1. Stuart Smith stuart.smith3@comcast.net
“Natural” Shapes
Stuart Smith

2. 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

3. 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.

4. 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.

5. Typical Wolfram Image

6. Typical Langlet Image

7. 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.

8. 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).

9. 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.

10. 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.

11. 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 user- selectable line, fill, and background colors.
The same can be done in Mathematica, Matlab, and other contemporary programming languages.

12. Analogy to Spirograph™

13. Simple Images

14. More complex Images

15. …and everything in between.

16. 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

22. 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.

23. 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.

24. The End