Hyperbolic Fibonacci and Lucas Functions, “Golden” Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s Fourth Problem—Part II. A New Geometric Theory of Phyllotaxis (Bodnar’s Geometry)

This article refers to the “Mathematics of Harmony” by Alexey Stakhov in 2009, a new interdisciplinary direction of modern science. The main goal of the article is to describe two modern scientific discoveries–New Geometric Theory of Phyllotaxis (Bodnar’s Geometry) and Hilbert’s Fourth Problem based on the Hyperbolic Fibonacci and Lucas Functions and “Golden” Fibonacci λ-Goniometry (λ > 0 is a given positive real number). Although these discoveries refer to different areas of science (mathematics and theoretical botany), however they are based on one and the same scientific ideas-the “golden mean,” which had been introduced by Euclid in his Elements, and its generalization—the “metallic means,” which have been studied recently by Argentinian mathematician Vera Spinadel. The article is a confirmation of interdisciplinary character of the “Mathematics of Harmony”, which originates from Euclid’s Elements.

Phyllotaxis-inspired nanosieves with multiplexed orbital angular momentum

Nanophotonic platforms such as metasurfaces,achieving arbitrary phase profiles within ultrathin thickness,emerge as miniaturized,ultracompact and kaleidoscopic optical vortex generators.However,it is often required to segment or interleave independent sub-array metasurfaces to multiplex optical vortices in a single nano-device,which in turn affects the device’s compactness and channel capacity.Here,inspired by phyllotaxis patterns in pine cones and sunflowers,we theoretically prove and experimentally report that multiple optical vortices can be produced in a single compact phyllotaxis nanosieve,both in free space and on a chip,where one meta-atom may contribute to many vortices simultaneously.The time-resolved dynamics of on-chip interference wavefronts between multiple plasmonic vortices was revealed by ultrafast time-resolved photoemission electron microscopy.Our nature-inspired optical vortex generator would facilitate various vortex-related optical applications,including structured wavefront shaping,free-space and plasmonic vortices,and high-capacity information metaphotonics.

Hyperbolic Fibonacci and Lucas Functions, “Golden” Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s Fourth Problem—Part III. An Original Solution of Hilbert’s Fourth Problem

This article refers to the “Mathematics of Harmony” by Alexey Stakhov [1], a new interdisciplinary direction of modern science. The main goal of the article is to describe two modern scientific discoveries—New Geometric Theory of Phyl-lotaxis (Bodnar’s Geometry) and Hilbert’s Fourth Problem based on the Hyperbolic Fibonacci and Lucas Functions and “Golden” Fibonacci -Goniometry ( is a given positive real number). Although these discoveries refer to different areas of science (mathematics and theoretical botany), however they are based on one and the same scien-tific ideas—The “golden mean,” which had been introduced by Euclid in his Elements, and its generalization—The “metallic means,” which have been studied recently by Argentinian mathematician Vera Spinadel. The article is a confirmation of interdisciplinary character of the “Mathematics of Harmony”, which originates from Euclid’s Elements.

Characterization of a <i>Tos</i>17 Insertion Mutant of Rice Auxin Signal Transcription Factor Gene, <i>OsARF</i>24

Auxin signaling plays a key role in the regulation of various growth and developmental processes in higher plants. Auxin response factors (ARFs) are transcription factors that regulate the expression of auxin-response genes. The osarf24-1 mutant contains a truncation of domain IV in the C-terminal dimerization domain of a rice ARF protein, OsARF24. This mutant showed auxin-deficient phenotypes and reduced sensitivity to auxin. However, OsARF24 protein contains an SPL-rich repression domain in its middle region and acts as a transcriptional repressor. These results imply that the C-terminal dimerization domain, especially the C-terminal half of domain IV, is essential for the proper regulation of OsARF24 function as a transcriptional repressor in rice.

Hyperbolic Fibonacci and Lucas Functions, “Golden” Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s——Part I. Hyperbolic Fibonacci and Lucas Functions and “Golden” Fibonacci Goniometry

This article refers to the “Mathematics of Harmony” by Alexey Stakhov in 2009, a new interdisciplinary direction of modern science. The main goal of the article is to describe two modern scientific discove-ries—New Geometric Theory of Phyllotaxis (Bodnar’s Geometry) and Hilbert’s Fourth Problem based on the Hyperbolic Fibonacci and Lucas Functions and “Golden” Fibonacci λ-Goniometry ( λ > 0 is a given positive real number). Although these discoveries refer to different areas of science (mathematics and theoretical botany), however they are based on one and the same scientific ideas—the “golden mean”, which had been introduced by Euclid in his Elements, and its generalization—the “metallic means”, which have been studied recently by Argentinian mathematician Vera Spinadel. The article is a confirmation of interdisciplinary character of the “Mathematics of Harmony”, which originates from Euclid’s Elements.

Hilbert’s Fourth Problem: Searching for Harmonic Hyperbolic Worlds of Nature

Recently the new unique classes of hyperbolic functions-hyperbolic Fibonacci functions based on the “golden ratio”, and hyperbolic Fibonacci l-functions based on the “metallic proportions” (l is a given natural number), were introduced in mathematics. The principal distinction of the new classes of hyperbolic functions from the classic hyperbolic functions consists in the fact that they have recursive properties like the Fibonacci numbers (or Fibonacci l-numbers), which are “discrete” analogs of these hyperbolic functions. In the classic hyperbolic functions, such relationship with integer numerical sequences does not exist. This unique property of the new hyperbolic functions has been confirmed recently by the new geometric theory of phyllotaxis, created by the Ukrainian researcherOleg Bodnar(“Bodnar’s hyperbolic geometry). These new hyperbolic functions underlie the original solution of Hilbert’s Fourth Problem (Alexey Stakhov and Samuil Aranson). These fundamental scientific results are overturning our views on hyperbolic geometry, extending fields of its applications (“Bodnar’s hyperbolic geometry”) and putting forward the challenge for theoretical natural sciences to search harmonic hyperbolic worlds of Nature. The goal of the present article is to show the uniqueness of these scientific results and their vital importance for theoretical natural sciences and extend the circle of readers. Another objective is to show a deep connection of the new results in hyperbolic geometry with the “harmonic ideas” of Pythagoras, Plato and Euclid.

LEAFY HEAD2,which encodes a putative RNA-binding protein,regulates shoot development of rice

During vegetative development, higher plants continuously form new leaves in regular spatial and temporal patterns. Mutants with abnormal leaf developmental patterns not only provide a great insight into understanding the regulatory mechanism of plant architecture, but also enrich the ways to its modification by which crop yield could be improved. Here, we reported the characterization of the rice leafy-head2 （lhd2） mutant that exhibits shortened plastochron, dwarfism, reduced tiller number, and failure of phase transition from vegetative to reproductive growth. Anatomical and histological study revealed that the rapid emergence of leaves in lhd2 was resulted from the rapid initiation of leaf primordia whereas the reduced tiller number was a consequence of the suppression of the tiller bud outgrowth. The molecular and genetic analysis showed that LHD2 encodes a putative RNA binding protein with 67% similarity to maize TEl. Comparison of genome-scale expression profiles between wild-type and lhd2 plants suggested that LHD2 may regulate rice shoot development through KNOXand hormone-related genes. The similar phenotypes caused by LHD2 mutation and the conserved expression pattern of LHD2 indicated a conserved mechanism in controlling the temporal leaf initiation in grass.

The Wheat Plastochron Mutant, fushi-darake, Shows Transformation of Reproductive Spikelet Meristem into Vegetative Shoot Meristem

In wheat plants at the vegetative growth stage, the shoot apical meristem (SAM) produces leaf primordia. When reproductive growth is initiated, the SAM forms an inflorescence meristem (IM) that differentiates a series of spikelet meristem (SM) as the branch. The SM then produces a series of floret meristem (FM) as the branch. To identify the mechanisms that regulate formation of the reproductive meristems in wheat, we have investigated a leaf initiation mutant, fushi-darake (fdk) which was developed by ion beam mutagenesis. The morphological traits were compared in wild type (WT) and fdk mutant plants grown in the experimental field. WT plants initiated leaves from SAM at regular intervals in spiral phyllotaxy, while fdk plants had 1/2 alternate phyllotaxy with rapid leaf emergence. The fdk plants have increased numbers of nodes and leaves compared with WT plants. The time interval between successive leaf initiation events (plastochron) was measured in plants grown in a growth chamber. The fdk plants clearly show the rapid leaf emergence, indicating a shortened plastochron. Each tiller in fdk plants branches at the upper part of the culm. The fine structure of organ formation in meristems of fdk plants was examined by scanning electron microscopy (SEM). The SEM analysis indicated that fdk plants show transformation of spikelet meristems into vegetative shoot meristems. In conclusion, the fdk mutant has a heterochronic nature, i.e., both reproductive and vegetative programs were simultaneously in operation during the reproductive phase, resulting in a shortened plastochron and transformation of reproductive spikelets into vegetative shoots.

Leaf morphogenesis:The multifaceted roles of mechanics

Plants produce a rich diversity of biological forms,and the diversity of leaves is especially notable.Mechanisms of leaf morphogenesis have been studied in the past two decades,with a growing focus on the interactive roles of mechanics in recent years.Growth of plant organs involves feedback by mechanical stress:growth induces stress,and stress affects growth and morphogenesis.Although much attention has been given to potential stress-sensing mechanisms and cellular responses,the mechanical principles guiding morphogenesis have not been well understood.Here we synthesize the overarching roles of mechanics and mechanical stress in multilevel and multiple stages of leaf morphogenesis,encompassing leaf primordium initiation,phyllotaxis and venation patterning,and the establishment of complex mature leaf shapes.Moreover,the roles of mechanics at multiscale levels,from subcellular cytoskeletal molecules to single cells to tissues at the organ scale,are articulated.By highlighting the role of mechanical buckling in the formation of three-dimensional leaf shapes,this review integrates the perspectives of mechanics and biology to provide broader insights into the mechanobiology of leaf morphogenesis.