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The Dream-Catcher. Draw a circle, and select an odd number of points, evenly spaced, on that circle--the base circle. Connect each of these nodes to every one of the other nodes with a line segment. The resulting figure is known to many Native American tribes as a Dream-Catcher; it is also commonly called a God's Eye. Physical models of this figure are sometimes constructed from a circle formed out of wire with taut pieces of string forming the line segments. A Dream-Catcher on n nodes requires n (n - 1)/2 line segments, so the number of nodes craftsmen put into such models is limited. The figure seen here has 31 nodes and is therefore rather more intricate than all but the most dedicated would attempt to construct from wire and string. All of the mandalas on this site are the result of applying one or more transformations to this basic figure.

A Poincare Dream-Catcher. The Dream-Catcher can be thought of as a collection of inter-connected collections of parallel lines in the Beltrami-Klein model of the hyperbolic plane. (But note that the nodes and the bounding circle itself are not part of the model--only those points interior to the circle count.) Each collection of segments emanating from the same node represents one family of parallels in this model. The mandala shown here is based on applying the Dream-Catcher construction to the Poincare model of the hyperbolic plane. In this model, the plane is again represented as the interior of a circle, but lines are represented by arcs of circles that are orthogonal to the base circle. So here we have again chosen 31 nodes equally spaced around the base circle, constructed a circle that passes through each pair of nodes in such a way that its tangent at that node is perpendicular to the tangent to the base circle at that same node, and then drawn only that portion of the arc the lies on or inside the base circle. This can be realized as a simple geometric transformation; see Euclidean and Non-Euclidean Geometries, M. J. Greenberg, 2nd Ed., W. H. Freeman and Co., New York, 1980, pg. 190.

A Node Inverted Dream-Catcher. Here we have chosen a circle, whose center lies directly below the center of the Dream-Cather's base circle, and which is orthogonal to that base circle. (It actually meets the base circle of the Dream-Catcher at points that just about divide the arc that forms the bottom half of that circle into thirds.) We have applied an inversion with respect to the new circle (see notes on Inversion with Respect to a Circle) to the base circle of our 31-node Dream-Catcher and followed its action on the Dream-Catcher's nodes. Then we have connected the transformed nodes, as with the original, with straight line segments. This is the resulting figure.

An Inverted Dream-Catcher. To produce this image from the basic Dream-Catcher, we applied the same transformation, inversion with respect to a certain circle orthogonal to the base circle, not just to the nodes this time, but to the line segments of the dream catcher as well. Circle inversion transforms the family of lines and circles into itself; in this case, the line segments of the Dream-Catcher become arcs of circles.

An Inverted Poincare Dream-Catcher. Here we've applied the same circle inversion to the Poincare Dream-Catcher. Because circle inversion is conformal, the arcs of orthogonal circles from which the Poincare Dream-Catcher is constructed are carried to arcs of other orthogonal circles. Thus, we obtain the same figure whether we apply the inversion only to the nodes, and then connect by arcs of orthogonal circles, on the one hand, or apply the inversion to the whole figure.

A Squared Dream-Catcher. Here we treat the basic Dream-Catcher as though it were embedded in the unit disk in the complex plane, the unit circle being the base circle. Then we apply the complex squaring function to the points of that disk. The result is quite pleasing, and contains a new level of complexity. Note the "condensation" at the center of the disk. It arises from the fact that the squaring function "attracts" the inner parts of the disk closer to the origin. Here is another view of the squaring function in the complex plane. If you have trouble with it, you can probably find out why by going to the Mathematics Animated page.

A Squared Poincare Dream-Catcher. Complex squaring, again; this time applied to the Poincare Dream-Catcher. Note how the sheaves of arcs entering the nodes have become "pointier". Note also, that the arcs connecting adjacent nodes seem to have disappeared. They're still there, but now must detour deeper into the figure, almost passing through the center, because the nodes that appear adjacent in this figure started out almost diametrically opposed in the original figure. The same effect is visible in the previous figure, though it doesn't seem as strong to my eye as it does here.

A Cubed Dream-Catcher. Here we've applied the complex cubing function to the Dream-Catcher. Note the heavy condensation near the center of the figure. The cubing function has a very strong attractor at the origin.

A Rescaled Cubed Dream-Catcher. Another application of the complex cubing function to the basic Dream-Catcher, but this time we've opened up the condensation at the center by dividing out the the modulus. That is, we applied the map that transforms the complex number z into the complex number (z^3)/|z|. The result displays very intricate structure

A Squared Squared Dream-Catcher. This one is the complex fourth-power function. It probably needs to be re-scaled, too. But I never got around to it.