To R- geometry of plants V.I. Moiseyev vimo@list.ru Moscow State University of Medicine and Dentistry by A.I. Evdokimov, Moscow, Russia The article gives a definition of the organic form as an infinite-similar and self-similar structure, which is based on the increased unity of the whole and the part - holomereological symmetry. A model of plant forms as organic forms is proposed, which are based on spherocylinders โ€“ infinite-similar cylindrical volumes joined with hemispheres at their ends. The data on the golden wurf in the metric of plant stems are presented. The cirrus leaf model in MathCad is briefly described. The problem of the organic form, the form of living organisms, is a long-discussed topic in biology, in particular, in morphology. There is a lot of empirical material relating to the description of a huge variety of biological forms. However, until now, the laws of the organic form and its specificity, in comparison with the forms of inorganic bodies, continue to raise more questions than answers. It is clear that organic forms are special, they have their own laws and types of morphogenesis. But what is the essence of these features, what is the logic of their organization, all this is still largely unclear. Key words: R-analysis, R-geometry, organic form, infinite-similarity, self-similarity, holomerological symmetry, spherocylinder, golden wurf โˆ’1 โˆ’1 ๏ฟฝ๐‘…๐‘…๐‘€๐‘€(๐‘Ÿ๐‘Ÿ) , ๐œ‘๐œ‘, ๐œƒ๐œƒ๏ฟฝ = (๐‘Ÿ๐‘Ÿ โˆ— , ๐œ‘๐œ‘, ๐œƒ๐œƒ), where ๐‘…๐‘…๐‘€๐‘€(๐‘Ÿ๐‘Ÿ) is a scalar R- 1. Introduction function that isomorphically compresses the real axis R The author develops a new approach to the into the interval (-M,+ M). interpretation of the essence of organic forms, suggesting For example: that these forms are small spaces with its symmetry, R+1M(x) = (2๐‘€๐‘€/๐œ‹๐œ‹) ๐‘ก๐‘ก๐‘ก๐‘ก (๐œ‹๐œ‹๐œ‹๐œ‹/2๐‘€๐‘€) is the direct R-function, topology and metric, featuring considerably from those of R-1M(x) = (2๐‘€๐‘€/๐œ‹๐œ‹) ๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž๐‘Ž (๐œ‹๐œ‹๐œ‹๐œ‹/2๐‘€๐‘€) is the inverse R- the environment. For the expression of such specificity, a function. new mathematical tool, so-called R-analysis (relativistic Similarly, a cylindrical volume C* = R-1C*(C) with analysis of quantity) is used, which is based on the radius M and height 2H can be built, where C is the space isomorphic mappings (R-functions) between the finite with the cylindrical coordinate system (ฯ, ฯ•, z), and structures and their infinite prototypes. On this basis, a โˆ’1 ๐‘…๐‘…๐ถ๐ถโˆ’1โˆ— (๐œŒ๐œŒ, ๐œ‘๐œ‘, ๐‘ง๐‘ง) = ๏ฟฝ๐‘…๐‘…๐‘š๐‘š(๐œŒ๐œŒ) โˆ’1 , ๐œ‘๐œ‘, ๐‘…๐‘…๐ป๐ป(๐‘ง๐‘ง) ๏ฟฝ = (๐œŒ๐œŒโˆ— , ๐œ‘๐œ‘, ๐‘ง๐‘ง โˆ— ). new interpretation of the basic constructions of To model plant forms we will further use a mathematical analysis is proposed, in particular, the spherocylinder - a cylindrical R-space C*, to the upper and calculus of infinitesimals [1-5]. lower bases of which the upper S+* and lower S-* R- Applications of the ideas of R-analysis to geometry hemispheres are attached, and they obtained from the R- allow to develop a new direction, which may be sphere S* by narrowing zenith angle ฮธ to values ฮธโˆˆ[0,ฯ€/2] conventionally designated as "R-geometry" [6]. It is based and ฮธโˆˆ[ฯ€/2,ฯ€] respectively (fig. 1). on the relation between infinite spaces and finite volumes isomorphic to them (R-spaces). It is assumed that the shape of living organisms is a multi-level R-space, compiled by protoforms, as repeated at different levels one typed structures [7]. In this paper, organic forms of plant organisms are chosen for modeling as simpler and more geometrized types of living forms. But it is assumed that the results obtained for plants can be further generalized for all organic forms. 2. Materials and methods When modeling organic forms, it is proposed to use isomorphic mappings (R-functions) Rยฑ1V* between finite Fig. 1. Spherocylinder 3D-volumes V* and their infinite prototypical spaces V, where V* = R-1V*(V), V = R+1V*(V*). In this case, for the 3. Literature review final volume V*, a coordinate system can be determined that most economically and organically expresses the The specificity of the organic form was noted for a internal geometry of the volume V*. Such a coordinate long time. For example, in the research of I.V. Goethe, a system can be called a natural (selected) coordinate theory of metamorphosis was proposed, i.e. the system for the R-space V*. In simpler cases, vector R- transformation of one plant organ into another. Goethe functions can be reduced to scalar ones. For example, if also believed that the protoform of the plant is the leaf the 3D-sphere S* with radius M>0 is considered as the from which its other organs originate: calyx, corolla, volume V*, then it is natural to consider the spherical stamens, and pestle. An attempt to express plant forms as formed on the basis of a single protoform is also vividly coordinate system (r, ฯ•, ฮธ) in the space S to be the selected presented in the so-called telome theory of the German system for S*, assuming that ๐‘…๐‘…๐‘†๐‘†โˆ’1 โˆ— (๐‘Ÿ๐‘Ÿ, ๐œ‘๐œ‘, ๐œƒ๐œƒ) = botanist V.Zimmermann [8], in accordance of which all Copyright ยฉ 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0) the basic forms of the plant are formed on the basis of finite. These kinds of invariants can be called fin- cylindrical structures: terminal (telomes) or intermediate infinites, or even shorter: finfinites. (mesomes). V.I.Vernadsky noted the peculiarity of organic In this case, in the most concise way, it can be argued forms associated with the presence of their own symmetry that the organic form is a finfinite holomerone. and local geometry. We can not pass the studies of morphogenesis in living organisms by D'Arcy Thompson, 5. Plant as an organic form presented in his famous work, "On Growth and Form" [9]. The abstract idea of an organic form, as defined above, He puts forward the idea of regular transformations of can be more specifically implemented in the mathematical some species forms into others, continuing the principles modeling of a generalized plant as a system of coordinated of metamorphosis on the supraorganismic level. The paper spherocylinders. of S.V.Petukhov [10] introduced the concept of the gold According to Zimmermann's telome theory, a plant is wurf W (a,b,c) = (a + b)(b + c)/b(a + b + c) = ฯ•2/2 โ‰ˆ a system of coordinated cylinders. However, the 1.309, where a, b, c are the values of three adjacent cylindrical coordinate system cannot generate, as its own divisions of the organic form, ฯ• โ‰ˆ 1. 618 is the proportion movements, a laterally deviation of one cylinder relative of the golden ratio. The reproducibility of the value of the to another. For such a deviation, it is necessary that the z golden wurf for many biological divisions, as well as the axis of the lateral cylinder deviates by a nonzero zenith importance of conformal and projective mappings in the angle relative to the z axis of the original cylinder, but transformations of the organic form, is noted in [10]. there is no zenith angle in the cylindrical coordinate Numerous examples of biological symmetry, attempts of system, it is only in the spherical one. Therefore, their generalizations are shown by N.A.Zarenkov, considering the plant form as an objectified own particularly in [11]. He develops the ideas of coordinate system, we must supplement the cylindrical "biosymmetry" proposed by Yu.A.Urmantsev in his structures of the telome theory with spherical structures. version of the general theory of systems (GTS). An The design of the spherocylinder described above, just indication of the advantages of the component approach allows you to implement such a deviation. Due to the when transferring a geometric shape from one medium to presence of hemispheres at the ends of the cylinder, lateral another is found, for example, in [12]. Interesting data on deviations of the z axis are possible here with the the coordination of parameters of organic and inorganic formation of a new spherocylinder. Thus, we can assume forms within the framework of temporal definitions can be that wherever plant structure is capable of branching, it can found in [13-15]. You can also mention the research do so only within a spherical component (hemispheres) of of A.A. Lyubishev, L.V. Belousov, dedicated to the spherocylinders (fig. 2). specificity of the organic form. But in general, it should be noted that the field of biological morphology still remains at a predominantly descriptive level and urgently needs the first theoretical generalizations. 4. The phenomenon of organic form The organic form can be defined as a spatial structure, which has the following essential properties: 1) infinite- similarity, i. e., in our case, this structure is a finite volume V*, which is isomorphic to the infinite space V, i. e. ๐‘‰๐‘‰ โˆ— = ๐‘…๐‘…๐‘‰๐‘‰โˆ’1โˆ— (๐‘‰๐‘‰ โˆ— ), where ๐‘…๐‘…๐‘‰๐‘‰โˆ’1โˆ— is an isomorphism, 2) self-similarity: in the form V*, there are parts v* that are similar to the whole V*, i.e. an isomorphic mapping ๐‘‰๐‘‰ โˆ— = ๐‘Ÿ๐‘Ÿ๐‘‰๐‘‰โˆ’1 โˆ— (๐‘‰๐‘‰ ) is โˆ— defined (note that self-similarity is a property of the Fig. 2. Docking of central and lateral spherocylinders whole V*), 3) holo-similarity, when a part of the whole is similar to the whole (holo-similarity is a property of part The fact that a part of the stem can be cut out of the v*), 4) the presence of protoform, i.e. such a form v0*, by plant, planted in the ground, and new shoots will start to transformations of which all the holo-similar parts v* of grow from its upper end, and the roots begin to grow the whole V* (including the whole V * itself) are formed. below, suggests that inside some spherocylinders there are All the described features of the organic form express potentially others that can be activated under certain one general principle: the increased interpenetration of conditions. This is the property of self-similarity of the the whole and the part when the third state of the whole- plant form as organic form. part arises (or holomerone, from the greek โ€œholosโ€ - the Morphologically, the lateral branchings of the plant whole, and โ€œmerosโ€ - the part), i.e. an invariant of stem realize themselves through the formation of leaves transformations between the whole and the part. This kind and the growth of the new shoot from the lateral bud, of invariance can also be called holomereological which is located in the sinus between the leaf stalk and the symmetry. stem. The leaf itself expresses the finale of plant growth, In addition, the organic form as a whole V* also has while the lateral bud contains the germ of a new shoot with infinite-similarity, i.e. it is an infinite space compressed its growth potential. But in order to form a new shoot in a into a finite volume, representing additional kind of new direction of growth, you need a leaf and a lateral bud invariance, the invariance between infinite and that appears near it. Like a leaf is a direction with a fixed value, the lateral bud and its future shoot are a value with the Fibonacci series and are characteristic of each type of a fixed direction. plant. For example, the leaf disposition in cereals, birch, It seems that it is very convenient to simulate by grapes is expressed by the formula ยฝ, in a tulip, alder - 1/3, changing of the vector X, where |X| is the value of X, and in pears, currants, plums - 2/5, in cabbage, radishes, flax - x is the unit vector of X, i.e. x = X/|X|. Then we get: 3/8, etc. ๐‘‘๐‘‘๐‘‘๐‘‘ ๐‘‘๐‘‘๐‘‘๐‘‘|๐‘‹๐‘‹| ๐‘‘๐‘‘๐‘‘๐‘‘ ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ|๐‘‹๐‘‹| In this case, we see the involvement of the azimuthal = = |๐‘‹๐‘‹| + . angle in the organization of spherocylinders. On the stem ๐‘‘๐‘‘๐‘‘๐‘‘ ๐‘‘๐‘‘๐‘‘๐‘‘ ๐‘‘๐‘‘๐‘‘๐‘‘ ๐‘‘๐‘‘๐‘‘๐‘‘ Here, the complete change of the vector in time is the of the plant upwards, there is a spiral organization, sum of two changes: 1) change in direction with the fixed expressed in lateral leaves (shoots). It should be noted that ๐‘‘๐‘‘๐‘‘๐‘‘ value, which is expressed by the term |๐‘‹๐‘‹| , 2) change in a cylindrical spiral is a fairly organic structure in the ๐‘‘๐‘‘๐‘‘๐‘‘ ๐‘ฅ๐‘ฅ๐‘ฅ๐‘ฅ|๐‘‹๐‘‹| cylindrical coordinate system. It is expressed by the form the value with the fixed direction . (ฯ0, ฯ•, z(ฯ•)), where for example z(ฯ•) = kฯ•. ๐‘‘๐‘‘๐‘‘๐‘‘ So in the dynamics of plant growth, these two Let us see how a lateral shoot is formed. In the area of components are morphologically separated. The moment the stem node, there is a cross section of the cylinder, of direction change with the fixed value is allotted to the which becomes the base of the upper hemisphere, due to leaf with the formation of a lateral bud, while the growth which the z axis can deviate, and a lateral spherocylinder of the lateral shoot from this bud expresses the moment of appears. This creates an uncompensated spherical polarity change in the value with the fixed direction. (in the plane (ฯ, ฯ•)), which must be compensated, and the As you know, the stem has negative geotropism (the emergence of a sequence of other side leaves (shoots) is tendency to grow up, against gravity), and the root has the intention to gradually compensate for the initial positive one (grow down, by gravity). This means that the polarity until the branch will come exactly above the initial stem and root carry their own coordinate systems, where one (on one orthostich). Similar relationships can be there is a vertical axis, and they coordinate this axis with expressed by polar analysis [3]. the direction of gravity. In the cylindrical coordinate Summing up, we can assume the plant structure as the system, there is such an axis, this is the z axis. But we need organic form having increased unity of the whole and also the asymmetry of the axis z to express geotropism, parts, as well as infinite-similarity whole, more what can be done through the introduction of two specifically expressed in the possibility of providing R- cylindrical coordinate system: (ฯ, ฯ•, zL) for stem spherocylinders as a base plant protoform, by various spherocylinder, and (ฯ, ฯ•, zG) for the root one, where zL = modifications and compositions of which derivative plant - zG. forms are generated. In this case, the plant appears as a But if there are essentially two oppositely polarized multi-level hierarchical R-space, which includes many cylinders with coordinates zL and zG in the cylinder of the spherocylindrical R-subspaces and their compositions. spherocylinder, then each of them is associated with only Each of the vertical R-spherocylinders includes two one hemisphere (upper for itself). As a result, we have two hemicylinders with opposite geotropy and their own hemispherocylinders in one spherocylinder. They can be cylindrical coordinate systems that are oppositely directed called, for example, as L - and G-hemispherocylinders along the z axis. (levitational and gravitational ones). Another interesting phenomenon of plant 6. Golden wurf in the proportion of plants morphogenesis is phyllotaxis, i.e. patterns of disposition of S.V. Petukhov introduced the concept of the golden leaves on the stem. It turns out that in the general case, the wurf and showed its implementation for many divisions of attachment points of leaf stalks to the stem form a spiral, organic forms [10]. The author investigated the where m turns of spiral have n leaves, if we go along the implementation of the golden wurf W(a, b, c) for stem (as z axis) from a leaf with a certain azimuth neighboring divisions a, b, c using the example of the angle ฯ• in the cylindrical coordinate system (ฯ, ฯ•, z) to the lengths of internodes of plant stems and obtained a good first leaf with the same angle. They say about such leaves agreement between the average values of the wurfs and that they lie on one vertical line - orthostichia. It is 1.3. Below is one of the tables for the chicory stalk (table remarkable that pairs of numbers m and n (usually they 1). After the table, there is a graph representing the values are written as the fraction m/n) in this case are elements of of the Wurfs from this table (fig. 3). Table 1. The lengths of neighboring internodes xk and the wurfs of their triples for the chicory stem k Length xk of neighboring internodes, cm Wurf W(xk,xk+1,xk+2) Wurf average 1 4.8 1.240 2 6.7 1.407 3 5.8 1.270 4 6.8 1.430 5 5.8 1.250 1.331 6 7.3 1.430 7 6 1.290 8 7.3 1.270 9 7.3 1.390 10 5.5 1.340 k Length xk of neighboring internodes, cm Wurf W(xk,xk+1,xk+2) Wurf average 11 5.4 1.280 12 5.4 1.380 13 4.2 1.340 14 4 1.270 15 4 1.450 16 2.9 1.270 17 3.4 1.250 18 2.9 1.570 19 1.7 1.200 20 2.3 1.240 21 1.5 1.400 22 0.7 0.5 Fig. 3. Graphical representation of the data from table 1 ๐‘€๐‘€ โˆ’ ๐‘ฅ๐‘ฅ โˆ— ๐›ฝ๐›ฝ 7. Cirrus leaf modeling ๐พ๐พ(๐‘ฅ๐‘ฅ โˆ— ) = ๐‘€๐‘€0 ๏ฟฝ ๏ฟฝ . ๐‘€๐‘€ In a number of my works [6, 7, 16], I described leaf Knowing the contour K (x*) and the point on the models with arc-like venation as a result of the action of central vein x*i , we can compress along the y axis, above the inverse R-function on the 2D-plane, justifying the this point, a segment of length from x*i to K(x*i ) by the infinite-similariy of the leaf as a special case of the organic inverse R-function R-1K (x*i) , for which it is necessary to form. Below is a brief description of the simulation in use again the parametric definition of functions, for program MathCad of more complicated case of cirrus leaf, example, from the same x. which can not be received by the effect of one R-mapping, Here, it will be necessary to determine the function and it has to connect this type of transformation of infinite along the y axis with respect to some argument Xi(x), structures to finite ones with external transformations of which will give the value x*i for all x. the finite structures, generally yielding a mixed strategy of Accurate hit on the contour, i.e. at the value of K(x*i), organic forming. is achieved in this case for the function R-1ะš(S(ั…*i))(ั…) = At the beginning, when constructing a mathematical 1/Rะš(ั…i)(ั…), and not for R-1ะš(ั…*i)(ั…), where S = R+1M is the model of cirrus leaf, it is carried out the compression by direct R-function. inverse R-function R-1M of the axis x, yielding the variable So, in MathCad, we are building the function x* = R-1M(x). Then, the contour of the sheet K(x*) is 2โ‹… K ( S( xi) ) ๏ฃถ x Ri( x) := ๏ฃซ๏ฃฌ ๏ฃซ ๏ฃถ ๏ฃท โ‹… atan ๏ฃฌ ฯ€ โ‹… 2โ‹… K ( S( xi) ) ๏ฃท , formed, taking, for example, a linear function, raising it to ๏ฃญ ฯ€ ๏ฃธ ๏ฃญ ๏ฃธ a power ฮฒ of less than 1, which will give the convexity of X0( x) := xi, the contour: where xi is understood as the value on the R-compressed ฮฒ M โˆ’ R( x) ๏ฃถ axis, which will correspond to S(xi) for the parameter x. K ( x) := M0โ‹… ๏ฃซ๏ฃฌ ๏ฃท . Therefore, we write not K(xi), but K(S(xi)). ๏ฃญ M ๏ฃธ Therefore we get the vertical segment over xi up to Here M0 is the maximum level of the sheet edge along K(xi). This segment expresses the orthogonal lateral vein the y axis, R(x) is the inverse R-function R-1M(x). at the point of separation xi from the central vein. In MathCad, we obtain the graph K(x*), expressing To express cirrus venation, we tilt this segment from K(x) along the axis y, and along the axis x we have value point xi to point x(i +1). To do this, we introduce a new x* = R(x) = R-1M(x). Here x acts as a parameter, on which parameter ai, for which we set the range of variation and a both the absciss and the ordinate depend. But the function linear function on this range: K is written for x as its argument. And to get a point of the ai := S( xi) , S( xi) + 0.1 .. S( xi + 1) , contour K(x*), we need to set the value of the argument x R( ai) โˆ’ xi ๏ฃน for the function K. It should be borne in mind that the Zi( ai) := ๏ฃฎ๏ฃฏ ๏ฃบ โ‹… K ( S( xi + 1) ) . functions K(x) and K(x*) are different! ๏ฃฐ ( xi + 1) โˆ’ xi๏ฃป Here again, it should be noted that we take We also supplement all the constructions for yโ‰ฅ0 on the parameter ai relative to the R-compressed scale x*, the region y<0 by putting a minus sign in front of the defining it in the interval from (๐‘†๐‘†(๐‘ฅ๐‘ฅ๐‘–๐‘– ))โˆ— = ๐‘ฅ๐‘ฅ๐‘–๐‘– to corresponding functions. As xi, points 0, 2 and 4 were (๐‘†๐‘†(๐‘ฅ๐‘ฅ๐‘–๐‘–+1 ))โˆ— = ๐‘ฅ๐‘ฅ๐‘–๐‘–+1 , therefore, the parameter ai itself must selected. The last inclined straight is ended at the point be determined with respect to the direct R-maps for this K(5.5) of the edge. As a result, we have the following segment. picture for M = 6 and M0 = 5 โ€“ fig. 4. Fig. 4. Cirrus leaf model Summing up, we can conclude that the laws of the plant form, as an important example of the living form in Acknowledgment general, completely correspond to the hypothesis of the The reported study was funded by RFBR according to organic form and infinite-similarity of biological systems, the research project โ„– 19-07-01024. striving in the limit to maximize fusion of the whole and the part in the state of holomereological symmetry, References: combining the properties of infinite-similarity and self- similarity, reinforced by penetration of a single [1] Moiseev V.I. 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R-analysis and the problem of modeling organic forms // SCVRT2018 Proceedings of the International Scientific Conference of the Moscow Institute of Physics and Technology (State University) of the Institute of Physicotechnical Informatics, November 20-23, 2018, TsarGrad, Moscow Region, Russia. - M.: Institute of Physical and Technical Informatics, 2018. - P.339-346. About the autor Moiseyev Vyacheslav I., PhD Hab, professor, head of the Department of Philosophy, Biomedethics and Humanitarian Sciences of Moscow State University of Medicine and Dentistry by A.I.Evdokimov. E-mail: vimo@list.ru.