2D and 3D Animation Using Rotations of a Jordan Curve

The authors discuss briefly a method and the art behind it to create the possible maximum number 2 of planar simple connected regions that can be distinguished with different binary string codes with length exactly n. This method uses the rotations of a single simple closed planar curve n times over 360/n degrees. The method creates planar diagrams which are called (rotational) symmetric Venn diagrams. The authors describe that how they created a 2D and 3D animation video to show this rotations for the case n=11. They also illustrate how this can be projected into the surface of a sphere creating marvelous spherical images. (A video will be presented at the 2007 Conference of Bridges Donostia, Mathematics, Music, Art, Architecture, Culture, and some images from it in this paper.) Introduction We discuss a method to create the maximum possible number 2 of planar or spherical simplyconnected regions that can be distinguished with different binary string codes of length exactly n. One of these methods is the following. Find and draw a simple closed curve in the plane (known as a Jordan curve) and a center of rotation in the interior but not on it. Rotate this curve first by 360/n degrees. Place the rotated curve on the top of the original curve. Some connected regions are created, see Figure 2. Some but few of the created regions will not be changed in the rest of the process. Mark those regions which will not be changed in the rest of the process. Then rotate the new curve again about the same center and by the same angle and place the double rotated curve on top of the two already drawn ones. (In fact this step can be considered such that second time the first curve is rotated about the given center over 2·360/n degrees.) Again some new regions are created. Some of them will not be changed by the rest of the process. Mark those new regions which will not be changed in the remaining part of the process. Repeat this n-1 times. (At the n rotation you rotate back to the original selected curve.) After each rotation draw the new curve on top of the already drawn curves and select those newly created regions which will not be changed in the rest of the process, (see Figure 3). At the end these are the regions that you label with different binary codes or with different colors. In this process one has created n copies of the Jordan curve; label them with numbers 1 through n. Now designate a length-n binary string code to each of these created regions; we call them n-binary string codes. It is clear that the maximum number of these regions cannot exceed 2. These string codes can be assigned to each of the region by writing a 1 in place k if the region is inside curve k and 0 if this region is outside of curve k, where k=1,2,...,n. Since we have n labeled curves we create an n long binary string code for each of the regions. This process is illustrated in Figures 1, 2, and 3. When you finish the creation of the planar diagram and its labeling, use the method of stereographic projection from the plane to the surface of a sphere to project the planar image to the surface of a sphere, (see Figure 4). By doing this you create a partition of the surface of a sphere into 2 regions and its labeling with 2 many length-n binary string codes. By coloring its orbits (see the definition of orbits below) you get marvelous spherical images, see Figures 10, 11. If we have to use all different binary string code with length-n for labeling the regions, then the diagram obtained above is an example of a so-called symmetrical planar Venn diagram with n curves. In an n-Venn diagram each of the n sets is represented as the interior of a Jordan curve. Furthermore, for each choice of any k sets, the regions inside the chosen sets (and outside of others) must be connected. An n-Venn diagram is called a (rotationally) symmetric Venn diagram if all the curves are obtained by rotating one of the curves about a common center by multiples 360/n degrees. (Diagrams in which some intersections fail to be connected are called independent families of curves; and diagrams in which some intersections fail to appear are called Euler diagrams. They first appeared in Euler’s famous Letters to a German Princess.) Figure 1: Rotation of a circle three times over 120 degrees, and the labeling of its regions results in the well known symmetric 3-Venn diagram. A symmetric 5-Venn diagram. Figure 2: Two curves and nine curves drawn in top of each other to create a symmetric 11-Venn diagram; the 11-Venn diagram is shown in the first image of Figure 6. For the history and for the definition of different type of Venn diagrams we refer the readers to papers [1, 2, 3, 4, 5, 7, 8, 9,] and [16]. For the details of the different type of constructions of symmetric Venn diagrams we refer the readers to papers [6, 10, 11, 2,] and [13]. In [12] it is said “It looks like just about anybody can draw n curves on the plane--and it might not even be hard to draw n non-self-intersecting curves to divide the plane into 2 connected regions. Then the challenge is a topological one, to get exactly one connected region for each possible n-digit code, so no in/out combination is missed, and none occurs in two regions. John Venn was the first, who showed that this could be done for every positive integer number n in 1880, [17]. He used a recursive process, starting from one curve, and then adding a new curve to the existing diagrams in each steps.” Later in the same paper it is said “To create a symmetric n-Venn diagram looks even simpler; you draw a curve, pick a center, rotate the curve to obtain all the other curves, and check that you indeed get 2 intersections and they are connected. The difficulty is that you do not know such curves. If you try with a curve you think will work, you often will not get all the intersections, and when you do they are often disconnected or the same binary code appears in two different regions.” For the complexity of the curves that are needed to create symmetric Venn diagrams see Figures 1 (second part), 4, and 7. Figure 3: The regions that are created exactly in the ninth rotation and in the first nine rotations in the process of drawing a symmetric 11-Venn diagram; this diagram is shows in the left of Figure 6. Figure 4: This curve creates this symmetric 7-Venn diagram, it has 128 regions. The illustration of the method of a stereographic projection. A mathematical method of finding a suitable curve for a symmetric p-Venn diagram The Boolean lattice Bn is the set of all subsets of {1,2,...,n}, partially ordered by inclusion. It can also be thought of as the n-dimensional hypercube, with vertices of n-tuples of 0’s and 1’s, in which two n-tuples are connected if and only if they differ exactly in one coordinate. Now it is clear that there is a one-to-one correspondence between the regions of the Venn diagram and the elements of the Boolean lattice and the n-hypercube, respectively. A symmetric chain is a path in the hypercube whose starting point has as many 0’s as its endpoint has 1’s, and in each step turns a 0 into a 1. This is equivalent to enlarging a subset in the Boolean lattice by one item in each step. A saturated chain is a chain with starting point and end point having different many 0’s and 1’s but does not skip any step in the enlarging process. A path, symmetric, or saturated chain partition separates the lattice into distinct paths or chains. Chain partitions and special paths are useful objects in the theory and application of Boolean lattice. Cipra writes: “In a paper titled “Doodles and Doilies, Non-Simple Symmetric Venn Diagrams,” presented in 1999 at a conference and published in 2002 in Discrete Mathematics, Hamburger introduced a new approach to the problem that enabled him to solve the n=11 casewith no computer search at all. Hamburger calls his rotationally symmetric diagrams “doilies” for the lacy crocheted pieces they resemble. His “doodles” are the key. Roughly speaking a doodle is a compact blueprint for a 2π/n wedge of the doily. Consisting of loops nested within loops, a doodle is derived from a study of symmetric chain decompositions of the Boolean lattice.” See [3] and Figure 5. Figure 5: The image of a symmetric and a saturated chain partition of the 7and the 11Boolean lattice. These chain decompositions are the blueprints for the diagrams in Figures 4 and the second image in Figure 6. The unconnected line segments must meet in the plane in infinity or at the South Pole on the sphere. Figure 6: A non-convex and a convex symmetric 11-Venn diagram. He continues later: “It’s not hard to turn a symmetric chain decomposition into a decomposition of the plane whose 2 regions correspond to subsets of {1, 2,..., n}. There are two tricky parts. One is to pick symmetric chain decomposition that produces a Venn diagram, rather than a jumble of disconnected duplicate intersections. The other is to get a diagram that is rotationally symmetric. But the real trick is to accomplish the two simultaneously.” “After a warm-up with n=7, Hamburger found a decomposition of B11 (actually a subposet thereof) into symmetric chains whose doodle produced a doily.” (See [3]). Hamburger’s method was enhanced with some additional techniques in [6] to show that this can be proved for all prime number p. Henderson showed that in 1963, symmetric Venn diagram cannot exist for composite n, see [14] and also [18] for a correction to the proof. Also it is easy to see that under the very specific rotation under which the diagram is symmetric a planar region that has a given bit string will become a region for which the bit string changes by shifting the last bit of the string to the first place and moving all the other bits by one place ahead. For example, the bit string 0111001 under one rotation becomes 1011100. This is called a shift. For a given fixed bit string the set of all shifts is called an orbit. In fact in a symmetric Venn diagram not only the cu