System Engineering based on Remarkable Geometric Properties of
Space
VOLODYMYR RIZNYK
Lviv Polytechnic National University,
79013, Lviv-13, Stepan Bandera Str., 12,
UKRAINE
Abstract: - In this paper, we regard designing systems based on remarkable geometric properties of space,
namely valuable rotational symmetry and asymmetry harmonious, using schematic and diagrammatic
presentations of the systems. Moreover, the relationships are a way to comprehend original information to serve
as a source of research and designing the systems. The objective of the future methodology is the advanced
study of spatial geometric harmony as profiting information for expansion fundamental and applied researches
for optimal solutions of technological problems in systems engineering. These systems engineering designs
make it possible to improve the quality indices of devices or systems concerning performance reliability, code
immunity, and the other operating indices of the systems. As examples, both up to 25% errors of lengths
correcting code and high-speed self-error-correcting vector data code formed under a toroidal coordinate
system are presented.
Key-Words: - symmetry and asymmetry harmonious relationships, generative rotational symmetry, Ideal Ring
Bundle, error-correcting cyclic code, vector monolithic coding system, manifold.
Received: August 26, 2023. Revised: December 27, 2023. Accepted: February 25, 2024. Published: April 15, 2024.
1 Introduction
The General Relativity Einstein`s theory describes
the Universe as a four-dimensional manifold
topology, [1], [2]. This research relates to a better
understanding of combinatorial properties of two-
and multidimensional geometric structures for
finding optimal solutions in system engineering,
using schematic and diagrammatic presentations of
the systems. There are innovative theoretical
investigations, using schematic modeling advanced
engineering systems based on combinatorial
configurations such as projective planes [3], cyclic
groups in extensions of the Galois field [4],
difference sets [5], and the other combinatorial
constructions, [6], [7].
2 Problem Formulation
The statement problem is to increase technical
variabilities in the system with an incomplete set of
elements and a number of bonds, using novel
designs based on combinatorial geometric
properties of space. Research involves configuring
the circuits and system engineering related to
finding the optimal placement of structural
elements in spatially or temporally distributed
systems, including the appropriate combinatorial
constructions for configuring innovative
engineering devices, schemes and systems in
spatially closed topologically systems.
3 Problem Solution
3.1 Symmetry and Asymmetry: Harmonious
Relationships
Symmetry is known form the foundation of the
geometrical construction of the Universe, [8]. We
refer to the process engineering for optimal
solutions to technological problems, using
harmonious penetration asymmetry into rotational
symmetry, and curvature structure of general
relativity, [9]. A complementary ensemble of 1-fold
(black line) and 2-fold (red lines) cyclic
asymmetries joined into 3-fold (S=3) rotational
symmetry illustrates Figure 1.
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Fig. 1: Complementary ensemble of 1-fold (black
line) and 2-fold (red lines) cyclic asymmetries
joined into 3-fold (S=3) rotational symmetry
Note, the chart (Figure 1) backgrounds the
ring-like models of the ideal ring protractor with
two (n=2) marks (red lines) placed into a dial as the
angular distance relation 1:2 (Figure 2).
Fig. 2: Chart of ideal ring protractor with two (n=2)
marks
It’s evident, that two S–1 = 2 angular distances
between marks (αmin=120o and αmax=240o) in the
protractor represent the perfect cyclic relationship
of integers. There are a lot of the underlying ideal
protractors generated from the rotational symmetry
of numerous orders. An example of 7-fold (S=7)
rotational symmetry is given in Figure 3. We call
this peculiarities “Generative Rotational Symmetry
” (GRS).
Fig. 3: Generative 7-fold (S=7) rotational symmetry
If we allow go round seven (S=7) lines of the 7-
fold rotational symmetry, moving clockwise
reference points H→A→R→M→O→N→Y (Figure
3), we can obtain a set of angular distances [α, 6α]
between distinct pairs of three (n1=3) black and four
(n2=4) red lines, α = 360º/ S =36 0º /7(T able 1).
Table 1. Angular distances [α, 6α] between distinct
pairs, moving clockwise reference points
H→A→R→M→O→N→Y of three (n1=3) black
and four (n2=4) red lines, α = 360º/S =360º/7
Hence, the ring scale reading system based on
7-fold (S=7) rotational symmetry allows on
partition of two-dimensional space perfectly for a
minimum number of intersections relative to the
reading point by spatial interval α = 360º/ 7
exactly once(R1=1) and/or twice (R2=2) by the
same interval. Easy to see, that the 7-fold rotational
symmetry creates an intelligent system as an
ensemble of two complementary numerical non-
uniform cyclic structures {1, 4, 2}, and {1, 1, 2, 3},
followed by H→A→N→H, and
R→M→O→Y→R cyclic sequences.
Parameters S, n1, n2, R1, R2 of GRSs for 3 ≤ S ≤ 31
are tabulated (Table 2).
Angle
Starting point
Final point
α
H
A
2α
N
H
3α
N
A
4α
A
N
5α
H
N
6α
A
H
Angle
Starting point
Final point
α
R
M
2α
R
O
3α
M
Y
4α
R
Y
5α
O
R
6α
M
R
Angle
Starting point
Final point
α
M
O
2α
O
Y
3α
Y
R
4α
Y
M
5α
Y
O
6α
O
M
O
1/3
1/3
1/3
2α
α
N
Y
O
M
R
α
A
H
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Table 2. Parameters S, n1, n2, R1, R2 of GRSs for 3
≤ S ≤ 31
In Table 2, we observe, that all S-fold GRSs are
completed from complementary pairs of ideal
protractors. Note, anyone of the given GRSs consists
of the protractors, each of them takes an even-fold
(n1), and odd-fold (n2) number of marks. The set of
all angular intervals [,360º] occurs exactly R1
(R2) times by step 360º / S. In the underlying
schematic demonstrations of rotational symmetry
and asymmetry harmonious relation phenomenon
it’s perfectly well understood remarkable geometric
properties of a space for comprehending original
information to serve as a source for research and
designing high-performance systems engineering.
3.2 Rotational Symmetry and Asymmetry
Relationships
From Table 2 and the underlying relationships
follow S = n (n – 1)/R +1, (1)
where S is the order of rotational symmetry, n -
number of asymmetrically diverged beams from, and
R - number times of enumerate the set of
commensurable angular distances between these
lines. Rotational symmetry-asymmetry relationships,
provide, essentially, new information about the
remarkable properties of GRSs as one more approval
hypothesis on the existing world-wide harmony of
the Universe, [8].
4 System Engineering based on
Generative Rotational Symmetry
4.1 Ideal Ring Bundles
Generally, IRBs are cyclic sequences of
ordered- chain sub-sequences of the sequence. The
modular sums of consecutive sub-sequences of an
Ideal Ring Bundle enumerate nodal points of a
manifold coordinate system exactly R-times. In a
particular case, the sums of one-dimensional IRB
enumerate the set of integers from 1 to S– 1 exactly
R-times.
For example, the IRB {1,4,2} formed by 7-fold
(S=7) rotational symmetry as three (n1=3) black
beams (Figure 3) enumerate the set of integers [1,6]
exactly once (R1=1):
1, 2, 3=2+1, 4, 5=1+4, 6=4+2.
At the same time the IRB {1, 1, 2, 3} follows from
four (n2=4) red beams - exactly twice (R2=2):
1, 1; 2, 2=1+1; 3, 3=1+2; 4=3+1, 4=1+1+2;
5=2+3, 5=3+1+1; 6=1+2+3, 6=2+3+1.
Therefore, we comprehend this harmonious
symmetry -asymmetry of order seven which
generates this creative ensemble.
Ideal Ring Bundles (IRBs) can be used for
optimal solutions to wide classes of technological
problems, using the applicability of one- and
multidimensional IRBs
4.2 IRBs Error Correcting Cyclic Codes
Designs of IRBs error correcting cyclic codes make
it possible to correct up to 25% of the code
combination lengths. For synthesis of the codes can
be used formula
xj–1≡∑𝑘𝑖
𝑗
𝑖=1 (𝑚𝑜𝑑 𝑆), j=1, 2,…, n (2)
xj – is numerical position of symbols “1” in an
initial code combination of the code combination
length S, where kj is i –th term of an IRB.
An example of error-correcting cyclic code based
on the IRB {1,4,2,1,2,1} with parameters S = 11,
n=6, R=3 is below (Table 3).
Table 3. Error correcting cyclic code based on the
IRB {1,4,2,1,2,1} with parameters S =11, n=6, R=3
Parameters of the GRSs
S
n1
n2
R1
R2
3
1
2
1
1
7
3
4
1
2
11
5
6
2
3
13
4
9
1
6
15
7
8
3
4
19
9
10
4
5
21
5
16
1
12
23
11
12
5
6
31
6
15
1
7
№
Code combinations
1
2
3
4
5
6
7
8
9
10
11
1
1
0
0
0
1
0
1
1
0
1
1
2
1
1
0
0
0
1
0
1
1
0
1
3
1
1
1
0
0
0
1
0
1
1
0
4
0
1
1
1
0
0
0
1
0
1
1
5
1
0
1
1
1
0
0
0
1
0
1
6
1
1
0
1
1
1
0
0
0
1
0
7
0
1
1
0
1
1
1
0
0
0
1
8
1
0
1
1
0
1
1
1
0
0
0
9
0
1
0
1
1
0
1
1
1
0
0
10
0
0
1
0
1
1
0
1
1
1
0
11
0
0
0
1
0
1
1
0
1
1
1
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The first code combination 10001011011 is
obtained by formula (2), while the rest
combinations of the code are fulfilled by its cyclic
shifting.
A number tcor of corrected errors for optimal
cyclic IRB-code to be depending n [10]:
tcor ≤ (n/2 – 1) (3)
We proceed from (3) that cyclic code (Table 3)
allows correct up to 2 errors.
The code size can be doubled if Table 3 is
added by the next table with opposite values of all
its binary characters (Table 4).
Table 4. Opposite values of binary characters of
Table 3 for doubled of the code size
Table of quality indexing the error- correcting
cyclic code based on the IRBs in ascending order of
its parameters n and S is below (Table 5).
Table 5. Quality indexing the error- correcting
cyclic code based on the IRBs
Table 5 evidents improving quality indexes tcor,
Q = tcor/S, and P of the error correcting codes based
on the IRBs approaching Q =25% errors of the
code length asymptotically if S → ∞.
4.3 Two-dimensional Ideal Ring Bundles
Definition. An n-stage cyclic sequence of
connected 2-stage sub-sequences of the sequence,
for which the set of all two-modular (t=2) vector-
sums (mod m1, mod m2) form two-dimensional grid
m1 × m2 over a torus surface is named two-
dimensional Ideal Ring Bundle (2-D IRB).
For example, two-dimensional four-stage
(n=4) cyclic sequence {(1,1), (0,1), (2,2), (2,1)},
where all vector-sums of connected sub-
sequences form two-dimensional cyclic grid 34,
taking modulo m1 = 3 for the first component of
the vector-sums, and modulo m2 = 4 for the
second ones, depicted in a chart (Figure 4).
Fig. 4: A chart of two-dimensional (t=2) 4-stage (n
=4) IRB {(1,1), (0,1), (2,2), (2,1)}
There are all two-dimensional vector-sums of
the connected 2-D vectors of the IRB {(1,1), (0,1),
(2,2), (2,1)}:
4mod,3mod
)3,0( (2,1)(0,1)(1,1)
(2,0)(2,2)(1,1)(2,1)
(1,0) (2,1)(2,2)(0,1)
)0,0( (2,2)(0,1)(1,1)
(0,2)(1,1)(2,1)
(1,3) (2,1)(2,2)
(2,3) (2,2)(0,1)
(1,2) (0,1)(1,1)
(4)
The set of two-dimensional vector-sums together with
vectors (1,1), (0,1), (2,2), (2,1) of the IRB complete
set of two-modular vector sums as follows:
(0, 0) (0, 1) (0, 2) (0, 3)
(1, 0) (1, 1) (1, 2) (1, 3)
(2, 0) (2, 1) (2, 2) (2, 3)
A vector ring diagram of the 2-D IRB {(1,1), (0,1),
(2,2), (2,1)} depicted as colored graph (Figure 5).
№
Code combinations
1
2
3
4
5
6
7
8
9
10
11
1
0
1
1
1
0
1
0
0
1
0
0
2
0
0
1
1
1
0
1
0
0
1
0
3
0
0
0
1
1
1
0
1
0
0
1
4
1
0
0
0
1
1
1
0
1
0
0
5
0
1
0
0
0
1
1
1
0
1
0
6
0
0
1
0
0
0
1
1
1
0
1
7
1
0
0
1
0
0
0
1
1
1
0
8
0
1
0
0
1
0
0
0
1
1
1
9
1
0
1
0
0
1
0
0
0
1
1
10
1
1
0
1
0
0
1
0
0
0
1
11
1
1
1
0
1
0
0
1
0
0
0
n
tcor
S
Q×100%
P
6
2
11
18
22
8
3
15
20
30
16
7
31
22,6
62
32
15
63
23,8
126
64
31
127
24,4
254
(2,1)
(1,1)
(0,1)
(2,2)
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Fig. 5: Vector ring diagram of the 2-D IRB{(1,1),
(0,1), (2,2), (2,1)}
The vector ring diagram (Figure 5)
demonstrates the interconnection of all two-modular
vector-sums from (0,0) to (2,3) inclusive formed on
the IRB {(1,1), (0,1), (2,2), (2,1)}.
4.4 Optimum Vector Monolithic Coding
Systems
An arbitrary vector monolithic coding system
unlike of ordinary ones are that all allowed code
words consist of no more than a single solid block
both of connected symbols “1”, and block “0” in
the words. The optimum vector monolithic coding
systems are formed on t-dimensional IRBs with
informative parameters n, S that makes it possible
to offer code size n2 – n in the minimised basis of
the system. Additionally, the set of all t-
dimensional vectors covers node points of manifold
reference system by t-axes taking t- modular (m1,
m2, …, mi,…, mn) manifold coordinate system,
allowing on binary encoding of vector signals
arranged as no more two sequences with the same
characters [10].
Optimum monolithic 2-D vector code based on the
Ideal Ring Bundle {(1,1), (2,3), (0,3), (3,3), (1,3)}
with parameters S=21, n=5, m1=4, m2=5, R=1 is
given in Tabl.6.
Table 6. Optimum monolithic 2-D vector code
based on the Ideal Ring Bundle {(1,1), (2,3), (0,3),
(3,3), (1,3)}
Table 6 contains S –1 = 20 two-dimensional 5-
digit (n=5) vector code combinations in a binary
coding system of 2-D (t =2) vector digit weights
{(1,1), (2,3), (0,3), (3,3), (1,3)}as the basis of the
toroid coordinate grid of sizes m1 × m2 = 4 × 5, m1 =
4, m2 = 5.
Optimum monolithic 3-D (t=3) vector coding
system formed under the IRB {(1,1,2), (0,2,2),
(1,0,3), (1,1,1), (0,1,0), (0,2,3)} with informative
parameters S=31, n=6, m1=2, m2=3, m3=5, and
R=1:
(0,0,0){(0,2,3)+(1,1,2)+(0,2,2)+(1,0,3)+(0,1,0)}
(0,0,1) {(0,2,2) + (1,0,3) + (1,1,1)}
(0,0,2) {(1,1,2) + (0,2,2) + (1,0,3)}
(0,0,3) {(0,2,3) + (0,1,0)}
(0,0,4){(0,2,2)+(1,0,3)+(1,1,1)+(0,1,0)+(0,2,3)}
(0,1,1) {(0,2,2)+ (1,0,3) + (1,1,1) + (0,1,0)}
(0,1,2) {(1,0,3)+ (1,1,1) + (0,1,0) + (0,2,3)}
(0,1,3){(1,1,1)+(0,1,0)+(0,2,3)+(1,1,2)+(0,2,2)}
(0,1,4) {(0,1,3) + (1,1,1)}
(0,2,0) {(0,2,3)+ (1,1,2) + (0,2,2) + (1,0,3)}
(0,2,1) {(1,1,1)+ (0,1,0) + (0,2,3) + (1,1,2)}
………………………………………………………….
Finally,
(1,2,4){(0,2,3)+(1,1,2)+(1,1,1)+(1,0,3)+ (0,1,0)}
The set of S –1=30 these three-dimensional
vectors form optimum coding system 2×3×5 in the
Vector
Digit weights
(1,1)
(2,3)
(0,3)
(3,3)
(1,3)
(0,0)
1
1
1
0
1
(0,1)
0
0
0
1
1
(0,2)
1
1
0
0
1
(0,3)
0
0
1
0
0
(0,4)
0
0
1
1
1
(1,0)
1
0
1
1
1
(1,1)
1
0
0
0
0
(1,2)
1
0
0
1
1
(1,3)
0
0
0
0
1
(1,4)
0
1
1
1
0
(2,0)
1
1
1
1
0
(2,1)
0
1
1
0
0
(2,2)
0
1
1
1
1
(2,3)
0
1
0
0
0
(2,4)
1
0
0
0
1
(3,0)
1
1
0
1
1
(3,1)
0
0
1
1
0
(3,2)
1
1
1
0
0
(3,3)
0
0
1
0
0
(3,4)
1
1
0
0
0
(0,3)
(0,0)
(1,2)
(1,1)
(0,1)
(0,2)
(2,3)
(2,1)
(2,2)
(1,0)
(1,3)
(2,0)
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minimized basis of manifold coordinate system
formed by three (t=3) annular axes m1=2, m2=3,
m3=5 with origin of the coordinates in common
point (0,0,0).
5 Discussion
Schematic presentations of remarkable geometric
properties of space are given in Figure 1, Figure 2
and Figure 3. Complementary ensemble of 1-fold
and 2-fold cyclic asymmetries joined into 3-fold
(S=3) symmetry illustrates Figure 1. Table 1
reveals the essence of the generative rotational
symmetry (GRS) as the “ideal” protractors, each of
them takes even-fold, and odd-fold number of
marks. Table 2 opens new information about
geometric relationships of optimal distributed
structural elements (events) as part of the “whole”
and logical interpretation of the phenomenon. The
7-fold rotational symmetry (Figure 3) creates an
intelligent system as an ensemble of two
complementary numerical non-uniform cyclic
structures {1, 2, 4}, and {1, 1, 2, 3} as the
Fibonacci numbers, [11]. In Table 3, Table 4 and
Table 5 we can observe improving quality indexes
of the optimized error-correcting codes based on
the IRBs approaching 25% errors of the code
length asymptotically if S → ∞. Unlike familiar
coding systems, the monolithic codes provide faster
self-correcting vector data signals for the
transmission of multidimensional information by
noise communication channels, as well as the
processing of the signals under a manifold
coordinate system. An example of an optimum
monolithic 2-D vector code of size m1 × m2 = 20,
m1=4, m2=5 illustrates in Table 1. The second
example demonstrates an optimum monolithic 3-D
vector coding system of code size m1 × m2 × m3=
30, where m1=2, m2=3, m3=5.
6 Conclusion
System engineering based on remarkable geometric
properties of space provides novel techniques for
improving the quality indices of engineering
devices and systems, using Ideal Ring Bundles
prospected from generative rotational symmetry.
The study of the generative rotational symmetry
(GRS) peculiarity allows expanding the underlying
combinatorial procedures for finding the optimal
placement of structural elements in spatially or
temporally distributed systems of numerous
engineering devices under development. Ideal Ring
Bundles (IRBs) can be used for optimal solutions to
of wide classes of technological problems, using
the applicability of one- and multidimensional
IRBs. It’s just what the generative rotational
symmetry and asymmetry provide mutual
penetration of existing eternal world intelligence of
the Universe.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
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