Mechanical Optimality Hidden in the Structure of Plants
Hiroyuki Shima
University of Yamanashi, Japanhshima@yamanashi.ac.jp

Plants of various morphology (shape) can be found in nature. These diverse shapes are believed tobe the result of the evolution of plants, which cannot move, to adapt to the surrounding environment.
In this talk, I will particularly focus on the shapes of three plant types: i) bamboo [1-4], ii) butterbur,and iii) muskmelon [5]. First, I will demonstrate how the shapes of bamboo and butterbur, which areboth tall and hollow plants, are optimal for supporting their own weight. Next, I will explain that therind of muskmelons develops a mesh pattern with a certain geometric property, which may be used byfarmers as a visual indicator of optimum sweetness of flesh; this is a consequence of a farmer seeking
the optimum sweetness of the flesh.
i) Optimal structure of bamboo
Bamboo is the fastest-growing plant worldwide. This rapid maturity stems from their hollow
structures, which require less material to develop, thereby promoting rapid growth. One critical
disadvantage of hollowness is the propensity to easily collapse under external forces such as wind,rain, and self-weight. To overcome this limitation, bamboos generate a sequence of “circular plates”
within the long cavity. The plates stiffen the culm locally; therefore, numerous plates from the bottom
to the top of the bamboo enhance the stability of the culm. However, this is not a case of “the more,the better”, because an excessive number of plates will inhibit the growth rate of the bamboo plant.
Therefore, the number of plates has to be “neither too many nor too few”. This trade-off between
stability and growth rate implies an optimum number of stiffening plates that bamboo has acquired
over millions of years of evolution.
Against this backdrop, our research group has explored the optimum solution to this issue by
investigating wild bamboo groves in Japan. Analysis of the measurement data showed that a single
universal law explaining the interdependence between the preferred number of stiffening plates andtheir ideal arrangement along the culm governs the growth of bamboo [3]. Specifically, the density at
which fibers are embedded increases as it gets closer to the outer skin. This universal law thus provides
a design criterion, endowing less material and high strength to wild bamboos, which are advantageous
for survival over competing tree species.
Surprisingly, another mechanism lurks in the cross-section of the bamboo to increase its stability
as a whole. It is bamboo fiber. Inside the bamboo, many long fibers are embedded in the longitudinal
direction. These fibers are called "vascular sheaths" in biological terminology, and they act as straws
that convey water and nutrients vertically. The structural mechanics theory revealed that this fiber
plays an important role in the mechanical properties of bamboo. If a part of the cross-section ofbamboo is cut and observed using an electron microscope, it can be seen that the fibers are not evenly
1distributed over the entire cross-section. Specifically, the closer it is to the outer skin, the more fibers
are embedded. We were able to prove within the scope of the elastic approximation theory that bygathering many reinforcing fibers near the outer skin of the bamboo, the bamboo as a whole is lesslikely to bend [4]. This result indicates that the bamboo makes its own body stronger by efficiently
arranging a limited number of reinforcing fibers inside the cross-section.
ii) Optimal cross-section of butterbur petiole:
There are many plant species with hollow stems and petioles, including those belonging to thefamily Poaceae. One of the reasons why many plant species have hollow tubular structures is thought
to be the structural stabilization provided by hollowness. A tube with a hollow cross-section deforms
less than a solid tube (filled) with the same cross-section. Thus, it can support its own weight by using
minimal resources. In addition, by choosing the optimal cross-sectional shape (circular, polygonal,etc.) according to the growth environment, the resistance to the most influential mechanical loads(crosswind, self-weight of stems, weight of petals, etc.) increases.
From this perspective, the plant that we focused on in this study was the Japanese butterbur, a typeof wild grass. A close examination of the cross-section of the stem of the Japanese butterbur reveals
that it has a shape similar to that of a horse's hoof, which is rarely seen in other plants. We hypothesizedthat this peculiar horseshoe-shaped stem cross-section is the optimal structure for supporting large
leaves at the tip of the stem. We have verified this hypothesis both experimentally and theoretically.
iii) Geometric law of mesh pattern of melon skin:
The surface of muskmelons, which is representative of high-grade melons, has a fine mesh pattern(net). It is empirically known that the appearance of this mesh pattern correlates with the quality ofmelons and is used as an important indicator for quality evaluation at production sites. However, only
a few studies have investigated the geometric properties of melon nets.
Our research group took photographs of several muskmelons sold at a retail store (for qualityassurance) and investigated the shape and size of the small green peel fragments separated by white
reticular stripes. Calculating the areas of these small peel fragments, obtaining the probability
distribution curves of the areas, and normalizing them, it was found that the distribution curves wereon a common curve regardless of the type and size of individual melons [5]. Furthermore, this commonprobability distribution curve was derived from fracture mechanics theory. This discovery suggests
the possibility of non-destructive identification and evaluation of melon quality using mesh pattern
imaging.

References[1] H. Shima, A. Inoue, and M. Sato: “Bamboo: A Mechanically Optimum Design in Nature”, Chapter
1 in Book: “Bamboo Science and Technology” (Springer, 2023) ISBN: 978-981-99-0014-5[2] A. Inoue and H. Shima: “Survival time of Princess Kaguya in an air-tight bamboo chamber”, Forma33, 1-5 (2018) [https://forma.katachi-jp.com/frame/33.html][3] H. Shima, M. Sato and A. Inoue: “Self-adaptive formation of uneven node spacings in a wildbamboo”, Physical Review E 93, 022406 (2016)[4] M. Sato, A. Inoue and H. Shima: “Bamboo-inspired optimal design of functionally graded hollowcylinders”, PLOS ONE 12, e0175029 (2017)[5] Y. Akiba, A. Ishibashi, M. Sato, and H. Shima: “Empirical rule of fruit rind fragmentation inmuskmelon epidermis netting”, Journal of the Physical Society of Japan 91, 104801 (2022)
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