we are left with a perfect circle. In this sense, a circle is merely a degenerate ellipse. See fig. 2, following page. focal points m + n = constant
7. 7. J fig. 2 Another way of thinking about an ellipse is to imagine a circle which is lilted along an axis through its center. The resulting curve is, again, an ellipse. The circle has been rotated and projected into the plane of the paper. See fig. 1 As outlined below, this is exactly how the arrangement of generating points for an elliptical Mandala* pattern is constructed: 1. Cut a square piece of paper whose sides are equal to the length of your desired background. Locate the centerpoint with diagonals as before and then draw two perpendicular lines dividing the sheet into four quadrants (refer to picture sequence below). 2. Draw an ellipse of desired proportions (See Appendix B) centered on the centerpoint as shown. Next draw a circle centered on the centerpoint. The circle must touch the ellipse at exactly two points, as shown. 3. Subdivide the circle in such a way that one of the vertices is on the quadrant line as indicated by arrow (a prime number). 4. Draw parallel lines connecting the vertices as shown. The generating points for the ellipse are defined by the intersections of the parallel lines with the ellips
8. 8. figure A figure B Designing A Starburst Template Many people consider the starburst ellipse the most interesting Mandala® geometric thread design. Its distorted symmetry implies a dynamic sense of motion; a fiery comet to some, the gliding of a fish to others. Regardless of what is seen in the completed starburst pattern, the arrangement of its set of generating points is intriguing. Returning to the topic of planetary motion, a step towards visualizing the peculiar arrangement of generating points in the starburst pattern may be taken by considering the orbit of Earth around the Sun. Imagine these two bodies as shown in figure A and that you, an observer, are stationed high above the orbital plane. You would see the planet pick up speed as it drew closer to the Sun, and slow down as it moved away from the Sun. Kepler, one of the early mathematician/astronomers, is credited with having first expressed this fact during his patient studies of the heavens. He pointed out that, during equal time intervals, areas swept out by the planet's orbit will be equal. If the progress of Earth was charted by exposing a plate of film repeatedly at set increments of time (say, by using a strobe light for a flash bulb), the photograph would resemble figure B. Not only is this the type of distribution used in the starburst pattern, but the layers of a starburst design represent elliptical orbits which might exist around one of the focal points used in locating the set of generating points. Now, how is this set of generating points drafted? There is certainly an easier way than obtaining a strobe-flash photograph of the Solar System! Steps for designing a starburst template: 1. Draw an ellipse of desired proportion. 2. Draw a circle centered on one of the two focal points. 3. Subdivide the circle into the desired number of divisions (a prime number). 4. Draw straight lines from the center of the circle through the divisions and onto the ellipse. The vertices of the starburst pattern are those points on the starburst which are crossed by the extended lines. Since the circle is centered at a focal point of the ellipse, this procedure gives rise to the correct distribution of vertices. See figure on following page. 11
9. 9. A beautiful aspect of Mandala® geometric thread design which you can now begin to realize is that the different patterns do not involve different threading methods, but arise entirely as a result of the arrangement of the outside generating points. This applies to the starburst pattern as well so that no special threading procedure or tricky counting is necessary. The sequence to follow in threading this design is identical to that used in threading circular designs. The smallest possible layer is formed by counting over (N — 1)/2 pins each time, the second smallest possible layer is formed by counting over (N — 3}/2 pins each time, etc. {Refer to Page 7.) There are certain features of the starburst pattern which may interfere with the effectiveness of the completed design unless carefully dealt with. One is that much of the thread becomes concentrated near the end with the higher density of points. By indiscriminately applying every possible layer of thread, the finished design appears crowded and imprecise. Another problem is that in this particular design, the smallest possible layer is usually so tiny that it creates an imbalance with the rest of the design. This tiny layer is also difficult to straighten due to the thick build-up of thread in such a small area. A general rule of thumb is to begin threading starburst designs with the second smallest possible layer, although this depends on the size of the design and the number of generating points involved. To alleviate the crowdedness that may result from applying many consecutive layers to the starburst design, it is best to skip some layers. A suggested layer sequence would consist of starting with the second smallest possible layer, then skipping to layer No. 4, 6, 8, and finishing with layer No. 11. That's five layers in all, and the results of this layer sequence with a 53-hole starburst pattern is shown on page 10. There are many different effects that can be achieved by manipulation of used and omitted layers, and it is fun to experiment around with these possibilities. 13
11. 11. Once a Mandala® pattern is threaded, a crucial finishing step is straightening the lines. Theoretically, the curves that the threads make should be perfect circles, or in the case of other patterns, ellipses. Due to the fact that we live in the real world, however, the threads won't be exactly in place. Using a small crochet hook, the threads can be moved to their correct positions and worked into a very precise arrangement. The effectiveness of the finished design is really a function of how well the threads are straightened; this step is vital. Straightening: before and after For artists who appreciate the effect of framing their creations, we recommend the use of clean, simple frame designs like the "floater" style. You want the frame to set off your Mandala® pattern, not distract from it, and one way of doing this is to use a smooth border, slightly offset from the piece as illustrated. 17
12. 12. variations of the basic patterns We've discussed the layout of the circular, elliptical, and starburst templates and how these desig threaded. Once these basic forms are understood, there are many more design possibilities. Some of the results obtained by combining the basic forms are shown on pages 6, 8, 14 and 16 double ellipse which suggests the whirling of electrons around the nucleus of an atom, is simply t intersecting ellipses. Combining four ellipses in a like manner yields the design on color page 8. Notice how each intersection of the ellipses on pages 8 and 14 is also a generating point. Beside being a magic trick in geometrical construction, this prevents the use of an odd number of genera points and therefore rules out all prime numbers. See appendix A for why this complicates the th of these designs, and keep this in mind before trying to do one. Ellipses of different "eccentricities" were used to create the effects shown on pages 6 and 16. "Disc in Rotation" depicts a disc in subsequent stages of rotation towards a light source. "Open Lissajou," named for a pattern frequently encountered in physics, simulates a similar dynamic effect. These are just a few examples of what can be done with the basic concepts explained earlier. Perhaps the most exciting Mandala® geometric thread designs are the three dimensional pieces : on pages 18 and 20. Spectacularly complex in appearance, these designs are the simple result of the different layers at various heights along poles inserted at the generating points. With this tech any pattern can be made three dimensional. The poles can be ordinary nails or, for a more profe finish, hardwood dowels cut to the desired length. An example of the steps in constructing a three dimensional Mandala® geometric thread design i given below for an ellipse: 1. Cut elliptical plywood base. 2. Drill 3/16" holes for dowels. 3. Insert dowels, pre-cut to desired length. 4. Paint. 5. Thread. 6. Straighten according to calculated spacings. 7. Glue thread in place. The locations of the generating points can be easily transferred from a paper template to the bas using a punch and a small hammer. 19
13. 13. You can tell from the 3-D's shown on pages 18 and 20 that the threads subtend a very smooth and symmetrical surface. The shape of the surface depends upon the spacing between layers. The inter!' of the circular 3-D on page 18 is spherical and the surface formed on page 20 outlines a paraboloi detailed account of how these spacings are determined is presented in Appendix C. It is very important that the dowels be perpendicular to the base. The best way of making sure of this to use a drill press for drilling the holes. Otherwise the posts will be out of alignment, resulting in a sloppy appearance. After the design is threaded, each layer is carefully adjusted to its predetermined height. Once the threads are in place, clear-drying white glue is brushed along the posts to hold the layers at their respective settings. 21