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Robert F. Kauffmann USA

1. Dragon Limit I, handcolored Serigraph, 64 x 30 cm / 18 x12 in. US$550 2. Dragon Limit II, handcolored Serigraph, 64 x 30 cm / 18 x12 in. US$500 3. TDragon Limit III, handcolored Serigraph, 64 x 30 cm / 18 x12 in. US$550 4. Dragon Limit IV, handcolored Serigraph, 64 x 30 cm / 18 x12 in. US$500 5. Magic Castle, Serigraph 51 x 41 cm / 20 x 16 in. US$400 6. Moth, Serigraph, 30 x 64 cm / 12 x18 in. US$350 7. Octagon Limit, Serigraph, 36 x 36 cm / 14 x 14 in. US$400 8. Magnetically Distorted City, Serigraph, 64 x 30 cm / 18 x12 in. US$400 9. Division of Plane with Dragon, Serigraph, 61 x 46 cm / 24 x18 in. US$400 10. Bifurcating Fish, Serigraph, 15 x 23 cm / 6 x 9 in. US$350 11. Tree, Serigraph, 30 x 64 cm / 12 x18 in. US$300 12. Sunflower, Serigraph, 61 x 46 cm / 24 x18 in. US$400 
1.4.The
Dragon Limit
series takes Escher's idea of approaching limits to infinity and
generalizes it to include the tiling of fractal figures which approach a
bounding limit which is also fractal. The fractal in this case is based on
constructions by Mandelbrot of a figure called a "Twin Dragon" which he
showed could be subdivided in order to tile a plane. The Dragon Limit
series consists of four variations arising from two different coloring
schemes and two different methods of dividing the space.
In this first example from the Dragon Limit series, design, the
dragontile approaches the border limit without ever actually touching it.
This can be seen as an illustration of Xeno's paradox. This also
illustrates the concept of "open sets" in set theory, which specifies an
area on a plane minus its border. (think of a solid object with its skin
removed.) The exact definition of an open set is actually somewhat less
intuitive. It is any set in which all of its points are a nonzero
"distance" from any point outside the set.
5.Magic Castle : This design is based on the "impossible perspective" prints by M.C. Escher (BELVEDERE, WATERFALL, ASCENDING AND DESCENDING). The design takes advantage of the loss of information inherent in a projection of 3D space to 2D space to create the optical illusion of a 3D structure which seems to recede into more than one direction at a time. Roger Penrose stated that such structures were impossible in three dimensionsimplying that such might be possible in higher dimensions. Penrose and his father were the first to experiment with "impossible perspectives."
6.Moth : An exercise in recursion, this moth has pictures of itself on its wings, which in turn have ever smaller images of the moth recursing to infinity. An exquisite work, the artist has managed to perfectly capture the delicacy and intricacy of both the moth and the leaf.
7.Octagon Limit : This work uses another of Kauffmann's designs, BIFURCATING FISH (see below), to create a complex recursive pattern which approaches infinity as it approaches the edge of the octagon.
8.Magnetically Distorted City : This ambitious design took ten years of thought to work out enough problems to actually execute. The execution took over one year. It portrays a cityscape whose perspective is projected onto magnetic lines of force from a point particle and their orthogonal trajectories (a set of curves crossing at right angles to a first set of curves at every point.) The central point serves as the vanishing point (VP) of north, south, and upwards. A pair of circles tangent to center was chosen as the horizon. The VP's for east and west lie on these 'horizon circles' opposite to the center. The VP for down is not a point, but rather the line running vertical through the center of the picture. The effect is similar to viewing a cityscape through a reflective ball, but the actual geodesics are different.
9.Division of Plane with Dragon : This piece started as an exercise in topology, the question being, "Can an interesting composition be created by drawing a single continuous line?" The eventual result was a single sinuous curve dividing a plane into two halves(red and white) which defines the image of a coiled dragon striking. A sideeffect of this design is that the difference between figure and ground becomes irreconcilably ambiguous.
10.Bifurcating Fish : This piece is named for its structural resemblance to a bifurcation diagram (a complicated fractal diagram used to map the behavior of certain nonlinear functions.) This design was also inspired by Escher's FISH AND SCALES.
11.Tree : A treelike shape forming a series of diminishing cells was a precursor, geometrically, of OCTAGON LIMIT and BIFURCATING FISH.
12.Sunflower : Based on an idea similar to TURTLE & SHELL (see below), each little cell in the center portrays a copy of the whole, recursing infinitely. The cells form spiral patterns radiating from the center, with counterspiral patterns intersecting at right angles. A striking pattern in itself, it becomes mesmerizing when combined with infinite recursion. The spiral pattern described above (minus recursion) can be observed in sunflowers and other species in nature. Escher's PATH OF LIFE series is based upon this geometry.
© Robert Kauffmann. All Rights Reserved.