This paper analyzes the geometric quantities that remain unchanged during parallel mapping （i.e., mapping from a reference curved surface to a parallel surface with identical normal direction）. The second gradient operator, the second class of integral theorems, the Gauss-curvature-based integral theorems, and the core property of parallel mapping are used to derive a series of parallel mapping invariants or geometrically conserved quantities. These include not only local mapping invariants but also global mapping invafiants found to exist both in a curved surface and along curves on the curved surface. The parallel mapping invariants are used to identify important transformations between the reference surface and parallel surfaces. These mapping invariants and transformations have potential applications in geometry, physics, biomechanics, and mechanics in which various dynamic processes occur along or between parallel surfaces.
This paper focuses on the interaction between a micro/nano curved surface and a particle located inside the surface (hereafter abbreviated as in-surface-particle).Based on the exponential pair potential (namely 1/R2k) between particles,the interaction potential between the micro/nano curved surface and the in-surface-particle is derived.The following results are shown:(a) For an even number of exponents in the pair potential,the interaction potential between the micro/nano curved surface and the in-surface-particle can be expressed as a unified function of the mean curvature and Gaussian curvature of the curved surface;(b) the curvatures and the gradients of curvatures of the micro/nano curved surface are the essential factors that dominate the driving force acting on the particle.
We verify the accuracy of the curvature-based potential.By means of the idealized numerical experiment,we show that the curvature-based potential is in good agreement with the numerical experiment,and the errors are within a reasonable range.Based on the curvature-based potential,the equipotential surfaces of particles are derived,and the intrinsic relations between the equipotential surfaces and Weingarten helicoids are shown.
Based on the recent research progress in fractal super fibers,the growth kinematics(or pattern kinematics) of(6+1) -circle fractal super fiber with snowflake-like cross section(abbreviated as"fractal super snowflake") is explored.The following results are obtained.(1) The fractal super snowflake obeys simply the straight-line growth mode.(2) At a given movement,the growth speed of the snowflake distributes uniformly in space.At a given point in space,the growth speed decreases along with the time.(3) The growth kinematics of the fractal super snowflake is intensively influenced by the self-similar ratio:If and only if the self-similar ratio is equal to 1/3,the macro speed equals to the micro speed,and the macro density equals to the micro density.If the self-similar ratio is smaller than 1/3,the micro speed is larger than the macro speed,and the micro density is larger than the macro density.If the self-similar ratio is larger than 1/3,the macro speed is larger than the micro speed,and the macro density is larger than the micro density.These results provide references for us to understand the complicated fractal growing phenomena in nature.
Through careful analysis on the cross-section of sisal fibers, it is found that the middle lamellae between the cell walls have clear geometric characteristics: between the cell walls of three neighboring cells, the middle lamellae form a three-way junction with 120° symmetry. If the neighboring three-way junctions are connected, a network of Steiner tree with angular symmetry and topological invariability is formed. If more and more Steiner trees are connected, a network of Steiner rings is generated. In another word, idealized cell walls and the middle lamellae are dominated by the Steiner geometry. This geometry not only depicts the geometric symmetry, the topological invariability and minimal property of the middle lamellae, but also controls the mechanics of sisal fibers.
Based on the concepts of fractal super fibers, the （3, 9＋2）-circle and （9＋2, 3）-circle binary fractal sets are abstracted form such prototypes as wool fibers and human hairs, with the （3）-circle and the （9＋2）-circle fractal sets as subsets. As far as the （9＋2） topological patterns are concerned, the following propositions are proved： The （9＋2） topological patterns accurately exist, but are not unique. Their total number is 9. Among them, only two are allotropes. In other words, among the nine topological patterns, only two are independent （or fundamental）. Besides, we demonstrate that the （3, 9＋2）-circle and （9＋2, 3）-circle fractal sets are golden ones with symmetry breaking.
The surfaces of the veins of dragonflies(Pantala flavesens Fabricius and Crocothemis servilia Drury) wings are observed through SEM,and interesting micro and nano structures and morphologies are discovered.On the surfaces of the veins,not only ripple wave morphologies are distributed,but also spikes are grown.Besides,on the surfaces of the spikes,straight stripe wave morphologies are grown along the generatrix.These marvelous micro and nano structures and morphologies may enable us to better understand the remarkable flying abilities of dragonflies.
