Sixty years ago in the lab adjacent to Fred Sanger (1958 Nobel Prize for protein chemistry), I discovered the cell surface hydroxyproline-rich glycoproteins. Nature keeps some of her secrets longer than others. It has taken many years to dissect the molecular function and biological role of extensins and arabinogalactan proteins (AGPs). Extensins template the formation of new cell walls. AGPs remained baffling and enigmatic until a Eureka moment when computer prediction of AGP calcium binding depicted paired glucuronic acid residues and thus the likely role of a cell surface AGP-Ca²⁺capacitor: In conjunction with the auxin-activated proton pump that releases bound Ca²⁺ it led us to formulate the Hechtian Growth Oscillator as A Global Paradigm with a pivotal role in Ca²⁺ homeostasis. The ramifications are profound. They cannot be shrugged off with sceptical disdain but demand critical reappraisal of current dogma. Phyllotaxis is an ancient problem; it involves an essential role for auxin and the auxin efflux “PIN” proteins together with mechanotransduction of stress-strain as phyllotactic determinants. However, a general explanation remains elusive despite much effort, particularly by mathematicians. Here we propose a novel biochemical algorithm: Hechtian oscillator transduction of cell wall stress generates phyllotactic patterns quite independent of a mathematical approach. Plants simply use different rules and follow a different route.

The article is devoted to applications of 2-dimensional hyperbolic numbers and their algebraic 2ⁿ-dimensional extensions in modeling some genetic and cultural phenomena. Mathematical properties of hyperbolic numbers and their bisymmetric matrix representations are described in a connection with their application to analyze the following structures: alphabets of DNA nucleobases; inherited phyllotaxis phenomena; Punnett squares in Mendelian genetics; the psychophysical Weber-Fechner law; long literary Russian texts (in their special binary representations). New methods of algebraic analysis of the harmony of musical works are proposed, taking into account the innate predisposition of people to music. The hypothesis is put forward that sets of eigenvectors of matrix representations of basis units of 2ⁿ-dimensional hyperbolic numbers play an important role in transmitting biological information. A general hyperbolic rule regarding the oligomer cooperative organization of different genomes is described jointly with its quantum-information model. Besides, the hypothesis about some analog of the Weber-Fechner law for sequences of spikes in single nerve fibers is formulated. The proposed algebraic approach is connected with the theme of the grammar of biology and applications of bisymmetric doubly stochastic matrices. Applications of hyperbolic numbers reveal hidden interrelations between structures of different biological and physical phenomena. They lead to new approaches in mathematical modeling genetic phenomena and innate biological structures.

In this paper, the sequential prime numbers are used as variables for the galactic spiral equations which were developed from the ROTASE model. Special spiral patterns are produced when prime numbers are treated with the unit of radian. The special spiral patterns produced with the first 1000 prime numbers have 20 spirals arranged in two groups. The two groups have perfect central symmetry with each other and are separated with two spiral gaps. The special spiral pattern produced with natural numbers from 1 to 7919 shows 6 spirals in the central area and 44 spirals in the outer area. The whole 7919 spiral points can be plotted with either 6-spiral pattern or 44-spiral pattern. For the spirals only produced by the prime numbers in the 6-spiral pattern plotting, the spiral 2 and spiral 3 each has only one spiral point produced by prime number 2 and 3, respectively, all other spiral points produced by other prime numbers are located on the spiral 1 and spiral 5. The special spiral pattern is well explained with careful analysis, it is concluded that all prime numbers greater than 3 must meet one of the equations: P₁ = 1 + 6 * n (n > 0) P₅ = 5 + 6 * n (n ≥ 0) In other words, every prime number greater than 3 is either a P₁ prime number or a P₅ prime number, no exception. Matching one of the equations is a necessary condition for a number to be a prime number, not a sufficient condition. Hope such sufficient condition can be found in the future. The number of P₁ prime numbers roughly equal the number of P₅ prime number in the first 2 billion prime numbers. The galactic spiral equations with golden angle can duplicate Vogel’s result for the simulation of sunflower seed head pattern, and a pinwheel pattern can be produced also with galactic spiral equations and 1 degree more than golden angle.

The promoter of the RECEPTOR-LIKE CYTOPLASMIC KINASE VI_A2 gene contains nine binding sites for the REPLUMLESS transcription factor. In agreement, the expression of the kinase gene was strongly downregulated in the rpl-4 mutant. Comparing phenotypes of loss-of-function mutants, it was revealed that both genes are involved in stem growth, phyllotaxis, organization of the vascular tissues, and the replum, highlighting potential functional interactions. Expression of the RLCKVI_A2 gene from the constitutive 35S promoter could not complement the rpl-4 phenotypes, but exhibited a dominant positive effect on stem growth and affected vascular differentiation and organization. The results also indicated that the number of vascular bundles is independently regulated from stem thickness. Although our study cannot demonstrate the direct link between the RPL and RLVKVI_A2 genes, it highlights the significance of the proper developmental regulation of the RLCKVI_A2 promoter for balanced stem development.

Voronoi mosaics inspired by the seed points placed on the Archimedes Spirals are reported. Voronoi entropy was calculated for these patterns. Equidistant and non-equidistant patterns are treated. Voronoi mosaics built from a sells of equal size which are of a primary importance for decorative arts are reported. The pronounced prevalence of hexagons is inherent for the patterns with an equidistant and non-equidistant distribution of points, when the distance between the seed points is of the same order of magnitude as the distance between the turns of the spiral. Penta- and heptagonal “defected” cells appeared in the Voronoi diagrams due to the finite nature of the pattern. The ordered Voronoi tessellations demonstrating the Voronoi entropy larger than 1.71, reported for the random 2D distribution of points, were revealed. The dependence of the Voronoi entropy on the total number of the seed points located on the Archimedes Spirals is reported. Voronoi tessellations generated by the phyllotaxis-inspired patterns are addressed. The aesthetic attraction of the Voronoi mosaics arising from seed points placed on the Archimedes Spirals is discussed.