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The Theory of Integrated Sticks

2025, Econographication Virtual Laboratory (EVL)

https://doi.org/10.13140/RG.2.2.14715.43047

This paper aims to present a new mathematical proof to demonstrate that the perfect interconnectivity of a series of sticks can enable them to float or move on any liquid or ice surface more efficiently and for longer durations. Furthermore, this research opens up new opportunities for applications in space exploration, particularly in non-gravitational environments. It proposes innovative designs for specialized sticks that can ensure optimal allocation and resistance at various altitudes and pressure levels without restrictions or constraints on their usability. Finally, we called to this mathematical framework the Theory of Integrated Sticks.

Full Copyrights under Econographication Virtual Laboratory (EVL) © 2024 The Theory of Integrated Sticks Mario Arturo Ruiz Estrada Econographication Virtual Laboratory (EVL) E-mail : [email protected] or [email protected] Website: www.econonographication.com Tel: +6012-6850293 Abstract This paper aims to present a new mathematical proof to demonstrate that the perfect interconnectivity of a series of sticks can enable them to float or move on any liquid or ice surface more efficiently and for longer durations. Furthermore, this research opens up new opportunities for applications in space exploration, particularly in non-gravitational environments. It proposes innovative designs for specialized sticks that can ensure optimal allocation and resistance at various altitudes and pressure levels without restrictions or constraints on their usability. Finally, we called to this mathematical framework the Theory of Integrated Sticks. Keywords: Econographicology, Theory of Integrated Sticks, multidimensional mathematical modelling, space bases basic structures, maritime transportation basic structures. 1. Introduction The concept of integrated sticks is critical from both engineering and technological perspectives. Its application spans solid and soft surfaces. Solid surfaces refer to solid soil, soft soil (sand), and mixed terrains, while soft surfaces pertain to liquid or aquatic environments. The core idea involves interconnecting sticks within the same physical space to create a consolidated structure capable of floating on any soft surface. This approach led us to propose a new theory, which we term the Theory of Integrated Sticks. This theory defines stick interconnectivity as the connection of a group of sticks with varying weights, angles, and Full Copyrights under Econographication Virtual Laboratory (EVL) © 2024 dimensions to construct one or more structures that can float on any liquid surface, at any time, and in any location. 2. An Introduction to the Theory of Integrated Sticks: Mathematical Modelling. This new theory seeks to demonstrate that stick interconnection can effectively works on soft surfaces (water) and hard liquid surfaces (ice). Entitled “Theory of Integrated Sticks,” this theory incorporates advanced multidimensional mathematical and graphical modeling (Ruiz Estrada, 2017) to design various stick forms, sizes, and angles within a unified 3D-printed prototype framework. Additionally, the theory considers the fundamental concept of multigravity points for all stick prototypes, ensuring optimal balance and functionality across different applications. The Theory of Integrated Sticks is formed by five large integrated types of sticks such as the main-sticks (MS), the integrated macro-sticks (IMAS), the integrated sub-sticks (ISS), the integrated micro-sticks (IMIS), the integrated nano-sticks (INS), and the integrated (IJIS) in different sizes (s) and angles (a) (See Expression 1). Finally, the Theory of Integrated Sticks is equal to the interconnectivity of all sticks from main to JI sizes (See Expression 2). The Theory of Integrated Sticks is applying the Dynamic Imbalance State (DIS) (Ruiz Estrada and Yap, 2013) and Omnia Mobilis Assumption (Ruiz Estrada, 2011) to fixed the uses of uses of the horizontal left and right sticks interconnectivity (╠ or ╣), neutral sticks interconnectivity (╬), and vertical top and bottom sticks interconnectivity (╦ or ╩) in the Theory of Integrated Sticks and a large number of MS, IMAS, ISS, IMIS, INS, and IJIS in different sizes (s) and angles (a) at the same structure. Moreover, the idea to consider the uses of the Theory of Integrated Sticks is to give enough support to build a single structure systematically. Basically, the Theory of Integrated Sticks shows different levels such as the main-sticks (MS), the integrated macro-sticks (IMAS), the integrated sub-sticks (ISS), the integrated micro-sticks (IMIS), the integrated nano-sticks (INS), and the integrated (IJIS) in high. Medium, and low level (See Figures 1, 2, 3, 4, 5, and 6). Finally, the interconnected sticks in 3D Printed is available possible to observe in two experiments in video 1 (https://www.youtube.com/watch?v=_n5oEJtuSOA) and video 2 (https://www.youtube.com/shorts/60dWSjoVbaw) in this research to prove the effectiveness of the Theory of Integrated Sticks. The Theory of Integrated Sticks assumes that the all JIS, nano, micro, sub, and main generic integrated sticks can experience different changes such as expansion or contraction according to the thermo-temperature and environmental changes. All these changes can experience different sticks are connected directly or indirectly in the same structure according to the behavior of different sizes rates (first and second partial derivatives) that are changing constantly from the integrated main-sticks (MS), the integrated macro-sticks (IMAS), the integrated sub-sticks (ISS), the integrated micro-sticks (IMIS), the general integrated nano- Full Copyrights under Econographication Virtual Laboratory (EVL) © 2024 sticks (INS), and the JI-Sticks (IJIS) in different sizes (s) and angles (a) (See Expression 1). In the process to draw and build the integrated sticks request the calculation of millions of growth rates simultaneously. Nevertheless, the size of each stick (s) and angle (a) depend on the growth rates (∆). The growth rates (∆) play an important role to define the diameters of each general integrated main-sticks (MS), the integrated macro-sticks (IMAS), the integrated sub-sticks (ISS), the integrated micro-sticks (IMIS), the integrated nano-sticks (INS), and the integrated JI-Sticks (IJIS) in different sizes (s) and angles (a) respectively. In the initial stage of Theory of Integrated Sticks assumes that each level of the sticks sizes from the integrated main-sticks (MS), the integrated macro-sticks (IMAS), integrated substicks (ISS), the integrated micro-sticks (IMIS), the integrated nano-sticks (INS), until arrives to the integrated JI-Sticks (IJIS) (See Expression 2). Additionally, the changes of all growth rates (∆) in different level of sticks are going affect directly the size of the full stick anytime and everywhere. The sticks sizes can determinate if exist expansion or contraction between different thermo-temperature periods, then we can observe how changes in different sticks levels can strikes in each of them. The Theory of Integrated Sticks also assumes that each general integrated main-stick (MS) has a single dependent growth rate and “n” number of thermos-temperature growth rates (sticks size), at the same time; each macro-stick size growth rate has “n” number of sub-sticks growth rates, “n” number of micro-independent size growth rates, “n” number of nano-sticks growth rates and “n” number of IJI-sticks size growth rates (See Expression 1). And the theory of the sticks interconnectivity is flexible by “n” number of millions or thousands of sticks with different sizes that always changing from the first period of time (To) in the bottom of the structure until arrive to the last period of time (Tn) on the top of the structure (See Figures 1, 2, 3, 4, 5, and 6). Hence, each integrated mega-stick (MS) has its horizontal left and right interconnectivity (╠ or ╣), neutral interconnectivity (╬), and vertical top and bottom interconnectivity (╦ or ╩) follow a logical correlation under a unique stick. All these sticks size growth rates are changing constantly in different magnitudes in time (See Expression 2). The Theory of Integrated Sticks also assumes that sticks are not linear; it has a non-linear random sticks sizes and forms always. Therefore, the Theory of Integrated Sticks will propose a strong and aerodynamic structure as a whole. Full Copyrights under Econographication Virtual Laboratory (EVL) © 2024 Full Level (∑([∆IJIS(S:A)] ╠…╦…╣ [∆IJIS(S+1:A+1)]) ║ (∑([∆INS(S:A:IJIS)] ╠ …╬…╣ [∆INS(S+1:A+1:IJIS+1)]) ║ (∑([∆IMIS(S:A:INS:IJIS)]╠…╬…╣ [∆IMIS(S+1:A+1: INS+1: IJIS+1)]) ║ (∑([∆ISS(S:A:IMIS: INS:IJIS)] ╠ …╬…╣ [∆ISS(S+1:A+1:IMIS+1: INS+1: IJIS+1)]) ║ (∑([∆IMAS(S:A:ISS: IMIS: INS:IJIS)] ╠ …╬…╣ [∆IMAS(S+1:A+1:ISS+1: IMIS+1: INS+1: IJIS+1)]) ║ (∑([∆MS(S:A: IMAS: ISS: IMIS: INS:IJIS)] ╠ …╬…╣ [∆MS(S+1:A+1:IMAS+1: ISS+1: IMIS+1: INS+1: IJIS+1)]) Where S = {0,1,…,∞}; A = {0,1,…,360’}; IIMAS = {0,1,…,∞}; IMAS = {0,1,…,∞}; ISS = {0,1,…,∞}; IMIS = {0,1,…,∞}; INS = {0,1,…,∞}; IJIS = {0,1,…,∞} Full Copyrights under Econographication Virtual Laboratory (EVL) © 2024 Therefore, we can identify a longer relationship in each stick level and angles from zero to infinity (See Expression 2). (∑([∆IJIS(0:0)] ╠…╦ …╦ [∆IJIS(∞:∞)]) ║ (∑(∆INS[(0:0:0)] ╠ …╬…╣[∆INS(∞:∞:∞)]) ║ (∑([∆IMIS (0:0:0:0)] ╠ …╬,,, ╣[∆IMIS (∞:∞:∞:∞)]) ║ (∑([∆ISS (0:0:0:0:0)] ╠…╬…╣ [∆ISS (∞:∞:∞:∞:∞)]) ║ (∑([∆IMAS (0:0:0:0:0:0)] ╠ …╬…╣ [∆IMAS (∞:∞:∞:∞:∞:∞)] ║ (∑([∆MAS(0:0:0:0:0:0:0)] ╠ …╬…╣[∆MAS(∞:∞:∞:∞:∞:∞:∞)]) ║ (∑([∆MS (0:0:0:0:0:0:0:0)] ╠…╩…╣ [∆MS (∞:∞:∞:∞:∞:∞:∞:∞)]) (2) Full Copyrights under Econographication Virtual Laboratory (EVL) © 2024 Hence, we have 3 levels of the integrated sticks is followed by L1 stick (see Expression 3), L2 stick (see Expression 4), L3 stick (see expression 5) The Integrated Sticks Level 1 (∑([∆IJIS(0:0)] ╠…╦ …╦ [∆IJIS(∞:∞)]) L1 ║ (∑(∆INS[(0:0:0)] ╠ …╬…╣[∆INS(∞:∞:∞)]) L1 ║ (∑([∆IMIS (0:0:0:0)] ╠ …╬,,, ╣[∆IMIS (∞:∞:∞:∞)]) L1 ║ (∑([∆ISS (0:0:0:0:0)] ╠…╬…╣ [∆ISS (∞:∞:∞:∞:∞)]) L1 ║ (∑([∆IMAS (0:0:0:0:0:0)] ╠ …╬…╣ [∆IMAS (∞:∞:∞:∞:∞:∞)] L1 ║ (∑([∆MAS(0:0:0:0:0:0:0)] ╠ …╬…╣[∆MAS(∞:∞:∞:∞:∞:∞:∞)]) L1 ║ (∑([∆MS (0:0:0:0:0:0:0:0)] ╠…╩…╣ [∆MS (∞:∞:∞:∞:∞:∞:∞:∞)]) L1 (3) Full Copyrights under Econographication Virtual Laboratory (EVL) © 2024 The The integrated sticks Level 2 (∑([∆IJIS(0:0)] ╠…╦ …╦ [∆IJIS(∞:∞)]) L2 ║ (∑(∆INS[(0:0:0)] ╠ …╬…╣[∆INS(∞:∞:∞)]) L2 ║ (∑([∆IMIS (0:0:0:0)] ╠ …╬,,, ╣[∆IMIS (∞:∞:∞:∞)]) L2 ║ (∑([∆ISS (0:0:0:0:0)] ╠…╬…╣ [∆ISS (∞:∞:∞:∞:∞)]) L2 ║ (∑([∆IMAS (0:0:0:0:0:0)] ╠ …╬…╣ [∆IMAS (∞:∞:∞:∞:∞:∞)] L2 ║ (∑([∆MAS(0:0:0:0:0:0:0)] ╠ …╬…╣[∆MAS(∞:∞:∞:∞:∞:∞:∞)]) L2 ║ (∑([∆MS (0:0:0:0:0:0:0:0)] ╠…╩…╣ [∆MS (∞:∞:∞:∞:∞:∞:∞:∞)]) L2 (4) Full Copyrights under Econographication Virtual Laboratory (EVL) © 2024 The integrated sticks Level 3 (∑([∆IJIS(0:0)] ╠…╦ …╦ [∆IJIS(∞:∞)]) L3 ║ (∑(∆INS[(0:0:0)] ╠ …╬…╣[∆INS(∞:∞:∞)]) L3 ║ (∑([∆IMIS (0:0:0:0)] ╠ …╬,,, ╣[∆IMIS (∞:∞:∞:∞)]) L3 ║ (∑([∆ISS (0:0:0:0:0)] ╠…╬…╣ [∆ISS (∞:∞:∞:∞:∞)]) L3 ║ (∑([∆IMAS (0:0:0:0:0:0)] ╠ …╬…╣ [∆IMAS (∞:∞:∞:∞:∞:∞)] L3 ║ (∑([∆MAS(0:0:0:0:0:0:0)] ╠ …╬…╣[∆MAS(∞:∞:∞:∞:∞:∞:∞)]) L3 ║ (∑([∆MS (0:0:0:0:0:0:0:0)] ╠…╩…╣ [∆MS (∞:∞:∞:∞:∞:∞:∞:∞)]) L3 (5) According to expression 3, 4, and 5. We applied the horizontal left stick and right stick interconnectivity (╠ or ╣), neutral stick interconnectivity (╬), and vertical top stick and bottom stick interconnectivity (╦ or ╩) on expression 3, 4, 5 to build the final structure sticks interconnectivity. The time of analysis is starting from angle zero to angle 360’ respectively. Full Copyrights under Econographication Virtual Laboratory (EVL) © 2024 Fig. 1 The Theory of Integrated Sticks Experiment on a Liquid Surface Source: Author Full Copyrights under Econographication Virtual Laboratory (EVL) © 2024 Fig.2 The Theory of Integrated Sticks Experiment on a Liquid Surface Source: Author Full Copyrights under Econographication Virtual Laboratory (EVL) © 2024 Fig. 3 The Theory of Integrated Sticks Experiment on a Liquid Surface Source: Author Full Copyrights under Econographication Virtual Laboratory (EVL) © 2024 Fig.4 The Theory of Integrated Sticks Experiment on a Liquid Surface Source: Author Full Copyrights under Econographication Virtual Laboratory (EVL) © 2024 Fig. 5 The Theory of Integrated Sticks Experiment on a Liquid Surface Source: Author Full Copyrights under Econographication Virtual Laboratory (EVL) © 2024 Fig. 6 The Theory of Integrated Sticks Experiment on a Solid Surface Source: Author Full Copyrights under Econographication Virtual Laboratory (EVL) © 2024 3. Conclussions The Theory of Integrated Sticks is grounded in the final growth rates at each level, proposing varying sizes and angles in the resulting structure. This research demonstrates that the Theory of Integrated Sticks offers a novel analytical tool and engineering model. It has potential applications in the construction of aerospace and maritime transportation systems. The primary structure derived from stick interconnectivity combines sticks of different sizes and angles, enabling them to interact and form a strong, dynamic framework. This adaptable structure is designed to function effectively in diverse environments and global locations, whether on Earth or in space. References Ruiz Estrada, M.A. (2011). Policy Modeling: Definition, Classification, and Evaluation. Journal of Policy Modeling, 33(4), 523-536. Ruiz Estrada, M.A. and Yap, S.F. (2013). The Origins and Evolution of Policy Modeling. Journal of Policy Modeling, 35(1), 170-182. Ruiz Estrada, M.A. (2017). An Alternative Graphical Modeling for Economics: Econographicology. Quality and Quantity, 51(5):2115-213.