=Paper= {{Paper |id=Vol-2763/CPT2020_paper_p-3 |storemode=property |title=To R- geometry of plants |pdfUrl=https://ceur-ws.org/Vol-2763/CPT2020_paper_p-3.pdf |volume=Vol-2763 |authors=Vyacheslav Moiseyev }} ==To R- geometry of plants== https://ceur-ws.org/Vol-2763/CPT2020_paper_p-3.pdf

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 – 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

−1 −1
�𝑅𝑅𝑀𝑀(𝑟𝑟) , 𝜑𝜑, 𝜃𝜃� = (𝑟𝑟 ∗ , 𝜑𝜑, 𝜃𝜃), where 𝑅𝑅𝑀𝑀(𝑟𝑟) 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) = (2𝑀𝑀/𝜋𝜋) 𝑡𝑡𝑡𝑡 (𝜋𝜋𝜋𝜋/2𝑀𝑀) is the direct R-function,
topology and metric, featuring considerably from those of R-1M(x) = (2𝑀𝑀/𝜋𝜋) 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 (𝜋𝜋𝜋𝜋/2𝑀𝑀) 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 (ρ, ϕ, z), and
structures and their infinite prototypes. On this basis, a −1
𝑅𝑅𝐶𝐶−1∗ (𝜌𝜌, 𝜑𝜑, 𝑧𝑧) = �𝑅𝑅𝑚𝑚(𝜌𝜌) −1
, 𝜑𝜑, 𝑅𝑅𝐻𝐻(𝑧𝑧) � = (𝜌𝜌∗ , 𝜑𝜑, 𝑧𝑧 ∗ ).
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 θ to values θ∈[0,π/2]
conventionally designated as "R-geometry" [6]. It is based and θ∈[π/2,π] 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) R±1V* 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, ϕ, θ) in the space S to be the selected
presented in the so-called telome theory of the German
system for S*, assuming that 𝑅𝑅𝑆𝑆−1
∗ (𝑟𝑟, 𝜑𝜑, 𝜃𝜃) =
botanist V.Zimmermann [8], in accordance of which all

Copyright © 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) = ϕ2/2 ≈ 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, ϕ ≈ 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. 𝑉𝑉 ∗ =
𝑅𝑅𝑉𝑉−1∗ (𝑉𝑉 ∗ ), where 𝑅𝑅𝑉𝑉−1∗ 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 𝑉𝑉 ∗ = 𝑟𝑟𝑉𝑉−1 ∗ (𝑉𝑉 ) is
∗

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 “holos” - the Morphologically, the lateral branchings of the plant
whole, and “meros” - 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 ½, 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.
𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑|𝑋𝑋| 𝑑𝑑𝑑𝑑 𝑥𝑥𝑥𝑥|𝑋𝑋| In this case, we see the involvement of the azimuthal
= = |𝑋𝑋| + . angle in the organization of spherocylinders. On the stem
𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑
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
𝑑𝑑𝑑𝑑
value, which is expressed by the term |𝑋𝑋| , 2) change in a cylindrical spiral is a fairly organic structure in the
𝑑𝑑𝑑𝑑
𝑥𝑥𝑥𝑥|𝑋𝑋| cylindrical coordinate system. It is expressed by the form
the value with the fixed direction . (ρ0, ϕ, z(ϕ)), where for example z(ϕ) = kϕ.
𝑑𝑑𝑑𝑑
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 (ρ, ϕ)), 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: (ρ, ϕ, zL) for stem spherocylinders as a base plant protoform, by various
spherocylinder, and (ρ, ϕ, 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 ϕ in the cylindrical coordinate system (ρ, ϕ, 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

𝑀𝑀 − 𝑥𝑥 ∗ 𝛽𝛽
7. Cirrus leaf modeling 𝐾𝐾(𝑥𝑥 ∗ ) = 𝑀𝑀0 � � .
𝑀𝑀
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-1К(S(х*i))(х) =
At the beginning, when constructing a mathematical 1/RК(хi)(х), and not for R-1К(х*i)(х), 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 2⋅ K ( S( xi) )  x
Ri( x) :=   
 ⋅ atan  π ⋅ 2⋅ K ( S( xi) )  ,
formed, taking, for example, a linear function, raising it to  π   
a power β 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
β
M − R( x)  axis, which will correspond to S(xi) for the parameter x.
K ( x) := M0⋅   . Therefore, we write not K(xi), but K(S(xi)).
 M 
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) − xi 
for the function K. It should be borne in mind that the Zi( ai) :=   ⋅ K ( S( xi + 1) ) .
functions K(x) and K(x*) are different!  ( xi + 1) − xi
Here again, it should be noted that we take We also supplement all the constructions for y≥0 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 (𝑆𝑆(𝑥𝑥𝑖𝑖 ))∗ = 𝑥𝑥𝑖𝑖 to corresponding functions. As xi, points 0, 2 and 4 were
(𝑆𝑆(𝑥𝑥𝑖𝑖+1 ))∗ = 𝑥𝑥𝑖𝑖+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 – 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 № 19-07-01024.
striving in the limit to maximize fusion of the whole and
the part in the state of holomereological symmetry, References:
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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.