Natural Shapes Stuart Smith stuart smith 3comcast net

“Natural” Shapes Stuart Smith stuart. smith 3@comcast. net

Scientific Modeling from Scratch • Wolfram[1], Langlet[2], Zaus[3] propose new ways to do scientific modeling: • Little or no use of continuous mathematics • Emphasis on Boolean operations • Extensive use of visualization

Visualization • Ubiquitous graphics programs allow informative and attractive presentation of the results of the new methods. • Wolfram’s A New Kind of Science a magisterial exposition of the new methods AND a great “coffee table” art book • Langlet’s paper Building the APL Atlas of Natural Shapes also shows the artistic potential of the new methods.

Wolfram vs. Langlet • Wolfram shows successive states in the evolution of a system, with time increasing down the image. • Langlet emphasizes the final state of a given process and presents it as a mandala, snowflake, or other symmetrical form.

Typical Wolfram Image

Typical Langlet Image

The Bit-String • Essential common feature of both Wolfram and Langlet: generation of a bit-string • Wolfram renders each bit-string as a line of black and white squares. Successive lines together form a rectangular picture of the system’s evolution. • Langlet repeatedly modifies one string and then uses the final modification to direct a graphical “turtle” to trace out the contours of the result image.

Ad Hockery in Langlet’s Approach • Wolfram’s method is simple and consistent, and it has plausible connections with actual natural processes • Langlet’s method involves several ad hoc modifications of the bit string that are hard to justify from a “natural” point of view. • Langlet also offers an “irregularity” option to obtain additional variety in the images generated. This is not only ad hoc, but it also requires floating-point arithmetic–a clear violation of Langletian principles (every operation must have an exact inverse).

Goal: generate Langlet’s pretty pictures with a consistent method • Retain Langlet’s geometric framework. • Substitute a version of Wolfram’s simple cellular automaton for Langlet’s L-system to generate the initial bit-string • Remove Langlet’s arbitrary bit-string modifications and “irregularity. ” • Allow the initial bit-string to be generated randomly. For complete consistency this can be done with the simple cellular automaton.

More on “Irregularity” • Wolfram’s simple cellular automaton can operate according to any of 256 different rules. • For many of these rules, step n in the evolution of the automaton cannot be calculated directly from the initial bit string. To obtain step n, it is necessary to calculate all of the preceding steps. • Therefore, there is no need for an “irregularity” feature to add variety or unpredictability. This feature is built right into the automaton.

Additional feature: color • Since the time of Langlet’s original paper, it has become trivial to render the images in color. • In Dyalog APL, the poly class will make sure that a sequence of points represents a closed contour and then render an image with userselectable line, fill, and background colors. • The same can be done in Mathematica, Matlab, and other contemporary programming languages.

Analogy to Spirograph™

Simple Images

More complex Images

…and everything in between.

Auditory display of input values • symmetry: the higher the order of symmetry the higher the frequency • rule: the greater the rule number the greater the depth of tremolo • length: the longer the contour bit-string the faster the rate of tremolo • duration: the greater the number of steps of the cellular automaton the longer the duration

Has the goal been reached? • We can make pretty pictures with Langlet+Wolfram, but… • Langlet wanted to show that the pictures somehow mirrored actual natural processes. This is doubtful. • What we have is an artistic diversion, a tool that can produce visually appealing patterns. • The programs are also a sales pitch for APL (Langlet constantly promoted the language). The main program is ~60 lines of APL code.

References [1] Stephen Wolfram. A New Kind of Science. Wolfram Media (2002). [2] Gérard Langlet. Building the APL Atlas of Natural Shapes. APL ’ 93 Proceedings of the International Conference on APL. New York, NY: ACM (1993). [3] Michael Zaus. Crisp and Soft Computing with Hypercubical Calculus: New Approaches to Modeling in Cognitive Science and Technology with Parity Logic, Fuzzy Logic, and Evolutionary Computing: Volume 27 of Studies in Fuzziness and Soft Computing. Physica-Verlag HD, 1999.

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