We present chemlambda (or the chemical concrete machine), an artificial chemistry with the following properties: (a) is Turing complete, (b) has a model of decentralized, distributed computing associated to it, (c) works at the level of individual (artificial) molecules, subject of reversible, but otherwise deterministic interactions with a small number of enzymes, (d) encodes information in the geometrical structure of the molecules and not in their numbers, (e) all interactions are purely local in space and time. This is part of a larger project to create computing, artificial chemistry and artificial life in a distributed context, using topological and graphical languages.

This paper considers the nature of the number in the light of second-order cybernetics.

The process equation At+1=At+gt* sin At formulates the basic principles of process theory: iteration models the discrete and unidirectional action temporal flow of energy ; the sine function models the cycling of complementary opposites. Demonstrating how the interaction of opposites generates novel and complex patterns, iteration generates convergence, periodicities, chaos, and bios. Novelty as defined and measured by the recurrence method differentiates bios from chaos. Bipolar feedback is necessary to generate novelty. Increasing the asymmetry of opposites in the modified equation At+1=At+g* (q+sin At), the complexity of pattern decreases as q increases. Bios occurs only when q > 0.05, and parabiotic patterns bios plus trend develop at larger q. Physiological and economic processes display biotic and parabiotic patterns, rather than chaos. These results illustrate and support the concept of cocreating opposites, an alternative to both deterministic and probabilistic scientific models.

The iteration of the simple equation A A g sin A generates t 1 t t fundamental numerical constants, ''biotic'' patterns with dynamic features observed in empirical recordings of physiological data but not in chaos , bifurcations, chaos, an infinite number of periodicities, and multiple flights toward infinity. For g 2, the equation converges to pi. At g 2, outcomes bifurcate and diverge. In a significant union of opposites, one path reaches the Fibonacci ratio describing spiral order when the opposite path achieves the Feigenbaum number describing chaos-inducing bifurcations. Chaotic patterns start when g approximates Feigenbaum's point 3.56 ... . Biotic patterns start at g 4.6 Feigenbaum's constant. Pointing to numerical cosmic forms, significant integer 2 n and irrational numbers occur as both outcomes and gain g. The equation embodies the basic postulates of process theory: 1 iteration models the linear flow of action i.e., time; 2 the sine function models the cycling of complementary opposites generating positive and negative feedback.

The article describes a tool for simplification and analysis of tangled configurations of mathematical knots. The proposed method addresses optimization issues common in energy based approaches to knot classification. In this class of methods, an initially tangled elastic rope is "charged" with an electrostatic like field which causes it to self repel, prompting it to evolve into a mechanically stable configuration. This configuration is believed to be characteristic for its knot type. We propose a physically based model to implicitly guard against isotopy violation during such evolution and suggest that a robust stochastic optimization procedure, simulated annealing, be used for the purpose of identifying the globally optimal solution. Because neither of these techniques depends on the properties of the energy function being optimized, our method is of general applicability, even though we applied it to a specific potential here. The method has successfully analyzed several complex tangles and is applicable to simplifying a large class of knots and links. Our work also shows that energy based techniques will not necessarily terminate in a unique configuration, thus we empirically refute a prior conjecture that one of the commonly used energy functions (J. Simon, 1994) is unimodal. Based on these results we also compare techniques that rely on geometric energy optimization to conventional algebraic methods with regards to their classification power.

A method for visualizing unknottedness of mathematical knots via energy optimization with simulated annealing is presented. In this method a potential field is formed around a tangled rope that causes it to self-repel. By allowing the rope to evolve in this field in search of an energy minimizing configuration we can determine the knot type of the initial configuration. In particular, it is natural to conjecture that if such a "charged rope" was not initially knotted, it will reach its minimal potential in a circular configuration, given a suitable energy functional. Because situations potentially arise in which the functional may not be strictly unimodal, we suggest it to be advantageous to use a robust stochastic optimization technique (simulated annealing), rather than a deterministic hill climber common in physically based approaches, to make sure that the evolving rope does not settle in a suboptimal configuration. The same method is applicable to simplifying arbitrary knots and links and for establishing knot equivalence. Aside from its theoretical appeal, the method promises to solve practical problems common in genetic research and polymer design.

Quaternions play a vital role in the representation of rotations in computer graphics, primarily for animation and user interfaces. Unfortunately, quaternion rotation is often left as an advanced topic in computer graphics education due to difficulties in portraying the four-dimensional space of the quaternions. One tool for overcoming these obstacles is the quaternion demonstrator, a physical visual aid consisting primarily of a belt. Every quaternion used to specify a rotation can be represented by fixing one end of the belt and rotating the other. Multiplication of quaternions is demonstrated by the composition of rotations, and the resulting twists in the belt depict visually how quaternions interpolate rotation. This article introduces to computer graphics the exponential notation that mathematicians have used to represent unit quaternions. Exponential notation combines the angle and axis of the rotation into concise quaternion expression. This notation allows the article to present more clearly a mechanical quaternion demonstrator consisting of a ribbon and a tag, and develop a computer simulation suitable for interactive educational packages. Local deformations and the belt trick are used to minimize the ribbon's twisting and simulate a natural-appearing interactive quaternion demonstrator.

The first half of a two-step quaternion Julia set visualization system is described. This step uses a quarternion square root function to adapt the classic inverse iteration algorithm to the quaternions. The augmented version produces a 3-D Julia set defined by a point cloud that can be interactively manipulated on a graphics workstation. Several cues are assigned to the point cloud to increase depth perception. Finally, a short theorem is proven that extends the domain of the inverse iteration method to a rotational family of quadratic quaternion Julia sets.>

As shown in 1982, Julia sets of quadratic functions as well as many other deterministic fractals exist in spaces of higher dimensionality than the complex plane. Originally a boundary-tracking algorithm was used to view these structures but required a large amount of storage space to operate. By ray tracing these objects, the storage facilities of a graphics workstation frame buffer are sufficient. A short discussion of a specific set of 3-D deterministic fractals precedes a full description of a ray-tracing algorithm applied to these objects. A comparison with the boundary-tracking method and applications to other 3-D deterministic fractals are also included.