Combinatorial Space Tiling

The present article studies combinatorial tilings of Euclidean or spherical spaces by polytopes, serving two main purposes: first, to survey some of the main developments in combinatorial space tiling; and second, to highlight some new and some old open problems in this area.Comment: 16 pages; to appear in "Symmetry: Culture and Science

Self-Reference, Biologic and the Structure of Reproduction

This paper concentrates on relationships of formal systems with biology. The paper is based on previous papers by the author. We have freely used texts of those papers where the formulations are of use, and we have extended the concepts and discussions herein considerably beyond the earlier work. We concentrate on formal systems not only for the sake of showing how there is a fundamental mathematical structure to biology, but also to consider and reconsider philosophical and phenomenological points of view in relation to natural science and mathematics. The relationship with phenomenology comes about in the questions that arise about the nature of the observer in relation to the observed that arise in philosophy, but also in science in the very act of determining the context and models upon which it shall be based.We examine the schema behind the reproduction of DNA. The DNA molecule consists of two interwound strands, the Watson Strand (W) and the Crick Strand (C). The two strands are bonded to each other via a backbone of base-pairings and these bonds can be broken by certain enzymes present in the cell. In reproduction of DNA the bonds between the two strands are broken and the two strands then acquire the needed complementary base molecules from the cellular environment to reconstitute each a separate copy of the DNA. At this level the situation can be described by a symbolism like this. DNA = -------> --------> = = DNA DNA. Here E stands for the environment of the cell. The first arrow denotes the separation of the DNA into the two strands. The second arrow denotes the action between the bare strands and the environment that leads to the production of the two DNA molecules. The paper considers and compares many formalisms for self-replication, including aspects of quantum formalism and the Temperley-Lieb algebra.Comment: LaTeX document, 71 pages, 33 figures. arXiv admin note: substantial text overlap with arXiv:quant-ph/020400