Image 109. Three independent logistic map processes, x = A x (1-x), provide coefficient values for the recurrence relation. The two thematic shapes are the result of an interplay between hyperbola-shaped and cardioid-shaped orbit traps.
Image 92. Modular arithmetic is used to produce a repeating border pattern. Activation of the orbit trap is delayed two iterations, and orbits are not trapped within an elliptical region around the origin, in order to reveal this motif, which otherwise would have been obscured by premature orbit termination.
Image 98-8. The eight figures in this image are created independently by the same computer code that differs only in a single three digit number specified in the first line of the code. This is the layout for the smaller 16″x12″ canvas piece.
Image 98-18. The eighteen figures in this image are created independently by the same computer code that differs only in a single three digit number specified in the first line of the code. This is the layout for the larger 24″x20″ canvas piece.
Image 91. Two nearly identical orbit traps produce the water and cloud shapes. A bit-shifting pseudo-random number generator provides coefficients for the cubic/quadratic recurrence relation.
Image 102. A sinusoidal function of the iteration counter provides a coefficient for one term in the numerator of the recurrence relation. A second sinusoidal function creates a wavy edge on one of the orbit trap shapes.
Image 93. The logistic map, x = A x (1-x), is used to generate coefficients for a cubic/quadratic recurrence. A sinusoidal orbit trap produces the rippled features of the landscape. The orbit trap is ignored on selected iterations to reveal additional features.
Image 94. A simple quadratic recurrence involving the inverse of the golden ratio, (√5-1)/2, is trapped by a braid-shaped orbit trap to produce the warm-colored shapes. The cool-colored shapes are produced by trapping orbits that wander very far (>1010) in the horizontal direction.
Image 51. The recurrence z = (3 z4 + c) / (4 z3), where c = -4 on the first iteration and c = 1 thereafter, is performed until an iterate is trapped by a narrow vertical or horizontal strip. The lower left corner of this image is the entire image 79.
Image 79. The recurrence z = (3 z4 + c) / (4 z3), where c = -4 on the first iteration and c = 1 thereafter, is performed until an iterate is trapped by a narrow vertical or horizontal strip. This image comes from the lower left corner of image 51.
Image 62. Irrational numbers are used to generate pseudo-random number sequences by shifting the bits with successive iterations. Orbits are trapped when they land within a narrow ring of unit radius, centered at the origin.
Image 77. The coefficients of a cubic/quadratic generator are updated using trigonometric functions of the iteration count. The orbit trap also uses a trig function to give the resulting shapes a distinctive waviness.
Image 78. A system of equations related to the stress analysis and optimal design of a civil engineering frame structure is solved by the method of steepest descent with a simplified Newton line search.
Image 80. The ratio of two successive Fibonacci numbers, which asymptotically approaches the golden ratio, is integral to this algorithm. Orbits are trapped by wedge and zipper shaped regions, yielding the wispy shapes and plant-like structures.
Image 84. The ratio of two successive Fibonacci numbers, which asymptotically approaches the golden ratio, is used as a coefficient in the recurrence. The orbit trap is designed to produce a woven pattern.
Image 88. The coefficients of a cubic/quadratic recurrence are updated using trigonometric functions of the iteration count. The orbit trap is designed to produce a woven pattern.
Image 83. The coefficients of this cubic/quadratic recurrence are updated using trigonometric functions of the iteration count. The orbit trap is designed to produce a woven pattern.
Image 67. The iteration count is used to define the ever-increasing degree of the recurrence relation. Orbits are trapped when they fall within a saw-tooth shaped region running diagonally across the plane.
Image 106. A function of the fractional part of the imaginary part of each iterate defines one part of the orbit trap used in this algorithm. The logistic map, x = A x (1-x), is used to generate a coefficient in the recurrence relation.
Image 105. This algorithm stands apart from all previous ones because its innermost loop, iteration of the recurrence relation, is missing. Instead, just one evaluation of the recurrence is sufficient to create the dark blue shape.
Image 104. The logistic map, x = A x (1-x), is known to generate a chaotic sequence for certain values of constant A. The recurrence relation in this algorithm makes use of successive values of a logistic map sequence to provide exponent values for two of its terms.