Chinese Science Bulletin © 2009
SCIENCE IN CHINA PRESS
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“Steiner trees” between cell walls of sisal LI GuanShi1, YIN YaJun2, LI Yan1 & ZHONG Zheng1† 1 2
School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China; School of Aerospace, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
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. sisal fiber, cell wall, middle lamella, network, Steiner tree
Networks widely exist in the natural world, among which are real networks and imagined networks as well. There are networks made up of massive carbon nanotubes[1], networks made by neurons in our brains[2], network models for studying the microscopic flow in rocks[3], and also artificial neural network used for vagetation change detection[4], to name a few. In a word, networks are everywhere. Recently, an interesting law was proved by Yin et al. during their studies on the biomembrane nanotube networks[5,6] and super carbon nanotube networks[7,8]. A mechanically stable equilibrium network of biomembrane nanotubes or super carbon nanotubes is geometrically equivalent to a Steiner minimal tree. This law provides the geometric foundation for the mechanics of biomembrane nanotube networks and super carbon nanotube networks. In another word, the geometrization for the mechanics of biomembrane nanotube networks and super carbon nanotube networks was realized. In this paper, it is further confirmed that Steiner network is the geometric foundation for the mechanics of vegetation fibers, such as the sisal fibers.
1 Micro-observations of sisal fibers A sisal fiber was bent broken after being left in liquid
nitrogen for a certain period of time so that a true morphology of the cross section can be obtained. The cross sections of sisal fibers so obtained were observed with the aid of scanning electronic microscopy (SEM). Interesting information about the geometry and topology of the cross section is disclosed. Figure 1 is the SEM photo with ×10000 magnification. We can clearly see that the middle lamellae between the three cell walls interact at a point and form a three-way junction. In Figure1 two three-way junctions are drawn between four neighboring cells. To one’s surprise, both the three-way junctions satisfy 120° symmetries, i.e. the angle between two arms is 120°. In geometry, such three-way junction is called the Steiner minimal tree[9–11] connecting three points (N=3). Inside a triangle ∆A1A2A3 with A1, A2 and A3 the three vertices (i.e. the “ ” in Figure 2), there is a unique Steiner point S (i.e. the “ ” in Figure 2) that satisfies A1SA2 A2 SA3 A3 SA1 3
120 and renders SAi a minimum value. In another i 1
Received April 30, 2009; accepted June 27, 2009 doi: 10.1007/s11434-009-0536-1 † Corresponding author (email:
[email protected]) Supported by the National Natural Science Foundation of China (Grant Nos. 10602040, 10872114)
Citation: Li G S, Yin Y J, Li Y, et al. “Steiner trees” between cell walls of sisal. Chinese Sci Bull, 2009, 54: 3220―3224, doi: 10.1007/s11434-009-0536-1
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the vertices of a quadrangle A1A2A3A4. Inside the quadrangle there are two pares of Steiner points (S1, S2),
(S′1 , S′2), forming two different Steiner trees Ts and Ts′ with full topologies. Both trees are the networks connecting the fixed four points with extreme value of total length. If A1CA2 A1CA4 , one of the networks is the
Figure 1
Figure 2
Three-way junction with 120° symmetry.
The three-point problem (N=3) for Steiner minimal tree.
word, the middle lamellae between three neighboring cell walls can be geometrically abstracted as a minimal network connecting three given points. Figure 3 is the SEM photo with ×8000 magnification. This magnification is lower than that of Figure 1. Thus more cells go into the visual field and the relations between the two neighbouring three-way junctions become more clear. If two neighboring three-way junctions are connected, then the Steiner tree with N=4 can be obtained[9–11]. Its abstracted description is shown in Figure 4: Four points A1, A2, A3 and A4 on the plane are taken as
is the Steiner minimal tree. According to this criterion, the middle lamella network in Figure 3 is an extreme network instead of a minimal network. Figure 5 is the SEM photo with ×6000 magnification. While more and more neighboring three-way junctions are connected, the topology of the network is changed: The tree-like networks are linked up and gradually ringlike networks appear. Similar to the Steiner trees, they are called Steiner rings. Since the Steiner trees are extreme networks, the Steiner rings are also extreme networks. In nature the most perfect Steiner rings are the carbon-atom-networks in graphite layers. In geometry, the perfect Steiner ring is the hexagon, and the perfect Steiner ring network is that bordered by identical hexagons. Figure 6 is the SEM photo with ×3000 magnification. It can be seen that as the number of cells increases, the symmetries of part of the three-way junctions are broken; although the topologies of the three-way junctions are still kept unchanged, the 120° symmetries are lost. Thus at the junctions, some angles are larger than 120°, and some angles are smaller than 120°. Correspondingly, both the trees and the rings are distorted; there are either quadrilateral, pentagonal or heptagonal rings.
2 Geometry and mechanics of sisal fibers How the phenomena revealed in the SEM photos can be explained? Are the phenomena related to the geometry and mechanics of sisal fibers? Both locally and globally, the idealized network of middle lamellae between cell walls tends to be the symmetric three-way junctions, the symmetric network of Steiner trees and the symmetric network of Steiner rings. In another word, the Steiner minimal trees and the Steiner minimal rings are the templates for the growth of cell walls. Hence we can say that there is geometry be
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minimal tree. The criterion[8] for the minimal tree is as follows: If A1CA2 A1CA4 (or S1S2 > S′1 S′2), then Ts
Figure 3 junction.
The Steiner tree connected by two symmetric three-way
Figure 4
The Steiner ring formed by Steiner trees.
The Steiner trees connected four points.
hind the cell walls. Its name is the Steiner geometry. This geometry has three important features: One is the geometric symmetry, i.e. the 120° symmetry. Another is the topological invariability, i.e. the three-way junction. The other is the extreme property, i.e. the total length of the network is of extreme or minimal property. This extreme property is also the optimization property. Thus from pure geometric viewpoint, we can answer an important question: why do the cell walls of sisal fibers “tend to” be Steiner geometry? The answer is very simple: once the cell walls grow along with the Steiner geometry, an optimization growth mode in which the 3222
Figure 5
Figure 6 The locally distorted or weighted Steiner trees and Steiner rings.
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A Ll ,
(1) where l is the total length of the middle lamella network. Thus, the total energy needed for generating all the cell walls is E A Ll , (2) where is the energy needed for forming unit surface area. The minimal (or extreme) energy principle is of universality in nature, and is valid for the growing process of vegetations. In another word, vegetations tend to complete their growing processes by minimal energies. We can minimize both sides of eq. (2):
Emin Llmin .
(3)
This means that a network with minimal energy is also a network with minimal length. Then it can be concluded that the growth of sisal fiber cell walls is dominated by Steiner geometry. The growing mechanics is the one in Steiner geometry. In modern industries, sisal fibers are often used to make ropes for carrying loads. Hence, except for the growth mechanics, there is also the loading mechanics. The cell walls not only determine the geometric spaces for their existence, but also regulate the substance spaces for the load-carrying fibers. Therefore, the load-carrying mechanics for sisal fibers is the one in Steiner geometry. If the load-carrying mechanics for sisal fibers is developed, the rules in Steiner geometry have to be followed and Steiner geometry is taken as the basis of the loadcarrying mechanics. It should be noticed that all the analyses and conclusions are made ideally. In real nature, however, this ideal circumstance definitely does not exist. Thus, just as shown in Figure 6, the middle lamella network is ideal Steiner network only in some sections. In most sections, there are distortions in the cell walls and the middle lamellae. In fact, distortions are inevitable: Not only is the out-
3 From sisal to general Can the results above be generalized? To answer this question, we further studied jute fibers briefly through optical microscope. Figure 7 is a microscopic photo of ×800. It can be clearly seen that there does exist a fine Steiner network. Obviously, the jute fibers fit Steiner geometry well. Thus, our supposition is that it is quite possible that all bast fibers fit.
Figure 7
Jute fibers (×800).
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side macro-environment changing during the growth of vegetation, the micro-environment for cell growth is also fluctuating every second. Besides, there are also inevitable deformations from the manufacture process. All these factors are directly or indirectly shown as forces, i.e. the external forces on vegetation, fibers and their cells, which are responsible for the distortions. The results of these distortions are the breakages in the symmetry of Steiner trees. But how can we count the influences of these distortions? It should be noticed that although the geometric symmetry is broken by distortions, the topologies are kept unchanged. Besides, symmetry breakages only occur locally rather than globally. Thus, we guess that it is possible to spread the ideal Steiner networks to weighted ones when the influences of the external forces mentioned above are counted (e.g. making external work counted when considering the total energy for cell wall growth). From a geometric point of view, a distorted Steiner network is equal to weighted one, whose weigh factors indicate the influences of external forces. This is just a guess and further verification and research are needed. In appearance, distortions seem to weaken the foundation of Steiner geometry. Hence the influences of distortions and symmetric breakages can be estimated by extending ideal Steiner networks to weighted ones. In this sense, the idealized Steiner geometry still has essential importance.
SOLID-STATE MECHANICS
“materials needed for architectures” are minimal is selected. Minimal materials for architectures refer to maximum spaces for cytoplasm. Minimal materials and maximum spaces are irresistibly attractive for the growth of the cells in sisal fibers. Except for the geometry, it is also necessary to explore the mechanics for sisal fibers. Take a segment of fiber with length L. In the cross section, the middle lamellae between cell walls form a network. Longitudinally, the cell walls form complicated cylindrical structures. The total surface area of the cylinders is estimated as
What about other vegetation fibers other than bast fibers? We further studied two other typical vegetation fibers. One of them is the root of wheat. Figure 8 is a wide spread illustration of transaction of wheat root. Here the picture selected is from a textbook of phytobiology. Although the picture is just an illustration, it still contains an impressive Steiner network that can be easily recognized. The second typical fiber is from bark of cork oak. Figure 9 is maybe one of the most famous microscopic photos——the cell structure of cork taken by Robert Hook more than 300 years ago. On the right side of this photo, the networks of cell walls are shown. It can be roughly seen that it is also a Steiner network. Until now, although it is impossible to check every vegetation fiber, it is possible to hypothesize based on fibers from bast fibers to root fibers of wheat, and then to fibers from cork that the Steiner symmetry observed in sisal fibers is a biological reality which exists in all vegetation fibers.
long time? Perhaps it is just because of the universality of this Steiner geometry in vegetation fibers which covered the specificity of the geometric form and topological structure up. They are seen so commonly that they can hardly catch people’s attention, needless to say its importance in vegetation fibers.
Figure 9
Cell structure of Cork, taken by Robert Hook.
4 Conclusions
Figure 8
Illustration of wheat root transaction.
Here we have another interesting question: It has been about 360 years since Fermat raised the question of minimal network, and it has been more than 300 years since Hook took his first photo of cells. Why did the relations between the two remain unveiled for such a 1
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The cell wall networks of sisal fibers are the products of natural selections. The geometry depicting such networks is the Steiner geometry with optimization properties. This result is important in two aspects. On the one hand, although this result does not involve the growth mechanics and load-carrying mechanics, it reveals the geometric foundations for the two types of mechanics. On the other hand, it may provide a universal paradigm, i.e. the Steiner geometry not only dominates the cell wall networks of sisal fibers, but also controls the cell walls of most of the vegetation fibers. If this judgment is true, then the result in this paper may be much more valuable. 6
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