INTRODUCTION
French biochemist and crystallographer Gérard Langlet (1940-1996) wrote a series of papers describing a
radically new, but simple, approach to scientific computing. This series culminated in his 1992 paper,
modestly subtitled “TOE” (i.e., Theory of Everything) in which he presented the fully worked-out version
of his computational system. Yet, today, a Google search will likely turn up none of his published works
and only one or two articles about him (below is a list of articles by or about Langlet). How could it be that
a charismatic professional scientist who wrote knowledgably about current issues in many scientific fields
is virtually unknown?
Perhaps the primary reason that Langlet is not better known is that he chose to write mainly for the APL
community. Within the community of APL implementers, application developers, and users, he was
regarded with almost the same respect and admiration as the creator of APL, Ken Iverson. But the APL
community has always been somewhat isolated from mainstream computer science. Dijkstra, for example,
dismissed APL with the gibe that “APL is a mistake, carried through to perfection. It is the language of the
future for the programming techniques of the past: it creates a new generation of coding bums.” This
overstates the typical view computer scientists have of APL, but not by much. Langlet tirelessly sought to
overcome this attitude and to show people what he perceived as the superiority of APL, especially for
computing in the sciences and engineering. Virtually every one of the articles he published in English is at
least partly a sales pitch for APL. This somewhat un-scholarly approach may have turned off computer
scientists accustomed to the conventions of traditional technical literature. It may also be that even within
the APL community Langlet’s single-minded devotion to APL began to look outdated by the early 1990’s
as a younger group of APL-influenced language designers and implementers started to create new
programming languages. Among these were Roger Hui’s “J,” Mike Jenkins’ “Q’Nial,” and a series of
powerful but very compact languages by Arthur Whitney. Each of these had features that were arguably
significant advances on APL, and they have consequently drawn increasing attention from the APL
community.
Another reason that Langlet is not better known is that the papers he published in English—as fascinating
as most of them are—rarely deliver a significant result or solution to a problem. For example, in his major
1992 paper, Towards the Ultimate APL-TOE, he suggests that his “Helical” transform can be used as a very
efficient signal transform, like the FFT. He shows the transform applied to a short text string but then says
nothing further about it. One is left to wonder what the transform has revealed about the string. The same
paper contains similarly short-circuited examples from physics, chemistry, and biology; however, the logic
of Langlet’s computational system is carefully worked out in the paper, which begins with the usual
mathematical framework of definitions, axioms, and theorems. But after reading the paper, one isn’t sure
what can actually be done with the system. After several years of on again/off again attempts to use
Langlet’s system to solve problems in what seemed to be obviously suitable application areas, it became
increasingly clear to me that the system is a solution in search of a problem. The near total absence of
citations of Langlet’s work bears this out.
A final reason that Langlet is not better known is that, after the publication of Stephen Wolfram’s A New
Kind of Science, his work was completely superseded by Wolfram’s and that of the many scientists and
scholars who have built on it. While Langlet and Wolfram shared some important new ideas about
computing in the sciences, Wolfram considered computation over a much wider range of applications and
with a much deeper knowledge of computer science. Although some reviewers of A New Kind of Science
have criticized Wolfram’s “experimental” mathematical methods, the book has nonetheless been influential
in many fields and it remains a force to be reckoned with. Many thousands of papers based on A New Kind
of Science have been published.
The few papers I’ve shared on this Web page represent my best attempts to understand and apply Langlet’s
system. Sierpinski and Thue-Morse: The Langlet Connection contains one of the few interesting results I
have obtained using Langlet’s system, namely that the Thue-Morse sequence and the Sierpinski Triangle
are connected by a very simple relationship that can be discovered with Langlet’s “Helical Transform.”
This problem was suggested by David Callan *, who described a much more complex relationship. Another
modestly interesting result is presented in the paper Texture Analysis with Langlet’s Transforms, where I
show how visual features hidden inside complex textures can be revealed with the aid of Langlet’s Helical
and Cognitive transforms. Analyzing Counterchange Patterns with Langlet Transforms explores some
related aspects of visual texture in connection with wallpaper and fabric pattern design.
Reflections on Langlet’s APL Theory of Human Vision was my first attempt to criticize some of the more
ambitious claims made for Langlet’s system, both by Langlet himself and by others. In Search of the
“World Formula” undertakes a critique of the foundations of Langlet’s entire system. Wolfram’s Rule 60
end Extended Additivity actually goes back to my first attempts to understand and apply Langlet’s ideas in
the late 1990’s. Like my “World Formula” paper, it is heavily indebted to Wolfram’s book, A New Kind of
Science. Finally, I included APL Wish Lists as an antidote to Langlet’s tunnel-visioned enthusiasm for APL.
This paper shows that during the time Langlet wrote his major papers there were already excellent
alternatives to APL within the overall array-based programming paradigm.
The Matlab Langlet Toolkit** I’ve provided will allow anyone who is interested to play with Langletian
ideas. Most of the basic functions are so small and simple that they can easily be translated into any
programming language. I’ve done so in APL2, C, C++, Q’Nial, J, and Q. I chose Matlab for the Toolkit
largely because it’s the language I’ve mostly worked in for the last decade and because it has nice built-in
graphics capabilities, which allow manipulation of the kinds of binary images that Langlet often used as
examples. If you get a significant result with Langlet’s system or have comments, please get in touch.
-Stu Smith
stuart.smith3@comcast.net
* David Callan. Sierpinski’s Triangle and the Prouhet-Thue-Morse Word.
http://www.stat.wisc.edu/~callan/notes/Thue_Sierpinski/Thue_Sierpinski.pdf
** Stuart Smith. Langlet Toolkit. www.cs.uml.edu/~stu
Publications by and about Langlet
Gérard Langlet. Paritons and Cognitons: Towards a New Theory of Information. Vector, 19:3, p.93.
Gérard Langlet. A Quite Different New Primitive. Vector, 12:1, p.93.
Gérard Langlet. Letter: More Thoughts on Dragons. Vector, 12:1, p.121.
Gérard Langlet. Letter: Non-syllogistic mathematical proof. Vector, 12:1, p.123.
Gérard Langlet. The APL Theory of Human Vision. Vector, 11:3, p.42.
Gérard Langlet. The Axiom Waltz – or When 1+1 make Zero. Vector, 11:3, p.101.
Gérard Langlet. APL94: Binary Algebra Workshop. Vector, 11:2, p.60.
Gérard Langlet. Chaotic behaviour revisited. Vector, 11:2, p.82.
Gérard Langlet. The Ultimate Turing Proof. Vector, 10:3, p.124.
Gérard Langlet. The Fractal MAGIC Universe. Vector, 10:1, p.137.
Gérard Langlet. From the Vital Execute to Fractals and 5-fold Symmetry. Vector, 9:3, p.91.
Gérard Langlet. Recreation with Transcendental Numbers. Vector, 8:2, p.25.
Gérard Langlet. The Steam Hammer and the Fly. Vector, 7:4, p.138.
Gérard Langlet. APL “RISC Programming Style”. Vector, 6:2, p.23.
Gérard Langlet Towards the Ultimate APL-TOE. ACM SIGAPL Quote Quad, 23(1) (July 1992).
Sylvia Camacho. Gérard Langlet: A Man of Distinction, 22:2, p.56.
Anthony Camacho, ed. Obituary: Memories of Gérard Langlet. Vector, 13:4, p.138.
Anthony Camacho. Editorial: Monsieur Langlet’s Enterprise. Vector, 11:2, p.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.