Allometric Scaling Inspection Through Figurate Numbers

Allometric Scaling Inspection Through Figurate Numbers | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Allometric Scaling Inspection Through Figurate Numbers Yekbun Adiguzel This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3050595/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Allometric scaling is a widely-studied, and discussed phenomena in biology that relates the parameters with a power function. There is still space for further investigation. Accordingly, the relationship is aimed to be explained for the first time in this work, through the figurate numbers, specifically the triangular, square, pentagonal, and hexagonal numbers. These mathematical series have not been considered in terms of a possible relation with allometric scaling, which is a mathematical relation in essence. We formulated the possible means of the relation of these figurate numbers with allometric scaling and suggested that their utilization can end up in different exponents and constant multipliers at varying scaling ranges. We also discussed further implications and exemplified the possible association with the blood flow-related metabolic rate adaptation. It is revealed that figurate numbers are having an inherent relation with the allometric scaling. Allometric relation triangular numbers square numbers cube numbers pentagonal numbers hexagonal numbers Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Introduction Allometric scaling correlate biological parameters like the drug clearances and interspecies physiological parameters (Huang et al., 2015), or the noncoding DNA and the protein size (Adiguzel, 2021), or the metabolic rate (Kozlowski et al., 2003). As stated by White and Seymour (White & Seymour, 2005), allometric relation was first suggested by Sarrus and Ramaeux (cited in (Brody, 1945)) and supported by Max Rubner. Kleiber (Kleiber, 1932; Kleiber, 1961) found that metabolic rate was proportional to an exponent (> 0.66) of body mass, and the exponent was later approved as 0.75. The allometric relation can be expressed with the equation y = aM b e , where y is the variable of interest, M b is the body mass, and e is the exponent. Allometric relationships can be investigated between different parameters, including the replacement of the M b with another parameter of interest. The exponent in the allometric relation of M b and the basal metabolic rate ( ml oxygen consumption per hour) is 0.686 ± 0.014, and it becomes a bit less (0.0675 ± 0.013) when it is corrected for body temperature, while it is 0.712 ± 0.013 for the thermoneutral resting metabolic rate (White & Seymour, 2005). Plotting the variation of the metabolic rate with the body masses of the terrestrial animals reveals an exponent of 0.73, which is commonly stated as 0.75 for the sake of simplicity. There can be certain adaptive variations like phylogenic, geographic, etc., but one can well assume that many common observations should be arousing from a variety of situations converging to an optimal solution. The basis of the relationships has been debated (West et al., 1997; West et al., 1999; West et al., 2001; Banavar et al., 2010; Razavi et al., 2018) and we believe that there is still space in this field. Therefore, we aim to make an explanation of the exponent 0.73 through figurate numbers, considering that the exponent of 0.73 is the result of converging solutions. Figurate numbers are appropriate to our purpose since they are classification of numbers “according to their geometric representation as sets of dots (Weaver, 1974)” (Kempen & Biehler, 2020). Methods We looked for association between the members of the figurate numbers (square, triangular, pentagonal, and hexagonal numbers) representing fold change in M b , and the allometric relation between the M b and metabolic rate, where the exponent is 0.73. Allometric relations are designated with the equation y = aM b e , where y is the dependent variable, a is the constant multiplier, M b is the independent variable, which is the body mass here, and e is the exponent. We took constant multiplier as 1. The resulting allometric relation ( y = M b e ), and the functions derived from it, became our reference model functions. Square numbers are the squares of the positive integers (i.e., n 2 ). Triangular numbers are the sum of positive integers until and including the i th position, where i is each consecutive member of the positive integers as the variable (i.e., \(\sum _{i=I}^{n}i\) ). Pentagonal numbers and hexagonal numbers are calculated through (3 n 2 - n )/2 and (3 n 2 - n ), respectively, where n is also the variable as the positive integers. As stated, we looked for the association between the members of the selected figurate numbers and the allometric scaling. We assumed that the relationship is through the squares of the numbers. We took the ratio of increase in the squares of the figurate numbers, through subtracting the squares of consecutive numbers and dividing the difference to the square of the latter number. Namely, if we write F n instead of a member of figurate number, the depicted calculation is (F n 2 -F n−1 2 )/ F n 2 , where F n is the n th figurate number. For comparison, the respective allometric relation, which was calculated by taking the power ( e = 0.73) of M b , was divided by M b . That calculation is M b 0.73 / M b , which is resulting in M b −0.27 . For better comparability, this calculation was performed for the fold changes in M b , where the fold changes corresponded to the triangular numbers. Comparison is performed through fitting the curves with power functions and comparing the exponents and the goodness of fit values. Microsoft Office Excel was utilized for the calculations and plots. We also displayed the evolution of the exponents (e) and the constant multipliers (a) of the fitting power functions, as the fold change in M b increased. This was performed not only from the beginning on, but also from the 55-fold change in M b on. These e- and a-values were compared with pairwise two-tail T-test, to see whether there was a statistically significant (p ≤ 0.05) difference in the means, or not. Results and discussion The first 20 numbers in the square, triangular, pentagonal, and hexagonal numbers are listed in Table 1 . Table 1 The first 20 numbers in the square, triangular, pentagonal, and hexagonal numbers. Square numbers Triangular numbers Pentagonal numbers Hexagonal numbers 1 1 1 1 4 3 5 6 9 6 12 15 16 10 22 28 25 15 35 45 36 21 51 66 49 28 70 91 64 36 92 120 81 45 117 153 100 55 145 190 121 66 176 231 144 78 210 276 169 91 247 325 196 105 287 378 225 120 330 435 256 136 376 496 289 153 425 561 324 171 477 630 361 190 532 703 400 210 590 780 As mentioned in the methods, we calculated the squares. Calculating the square of edge length gives the area in squares, which will be mentioned at the end, and taking the square also gets the maximum unidirectional connectivity between the dots representing the components of a number. An example of this interpretation is illustrated in Fig. 1 . As stated in the methods, we took the ratio of increase in the results of squares by subtracting the squares of consecutive numbers and dividing the differences to the latter numbers. These calculations for the first twenty numbers of the triangular numbers are displayed in Table 2 . Ratios of increases in the squares of figurate numbers, which are the first twenty triangular numbers, are compared with the ratios of metabolic rate changes to fold changes in body mass ( M b ), where the fold changes in M b are triangular numbers, for comparability (Table 2 ). Table 2 The first twenty numbers in the triangular numbers together with the relevant calculations for the plotted ratio-data. Metabolic rate changes and squares of triangular numbers are rounded. Triangular numbers Metabolic rate change Ratio of metabolic rate change to fold change in M b (reference) Square of triangular numbers Ratio of increase in the square of triangular numbers 1 1 1.00 1 3 4 0.62 14 0.93 6 7 0.48 52 0.74 10 11 0.41 130 0.60 15 16 0.36 259 0.50 21 21 0.32 453 0.43 28 27 0.30 725 0.37 36 33 0.27 1085 0.33 45 39 0.26 1548 0.30 55 46 0.24 2123 0.27 66 53 0.23 2824 0.25 78 61 0.22 3662 0.23 91 68 0.21 4649 0.21 105 76 0.20 5796 0.20 120 84 0.19 7115 0.19 136 93 0.19 8618 0.17 153 102 0.18 10315 0.16 171 111 0.18 12219 0.16 190 120 0.17 14340 0.15 210 129 0.17 16690 0.14 It can be seen in Fig. 2 that all cases had high the goodness of fit values. The ratios of metabolic rate changes to the fold changes in M b fitted perfectly by a power function, as should be the case. Its exponent is the closest to the exponents of the ratios of increases in the squares of pentagonal and hexagonal numbers (Fig. 2 ). However, an immediate interpretation of their being the best approximation among the others can be misleading. The biologically relevant and the readily comparable changes in the body masses and the associated metabolic rates would be spanning a narrower range than that is viewed in Fig. 2 . This is elaborated below. Although we took constant multiplier as 1, we compared the evolution of constant multipliers (a) and the exponents (e) of the fitting power functions, as the fold change in M b increased. This comparison was performed in two ranges, one from the first data on (Fig. 3a) and the second from the 55-fold change in M b on (Fig. 3b). Figure 3 revealed that the e- and a-values were relying on the ranges of the fitted-values. The e- and a-values were significantly different within the same range, when they were calculated in the given range only (i.e., from the 55-fold change in M b on), compared to that from the first data on. In the former situation (Fig. 3a), e-value evolution curves were crossing the data derived from the reference model ( y = aM b 0.73 , a = 1 ) at different fold changes in M b . Accordingly, the best model was relying on the range of interest of the fold changes in M b and the whole range of the fold changes in M b under study. Varying constant multipliers (White & Gould, 1965) was also supporting the dynamic variations in the experimental data (Fig. 4), which will not be discussed further here. It was indicated that the e-value evolution curves were crossing the data derived from the reference model at different fold changes in M b (Fig. 3a). Those fold changes in M b are 10, 36, 70, and 91, when the calculations were performed with the triangular numbers, square numbers, pentagonal numbers, and hexagonal numbers, as the fold changes in M b , respectively (Fig. 5 ). It means that the exponent in the reference relationship (i.e., y = M b 0.73 ) is obtained through the relationship derived by the triangular numbers, when there is a fold change in M b up to 10. Similarly, the exponent 0.73 is obtained through the relationship derived by the square numbers when there is a fold change in M b up to 36, while it is obtained through the relationship derived by the pentagonal and hexagonal numbers when there is a fold change in M b up to 70 and 91, respectively. Here, the e-value relies on the relationship derived through the figurate numbers, up to the selected range of fold change in M b . We set the reference to a specific e-value here, but allometric relations can have different e-values. Therefore, this study is bringing about a distinct approach to study the varying types of allometric relations. Now we will discuss a relevant feature of the triangular numbers. It was mentioned in the beginning that calculating the square of the edge length gives the area in squares. The sum of the cubes of the positive integer numbers from 1 to n is equal to the square of the sum of those numbers (i.e., \({\sum }_{i=1}^{n}{i}^{3}\) = ( \({\sum }_{i=1}^{n}i\) ) 2 ). This is visually illustrated in Fig. 6 . It should be noted here that the cubes of the positive integers are the cube numbers. The inter-related calculation that we performed in this study was subtracting the squares of consecutive triangular numbers and dividing the difference to the square of the latter number (i.e., (T n 2 -T n−1 2 )/ T n 2 , where T n is the n th triangular number). As observed, in Fig. 6 , consecutive side-lengths of the squares with the same origin are the triangular numbers. Accordingly, with each increase in the size of the square displayed in Fig. 6 , we divided the increase in the square-area to the total area. This is also the same as dividing the volume of the largest cube in Fig. 6 , sequentially, to the total area of the sum of the volumes of the cubes with unit increments in the edge-lengths. In other words, the n th cube number is divided to the sum of the cube numbers from 1 to n (i.e., n 3 / \({\sum }_{i=1}^{n}{i}^{3}\) ). When our calculation is viewed from this perspective, it is anticipated to be in accordance with the compared-ratio derived from the allometric relation, which was M b 0.73 / M b . Therefore, our approach introduces a geometric reasoning to the allometric relation with a means of treating the varying observations. The feature described above for the triangular numbers are correlated with the other figurate numbers we dealt with since figurate numbers are inter-related. Square numbers are the sum of the consecutive triangular numbers (i.e., S n =T n +T n−1 , where S n is the n th square number and T n is the n th triangular number). Also, the n th pentagonal number P n is equal to T n−1 +n 2 , or T n +2T n−1 , or T 2n − 1 -T n−1 , where n is the positive integer and T n is the n th triangular number. On the other hand, hexagonal numbers are the differences of consecutive cube numbers (i.e., H n =C n -C n−1 , where H n is the n th hexagonal number and C n is the n th cube number). The inter-relations that we worked out in this study are a conceivable basis of the correlation with the allometric relation, which we derived through the triangular numbers. A hypothetical example can be the number of molecules produced for enlarging volumes when the molecule is transported at different rates to each volume-layer, while being consumed or reacted at certain time points. Another example is the distance travelled at the enlarged surface area leading to a reduction in the supply due to increased resistance. Specifically, blood flow is the source of oxygen delivery to the cells and tissues, which is regulated by the metabolic needs. Adaptation to lower metabolic rates due to flow changes can be another phenomenon, which is evaluated further in a different manner than the present literature. We will treat increase in the size (i.e., M b ) to be leading to parallel branches emerging from the main vessel with a unit increase in the length of each consecutive new branches. Namely, the first, single branch having the length l , is followed by the addition of new branches having the sizes 2 l , 3 l , going up to the length n l , where each new branching up to the number n is indeed adding onto the present vasculature like the increased area of the square depicted in Fig. 6 . Flow through the parallel branches follow the Ohm's Law where the inverse of the total resistance is the sum of the inverses of the resistances in the branches (i.e., \({\sum }_{i=1}^{n}1/Ri\) =1/R 1 +1/R 2 +…+1/R n ). Flow through each branch is driven by the same pressure difference and inversely proportional with the resistance of the branch. Therefore, each new longer branch can be assumed to have lowered flow when the other flow-parameters are the same. If we calculate this difference in flow for each added branch with unit increment in the length, flow in the added branch compared to the flow in the single vessel (i.e., no branch) system is displayed in Table 3 for the first ten data. Table 3 Calculation and the first ten results of flow in the added branch with unit increments in length. Added branch length (i.e., n l ) 1/R n Σ(1/R n ) ΣR n : 1/Σ(1/R n ) ΔP = QxR ◊ Q = ΔP/R, ΔP constant Q at the n th branch (ΔP/(ΣR n xn l )) 1.00 1.00 1.00 ΣR 1 : 1.00 Q at the 1. branch (ΔP/(ΣR 1 x1 l )) 1.00 2.00 0.50 1.50 ΣR 2 : 0.67 Q at the 2. branch (ΔP/(ΣR 2 x2 l )) 0.75 3.00 0.33 1.83 ΣR 3 : 0.55 Q at the 3. branch (ΔP/(ΣR 3 x3 l )) 0.61 4.00 0.25 2.08 ΣR 4 : 0.48 Q at the 4. branch (ΔP/(ΣR 4 x4 l )) 0.52 5.00 0.20 2.28 ΣR 5 : 0.44 Q at the 5. branch (ΔP/(ΣR 5 x5 l )) 0.46 6.00 0.17 2.45 ΣR 6 : 0.41 Q at the 6. branch (ΔP/(ΣR 6 x6 l )) 0.04 7.00 0.14 2.59 ΣR 7 : 0.39 Q at the 7. branch (ΔP/(ΣR 7 x7 l )) 0.37 8.00 0.13 2.72 ΣR 8 : 0.37 Q at the 8. branch (ΔP/(ΣR 8 x8 l )) 0.34 9.00 0.11 2.83 ΣR 9 : 0.35 Q at the 9. branch (ΔP/(ΣR 9 x9 l )) 0.31 10.00 0.10 2.93 ΣR 10 : 0.34 Q at the 10. branch (ΔP/(ΣR 10 x10 l )) 0.29 Abbreviations with notes: ΔP is the pressure difference, which is the constant driving force; R is the resistance, which is depending only on the length l here, assuming all the other parameters equal; Q is the flow, which is influencing the metabolic rate; l is the unit length, n is positive integers representing unit increments in length when multiplied with the length l . We plotted these lowered flows in the added branches together with the graphs in Fig. 2 , assuming the diminished flow as the bottleneck of the oxygen delivery, and hence the metabolism. There is agreement with the other plots, so with our calculations and the reference data (Fig. 7 ). Conclusion The allometric relationship is explained for the first time through the triangular, square, pentagonal, and hexagonal numbers. Their utilization can end up in different exponents and constant multipliers at varying scaling ranges, applicable to different fields and aspects in biology. Declarations Acknowledgment: Ecology and Evolutionary Biology Society of Turkey is acknowledged. Conflicts of interests: Nothing to declare. Funding: None Ethical permissions: Not required References Adiguzel, Y. (2021). Information- theoretic approach in allometric scaling relations of DNA and proteins. Chem Biol Drug Des, 0 , 1–13. doi:10.1111/cbdd.13988 Banavar, J. R., Moses, M. E., Brown, J. H., Damuth, J., Rinaldo, A., Sibly, R. M., & Maritan, A. (2010). A general basis for quarter-power scaling in animals. Proc Nat Acad Sci, 107 , 15816–15820. doi:10.1073/pnas.1009974107 Brody, S. (1945). Bioenergetics and Growth. New York: Reinhold Publishing Corporation. Huang, W., Geng, L., Deng, R., Lu, S., Ma, G., Yu, J.,. .. Lu, X. (2015). Prediction of Human Clearance Based on Animal Data and Molecular Properties. Chem. Biol. Drug Des., 86 , 990–997. Kempen, L., & Biehler, R. (2020). Using figurate numbers in elementary number theory - discussing a 'useful' herustic from the perspectives of semiotics and cignitive psychology. Front Psychol, 11 , 1180. doi:10.3389/fpsyg.2020.01180 Kleiber, M. (1932). Body size and metabolism. Hilgardia, 6 , 315–353. Kleiber, M. (1961). The Fire of Life. New York, London: John Wiley and Sons, Inc. Kozlowski, J., Konarzewski, M., & Gawelczyk, A. T. (2003). Cell size as a link between noncoding DNA and metabolic rate scaling. Proc. Natl. Acad. Sci. U.S.A., 100 (24), 14080–14085. Razavi, M. S., Shirani, E., & Kassa, G. S. (2018). Scaling laws of flow rate, vessel blood volume, lengths, and transit times with number of capillaries. Front Physiol, 9 , 581. doi:10.3389/fphys.2018.00581 Weaver, C. (1974). Figurate numbers. Math Teach, 67 , 661–666. West, G. B., Brown, J. H., & Enquist, B. J. (1997). A general model for the origin of allometric scaling laws in biology. Science, 276 , 122–126. West, G. B., Brown, J. H., & Enquist, B. J. (2001). A general model for ontogenetic growth. Nature, 413 , 628–631. West, G. B., Enquist, J. H., & Enquist, B. J. (1999). The fourth dimension of life: fractal geometry and allometric scaling of organisms. Science, 284 , 1677–1679. White, C. R., & Seymour, R. S. (2005). Allometric scaling of mammalian metabolism. The Journal of Experimental Biology, 208 , 1611–1619. doi:10.1242/jeb.01501 White, J. F., & Gould, S. J. (1965). Interpretatıon of the coefficient in the allometric equation. The American Naturalist, 99 , 5–18. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3050595","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":212139009,"identity":"0d5f4ccf-f04e-4f4c-b6d9-b1767242e4d6","order_by":0,"name":"Yekbun Adiguzel","email":"data:image/png;base64,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","orcid":"","institution":"Atilim University","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Yekbun","middleName":"","lastName":"Adiguzel","suffix":""}],"badges":[],"createdAt":"2023-06-12 01:44:21","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3050595/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3050595/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":38998853,"identity":"37035bac-a790-4d31-977a-e0ba92400a3d","added_by":"auto","created_at":"2023-06-24 01:23:26","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":56559,"visible":true,"origin":"","legend":"\u003cp\u003eExample of maximum unidirectional connectivity between spots representing numbers. Accordingly, a total number of 1, 4, 9, and 16 unidirectional connectivity is displayed between the 1 (a), 2 (b), 3 (c), and 4 (d) spots, respectively. Maximum unidirectional connectivity represented by the arrows is the square of the number of spots.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-3050595/v1/23984ec1fe86f4bc8d0cec3a.png"},{"id":38998400,"identity":"cab10cc3-db2e-4a51-b91a-3cf0adb3a05b","added_by":"auto","created_at":"2023-06-24 01:15:26","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":151799,"visible":true,"origin":"","legend":"\u003cp\u003eRatios of increases in the squares of figurate numbers compared to the ratios of metabolic rate changes to the fold changes in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e.\u0026nbsp; Fold changes in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e are figurate numbers, which are the square numbers, triangular numbers, pentagonal numbers, and hexagonal numbers.\u0026nbsp; Reference data is in blue.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-3050595/v1/60a7fb5829110e93fc857111.png"},{"id":38997767,"identity":"d3d6d995-40da-4b21-9fd9-240d7363a606","added_by":"auto","created_at":"2023-06-24 01:07:27","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":162878,"visible":true,"origin":"","legend":"\u003cp\u003eEvolution of the e, exponents, of the fitting power functions, as the fold changes in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e increased.\u0026nbsp; This comparison was performed in two ranges, one from the first data on (a), the second from the 55-fold change in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e on (b).\u0026nbsp; Reference data is in blue.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-3050595/v1/f84f811c7b49ff88fea61414.png"},{"id":38998399,"identity":"8749541a-b8b7-41ef-9fff-3072dead2abf","added_by":"auto","created_at":"2023-06-24 01:15:26","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":157574,"visible":true,"origin":"","legend":"\u003cp\u003eEvolution of the a, constant multipliers, of the fitting power functions, as the fold changes in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e increased.\u0026nbsp; This comparison was performed in two ranges, one from the first data on (a), the second from the 55-fold change in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e on (b).\u0026nbsp; Reference data is in blue.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-3050595/v1/d03f352761e68c9250656798.png"},{"id":38997763,"identity":"31223fa6-9d53-4d15-9432-cdcd595bd3d7","added_by":"auto","created_at":"2023-06-24 01:07:26","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":116503,"visible":true,"origin":"","legend":"\u003cp\u003eEvolution of the e, exponents, of the fitting power functions, as the fold changes in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e increased, displayed until 100-fold change in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e.\u0026nbsp; Fold change range starts from the first data on.\u0026nbsp; Reference data is in blue and the data in the compared series that have the same e-value as that of the reference is shown.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-3050595/v1/671e847e7abefe98ddcdc6b8.png"},{"id":38998402,"identity":"53c68cb7-051b-4ca3-909a-d5bf425d4675","added_by":"auto","created_at":"2023-06-24 01:15:27","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":257803,"visible":true,"origin":"","legend":"\u003cp\u003eCubes can be formed through the added pieces that form the areas of squares, where the side-lengths of the squares are increasing as triangular numbers up to 15. Thus, the side lengths of the squares are 1, 3, 6, 10, and 15, consecutively, while the side lengths of the cubes are 1, 2, 3, 4, and 5.\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-3050595/v1/62dfb6d9f345c81604b6b7e5.png"},{"id":38997768,"identity":"16d9e6ea-5391-44b9-898f-0e79cb7e3e57","added_by":"auto","created_at":"2023-06-24 01:07:27","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":182336,"visible":true,"origin":"","legend":"\u003cp\u003eRatios of increases in the squares of figurate numbers compared to the ratios of metabolic rate changes to the fold changes in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e and to the diminished flow in the added branch to flow, with unit increment in lengths of the added branches.\u0026nbsp; Fold changes in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e are figurate numbers, which are the square numbers, triangular numbers, pentagonal numbers, and hexagonal numbers.\u0026nbsp; Reference data is in blue.\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-3050595/v1/21bbceeb87c1ee8e9be04798.png"},{"id":38998854,"identity":"14b0d437-61fc-45c3-8dd7-6ed25872d6d0","added_by":"auto","created_at":"2023-06-24 01:23:33","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1235590,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3050595/v1/95e1fb63-6a37-42e0-811d-a9f96f659eb6.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Allometric Scaling Inspection Through Figurate Numbers","fulltext":[{"header":"Introduction","content":"\u003cp\u003eAllometric scaling correlate biological parameters like the drug clearances and interspecies physiological parameters (Huang et al., 2015), or the noncoding DNA and the protein size (Adiguzel, 2021), or the metabolic rate (Kozlowski et al., 2003). As stated by White and Seymour (White \u0026amp; Seymour, 2005), allometric relation was first suggested by Sarrus and Ramaeux (cited in (Brody, 1945)) and supported by Max Rubner. Kleiber (Kleiber, 1932; Kleiber, 1961) found that metabolic rate was proportional to an exponent (\u0026gt;\u0026thinsp;0.66) of body mass, and the exponent was later approved as 0.75. The allometric relation can be expressed with the equation \u003cem\u003ey\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eaM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e\u003csup\u003e\u003cem\u003ee\u003c/em\u003e\u003c/sup\u003e, where y is the variable of interest, \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e is the body mass, and e is the exponent. Allometric relationships can be investigated between different parameters, including the replacement of the \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e with another parameter of interest. The exponent in the allometric relation of \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e and the basal metabolic rate (\u003cem\u003eml\u003c/em\u003e oxygen consumption per hour) is 0.686\u0026thinsp;\u0026plusmn;\u0026thinsp;0.014, and it becomes a bit less (0.0675\u0026thinsp;\u0026plusmn;\u0026thinsp;0.013) when it is corrected for body temperature, while it is 0.712\u0026thinsp;\u0026plusmn;\u0026thinsp;0.013 for the thermoneutral resting metabolic rate (White \u0026amp; Seymour, 2005). Plotting the variation of the metabolic rate with the body masses of the terrestrial animals reveals an exponent of 0.73, which is commonly stated as 0.75 for the sake of simplicity. There can be certain adaptive variations like phylogenic, geographic, etc., but one can well assume that many common observations should be arousing from a variety of situations converging to an optimal solution. The basis of the relationships has been debated (West et al., 1997; West et al., 1999; West et al., 2001; Banavar et al., 2010; Razavi et al., 2018) and we believe that there is still space in this field. Therefore, we aim to make an explanation of the exponent 0.73 through figurate numbers, considering that the exponent of 0.73 is the result of converging solutions. Figurate numbers are appropriate to our purpose since they are classification of numbers \u0026ldquo;according to their geometric representation as sets of dots (Weaver, 1974)\u0026rdquo; (Kempen \u0026amp; Biehler, 2020).\u003c/p\u003e"},{"header":"Methods","content":"\u003cp\u003eWe looked for association between the members of the figurate numbers (square, triangular, pentagonal, and hexagonal numbers) representing fold change in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e, and the allometric relation between the \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e and metabolic rate, where the exponent is 0.73. Allometric relations are designated with the equation \u003cem\u003ey\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eaM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e\u003csup\u003e\u003cem\u003ee\u003c/em\u003e\u003c/sup\u003e, where y is the dependent variable, a is the constant multiplier, \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e is the independent variable, which is the body mass here, and e is the exponent. We took constant multiplier as 1. The resulting allometric relation (\u003cem\u003ey\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e\u003csup\u003e\u003cem\u003ee\u003c/em\u003e\u003c/sup\u003e), and the functions derived from it, became our reference model functions. Square numbers are the squares of the positive integers (i.e., \u003cem\u003en\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e). Triangular numbers are the sum of positive integers until and including the \u003cem\u003ei\u003c/em\u003e\u003csup\u003e\u003cem\u003eth\u003c/em\u003e\u003c/sup\u003e position, where \u003cem\u003ei\u003c/em\u003e is each consecutive member of the positive integers as the variable (i.e., \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\sum _{i=I}^{n}i\\)\u003c/span\u003e\u003c/span\u003e). Pentagonal numbers and hexagonal numbers are calculated through (3\u003cem\u003en\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e-\u003cem\u003en\u003c/em\u003e)/2 and (3\u003cem\u003en\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e-\u003cem\u003en\u003c/em\u003e), respectively, where \u003cem\u003en\u003c/em\u003e is also the variable as the positive integers. As stated, we looked for the association between the members of the selected figurate numbers and the allometric scaling. We assumed that the relationship is through the squares of the numbers. We took the ratio of increase in the squares of the figurate numbers, through subtracting the squares of consecutive numbers and dividing the difference to the square of the latter number. Namely, if we write F\u003csub\u003en\u003c/sub\u003e instead of a member of figurate number, the depicted calculation is (F\u003csub\u003en\u003c/sub\u003e\u003csup\u003e2\u003c/sup\u003e-F\u003csub\u003en\u0026minus;1\u003c/sub\u003e\u003csup\u003e2\u003c/sup\u003e)/ F\u003csub\u003en\u003c/sub\u003e\u003csup\u003e2\u003c/sup\u003e, where F\u003csub\u003en\u003c/sub\u003e is the n\u003csup\u003eth\u003c/sup\u003e figurate number. For comparison, the respective allometric relation, which was calculated by taking the power (\u003cem\u003ee\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.73) of \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e, was divided by \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e. That calculation is \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e\u003csup\u003e0.73\u003c/sup\u003e/\u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e, which is resulting in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e\u003csup\u003e\u0026minus;0.27\u003c/sup\u003e. For better comparability, this calculation was performed for the fold changes in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e, where the fold changes corresponded to the triangular numbers. Comparison is performed through fitting the curves with power functions and comparing the exponents and the goodness of fit values. Microsoft Office Excel was utilized for the calculations and plots.\u003c/p\u003e \u003cp\u003eWe also displayed the evolution of the exponents (e) and the constant multipliers (a) of the fitting power functions, as the fold change in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e increased. This was performed not only from the beginning on, but also from the 55-fold change in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e on. These e- and a-values were compared with pairwise two-tail T-test, to see whether there was a statistically significant (p\u0026thinsp;\u0026le;\u0026thinsp;0.05) difference in the means, or not.\u003c/p\u003e"},{"header":"Results and discussion","content":"\u003cp\u003eThe first 20 numbers in the square, triangular, pentagonal, and hexagonal numbers are listed in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\u0026nbsp;\u003ctable id=\"Tab1\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eThe first 20 numbers in the square, triangular, pentagonal, and hexagonal numbers.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003ccolgroup cols=\"4\"\u003e\u003c/colgroup\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eSquare numbers\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eTriangular numbers\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003ePentagonal numbers\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eHexagonal numbers\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e9\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e12\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e15\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e16\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e22\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e28\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e25\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e15\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e35\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e45\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e36\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e21\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e51\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e66\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e49\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e28\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e70\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e91\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e64\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e36\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e92\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e120\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e81\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e45\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e117\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e153\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e100\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e55\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e145\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e190\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e121\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e66\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e176\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e231\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e144\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e78\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e210\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e276\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e169\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e91\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e247\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e325\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e196\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e105\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e287\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e378\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e225\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e120\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e330\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e435\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e256\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e136\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e376\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e496\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e289\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e153\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e425\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e561\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e324\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e171\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e477\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e630\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e361\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e190\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e532\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e703\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e400\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e210\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e590\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e780\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eAs mentioned in the methods, we calculated the squares. Calculating the square of edge length gives the area in squares, which will be mentioned at the end, and taking the square also gets the maximum unidirectional connectivity between the dots representing the components of a number. An example of this interpretation is illustrated in Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eAs stated in the methods, we took the ratio of increase in the results of squares by subtracting the squares of consecutive numbers and dividing the differences to the latter numbers. These calculations for the first twenty numbers of the triangular numbers are displayed in Table \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e. Ratios of increases in the squares of figurate numbers, which are the first twenty triangular numbers, are compared with the ratios of metabolic rate changes to fold changes in body mass (\u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e), where the fold changes in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e are triangular numbers, for comparability (Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e).\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\u0026nbsp;\u003ctable id=\"Tab2\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eThe first twenty numbers in the triangular numbers together with the relevant calculations for the plotted ratio-data. Metabolic rate changes and squares of triangular numbers are rounded.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003ccolgroup cols=\"5\"\u003e\u003c/colgroup\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eTriangular numbers\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMetabolic rate change\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eRatio of metabolic rate change to fold change in M\u003csub\u003eb\u003c/sub\u003e (reference)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eSquare of triangular numbers\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eRatio of increase in the square of triangular numbers\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1.00\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.62\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e14\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.93\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.48\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e52\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.74\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e11\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.41\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e130\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.60\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e15\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e16\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.36\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e259\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.50\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e21\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e21\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.32\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e453\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.43\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e28\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e27\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e725\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.37\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e36\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e33\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.27\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1085\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.33\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e45\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e39\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.26\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1548\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.30\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e55\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e46\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.24\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2123\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.27\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e66\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e53\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.23\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2824\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.25\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e78\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e61\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.22\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e3662\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.23\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e91\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e68\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.21\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e4649\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.21\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e105\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e76\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.20\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e5796\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.20\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e120\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e84\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.19\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e7115\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.19\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e136\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e93\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.19\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e8618\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.17\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e153\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e102\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.18\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e10315\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.16\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e171\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e111\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.18\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e12219\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.16\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e190\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e120\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.17\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e14340\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.15\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e210\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e129\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.17\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e16690\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.14\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eIt can be seen in Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e that all cases had high the goodness of fit values. The ratios of metabolic rate changes to the fold changes in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e fitted perfectly by a power function, as should be the case. Its exponent is the closest to the exponents of the ratios of increases in the squares of pentagonal and hexagonal numbers (Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e). However, an immediate interpretation of their being the best approximation among the others can be misleading. The biologically relevant and the readily comparable changes in the body masses and the associated metabolic rates would be spanning a narrower range than that is viewed in Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e. This is elaborated below.\u003c/p\u003e\n\u003cp\u003eAlthough we took constant multiplier as 1, we compared the evolution of constant multipliers (a) and the exponents (e) of the fitting power functions, as the fold change in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e increased. This comparison was performed in two ranges, one from the first data on (Fig. 3a) and the second from the 55-fold change in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e on (Fig. 3b).\u003c/p\u003e\n\u003cp\u003eFigure 3 revealed that the e- and a-values were relying on the ranges of the fitted-values. The e- and a-values were significantly different within the same range, when they were calculated in the given range only (i.e., from the 55-fold change in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e on), compared to that from the first data on. In the former situation (Fig. 3a), e-value evolution curves were crossing the data derived from the reference model (\u003cem\u003ey\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eaM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e\u003csup\u003e\u003cem\u003e0.73\u003c/em\u003e\u003c/sup\u003e, \u003cem\u003ea\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003e1\u003c/em\u003e) at different fold changes in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e. Accordingly, the best model was relying on the range of interest of the fold changes in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e and the whole range of the fold changes in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e under study. Varying constant multipliers (White \u0026amp; Gould, 1965) was also supporting the dynamic variations in the experimental data (Fig. 4), which will not be discussed further here.\u003c/p\u003e\n\u003cp\u003eIt was indicated that the e-value evolution curves were crossing the data derived from the reference model at different fold changes in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e (Fig. 3a). Those fold changes in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e are 10, 36, 70, and 91, when the calculations were performed with the triangular numbers, square numbers, pentagonal numbers, and hexagonal numbers, as the fold changes in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e, respectively (Fig. \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e). It means that the exponent in the reference relationship (i.e., \u003cem\u003ey\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e\u003csup\u003e\u003cem\u003e0.73\u003c/em\u003e\u003c/sup\u003e) is obtained through the relationship derived by the triangular numbers, when there is a fold change in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e up to 10. Similarly, the exponent 0.73 is obtained through the relationship derived by the square numbers when there is a fold change in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e up to 36, while it is obtained through the relationship derived by the pentagonal and hexagonal numbers when there is a fold change in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e up to 70 and 91, respectively. Here, the e-value relies on the relationship derived through the figurate numbers, up to the selected range of fold change in \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e. We set the reference to a specific e-value here, but allometric relations can have different e-values. Therefore, this study is bringing about a distinct approach to study the varying types of allometric relations.\u003c/p\u003e\n\u003cp\u003eNow we will discuss a relevant feature of the triangular numbers. It was mentioned in the beginning that calculating the square of the edge length gives the area in squares. The sum of the cubes of the positive integer numbers from 1 to \u003cem\u003en\u003c/em\u003e is equal to the square of the sum of those numbers (i.e., \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sum }_{i=1}^{n}{i}^{3}\\)\u003c/span\u003e\u003c/span\u003e = (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sum }_{i=1}^{n}i\\)\u003c/span\u003e\u003c/span\u003e)\u003csup\u003e2\u003c/sup\u003e). This is visually illustrated in Fig. \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e. It should be noted here that the cubes of the positive integers are the cube numbers. The inter-related calculation that we performed in this study was subtracting the squares of consecutive triangular numbers and dividing the difference to the square of the latter number (i.e., (T\u003csub\u003en\u003c/sub\u003e\u003csup\u003e2\u003c/sup\u003e-T\u003csub\u003en\u0026minus;1\u003c/sub\u003e\u003csup\u003e2\u003c/sup\u003e)/ T\u003csub\u003en\u003c/sub\u003e\u003csup\u003e2\u003c/sup\u003e, where T\u003csub\u003en\u003c/sub\u003e is the n\u003csup\u003eth\u003c/sup\u003e triangular number). As observed, in Fig. \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e, consecutive side-lengths of the squares with the same origin are the triangular numbers. Accordingly, with each increase in the size of the square displayed in Fig. \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e, we divided the increase in the square-area to the total area. This is also the same as dividing the volume of the largest cube in Fig. \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e, sequentially, to the total area of the sum of the volumes of the cubes with unit increments in the edge-lengths. In other words, the n\u003csup\u003eth\u003c/sup\u003e cube number is divided to the sum of the cube numbers from 1 to n (i.e., n\u003csup\u003e3\u003c/sup\u003e/\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sum }_{i=1}^{n}{i}^{3}\\)\u003c/span\u003e\u003c/span\u003e). When our calculation is viewed from this perspective, it is anticipated to be in accordance with the compared-ratio derived from the allometric relation, which was \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e\u003csup\u003e\u003cem\u003e0.73\u003c/em\u003e\u003c/sup\u003e/\u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e. Therefore, our approach introduces a geometric reasoning to the allometric relation with a means of treating the varying observations.\u003c/p\u003e\n\u003cp\u003eThe feature described above for the triangular numbers are correlated with the other figurate numbers we dealt with since figurate numbers are inter-related. Square numbers are the sum of the consecutive triangular numbers (i.e., S\u003csub\u003en\u003c/sub\u003e=T\u003csub\u003en\u003c/sub\u003e+T\u003csub\u003en\u0026minus;1\u003c/sub\u003e, where S\u003csub\u003en\u003c/sub\u003e is the n\u003csup\u003eth\u003c/sup\u003e square number and T\u003csub\u003en\u003c/sub\u003e is the n\u003csup\u003eth\u003c/sup\u003e triangular number). Also, the n\u003csup\u003eth\u003c/sup\u003e pentagonal number P\u003csub\u003en\u003c/sub\u003e is equal to T\u003csub\u003en\u0026minus;1\u003c/sub\u003e+n\u003csup\u003e2\u003c/sup\u003e, or T\u003csub\u003en\u003c/sub\u003e+2T\u003csub\u003en\u0026minus;1\u003c/sub\u003e, or T\u003csub\u003e2n\u0026thinsp;\u0026minus;\u0026thinsp;1\u003c/sub\u003e-T \u003csub\u003en\u0026minus;1\u003c/sub\u003e, where n is the positive integer and T\u003csub\u003en\u003c/sub\u003e is the n\u003csup\u003eth\u003c/sup\u003e triangular number. On the other hand, hexagonal numbers are the differences of consecutive cube numbers (i.e., H\u003csub\u003en\u003c/sub\u003e=C\u003csub\u003en\u003c/sub\u003e-C\u003csub\u003en\u0026minus;1\u003c/sub\u003e, where H\u003csub\u003en\u003c/sub\u003e is the n\u003csup\u003eth\u003c/sup\u003e hexagonal number and C\u003csub\u003en\u003c/sub\u003e is the n\u003csup\u003eth\u003c/sup\u003e cube number).\u003c/p\u003e\n\u003cp\u003eThe inter-relations that we worked out in this study are a conceivable basis of the correlation with the allometric relation, which we derived through the triangular numbers. A hypothetical example can be the number of molecules produced for enlarging volumes when the molecule is transported at different rates to each volume-layer, while being consumed or reacted at certain time points. Another example is the distance travelled at the enlarged surface area leading to a reduction in the supply due to increased resistance. Specifically, blood flow is the source of oxygen delivery to the cells and tissues, which is regulated by the metabolic needs. Adaptation to lower metabolic rates due to flow changes can be another phenomenon, which is evaluated further in a different manner than the present literature. We will treat increase in the size (i.e., \u003cem\u003eM\u003c/em\u003e\u003csub\u003e\u003cem\u003eb\u003c/em\u003e\u003c/sub\u003e) to be leading to parallel branches emerging from the main vessel with a unit increase in the length of each consecutive new branches. Namely, the first, single branch having the length \u003cem\u003el\u003c/em\u003e, is followed by the addition of new branches having the sizes 2\u003cem\u003el\u003c/em\u003e, 3\u003cem\u003el\u003c/em\u003e, going up to the length n\u003cem\u003el\u003c/em\u003e, where each new branching up to the number n is indeed adding onto the present vasculature like the increased area of the square depicted in Fig. \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e. Flow through the parallel branches follow the Ohm\u0026apos;s Law where the inverse of the total resistance is the sum of the inverses of the resistances in the branches (i.e., \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sum }_{i=1}^{n}1/Ri\\)\u003c/span\u003e\u003c/span\u003e=1/R\u003csub\u003e1\u003c/sub\u003e+1/R\u003csub\u003e2\u003c/sub\u003e+\u0026hellip;+1/R\u003csub\u003en\u003c/sub\u003e). Flow through each branch is driven by the same pressure difference and inversely proportional with the resistance of the branch. Therefore, each new longer branch can be assumed to have lowered flow when the other flow-parameters are the same. If we calculate this difference in flow for each added branch with unit increment in the length, flow in the added branch compared to the flow in the single vessel (i.e., no branch) system is displayed in Table \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e for the first ten data.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\u0026nbsp;\u003ctable id=\"Tab3\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eCalculation and the first ten results of flow in the added branch with unit increments in length.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003ccolgroup cols=\"7\"\u003e\u003c/colgroup\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eAdded branch length (i.e., n\u003cem\u003el\u003c/em\u003e)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e1/R\u003csub\u003en\u003c/sub\u003e\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e\u0026Sigma;(1/R\u003csub\u003en\u003c/sub\u003e)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e\u0026Sigma;R\u003csub\u003en\u003c/sub\u003e:\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e1/\u0026Sigma;(1/R\u003csub\u003en\u003c/sub\u003e)\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e\u0026Delta;P\u0026thinsp;=\u0026thinsp;QxR \u0026loz; Q\u0026thinsp;=\u0026thinsp;\u0026Delta;P/R,\u003c/p\u003e\n \u003cp\u003e\u0026Delta;P constant\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eQ at the n\u003csup\u003eth\u003c/sup\u003e branch (\u0026Delta;P/(\u0026Sigma;R\u003csub\u003en\u003c/sub\u003exn\u003cem\u003el\u003c/em\u003e))\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.00\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1.00\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1.00\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u0026Sigma;R\u003csub\u003e1\u003c/sub\u003e:\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1.00\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eQ at the 1. branch (\u0026Delta;P/(\u0026Sigma;R\u003csub\u003e1\u003c/sub\u003ex1\u003cem\u003el\u003c/em\u003e))\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1.00\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2.00\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.50\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1.50\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u0026Sigma;R\u003csub\u003e2\u003c/sub\u003e:\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.67\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eQ at the 2. branch (\u0026Delta;P/(\u0026Sigma;R\u003csub\u003e2\u003c/sub\u003ex2\u003cem\u003el\u003c/em\u003e))\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.75\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e3.00\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.33\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e1.83\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u0026Sigma;R\u003csub\u003e3\u003c/sub\u003e:\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.55\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eQ at the 3. branch (\u0026Delta;P/(\u0026Sigma;R\u003csub\u003e3\u003c/sub\u003ex3\u003cem\u003el\u003c/em\u003e))\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.61\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e4.00\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.25\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2.08\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u0026Sigma;R\u003csub\u003e4\u003c/sub\u003e:\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.48\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eQ at the 4. branch (\u0026Delta;P/(\u0026Sigma;R\u003csub\u003e4\u003c/sub\u003ex4\u003cem\u003el\u003c/em\u003e))\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.52\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e5.00\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.20\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2.28\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u0026Sigma;R\u003csub\u003e5\u003c/sub\u003e:\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.44\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eQ at the 5. branch (\u0026Delta;P/(\u0026Sigma;R\u003csub\u003e5\u003c/sub\u003ex5\u003cem\u003el\u003c/em\u003e))\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.46\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e6.00\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.17\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2.45\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u0026Sigma;R\u003csub\u003e6\u003c/sub\u003e:\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.41\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eQ at the 6. branch (\u0026Delta;P/(\u0026Sigma;R\u003csub\u003e6\u003c/sub\u003ex6\u003cem\u003el\u003c/em\u003e))\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.04\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e7.00\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.14\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2.59\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u0026Sigma;R\u003csub\u003e7\u003c/sub\u003e:\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.39\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eQ at the 7. branch (\u0026Delta;P/(\u0026Sigma;R\u003csub\u003e7\u003c/sub\u003ex7\u003cem\u003el\u003c/em\u003e))\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.37\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e8.00\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.13\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2.72\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u0026Sigma;R\u003csub\u003e8\u003c/sub\u003e:\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.37\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eQ at the 8. branch (\u0026Delta;P/(\u0026Sigma;R\u003csub\u003e8\u003c/sub\u003ex8\u003cem\u003el\u003c/em\u003e))\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.34\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e9.00\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.11\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2.83\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u0026Sigma;R\u003csub\u003e9\u003c/sub\u003e:\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.35\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eQ at the 9. branch (\u0026Delta;P/(\u0026Sigma;R\u003csub\u003e9\u003c/sub\u003ex9\u003cem\u003el\u003c/em\u003e))\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.31\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e10.00\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e2.93\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u0026Sigma;R\u003csub\u003e10\u003c/sub\u003e:\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.34\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eQ at the 10. branch (\u0026Delta;P/(\u0026Sigma;R\u003csub\u003e10\u003c/sub\u003ex10\u003cem\u003el\u003c/em\u003e))\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.29\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eAbbreviations with notes: \u0026Delta;P is the pressure difference, which is the constant driving force; R is the resistance, which is depending only on the length \u003cem\u003el\u003c/em\u003e here, assuming all the other parameters equal; Q is the flow, which is influencing the metabolic rate; \u003cem\u003el\u003c/em\u003e is the unit length, n is positive integers representing unit increments in length when multiplied with the length \u003cem\u003el\u003c/em\u003e.\u003c/p\u003e\n\u003cp\u003eWe plotted these lowered flows in the added branches together with the graphs in Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e, assuming the diminished flow as the bottleneck of the oxygen delivery, and hence the metabolism. There is agreement with the other plots, so with our calculations and the reference data (Fig. \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e).\u003c/p\u003e"},{"header":"Conclusion","content":"\u003cp\u003eThe allometric relationship is explained for the first time through the triangular, square, pentagonal, and hexagonal numbers. Their utilization can end up in different exponents and constant multipliers at varying scaling ranges, applicable to different fields and aspects in biology.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAcknowledgment:\u003c/strong\u003e\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eEcology and Evolutionary Biology Society of Turkey is acknowledged.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConflicts of interests:\u003c/strong\u003e\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eNothing to declare.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding:\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNone\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eEthical permissions:\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNot required\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003e\u003cspan\u003eAdiguzel, Y. (2021). Information- theoretic approach in allometric scaling relations of DNA and proteins. \u003cem\u003eChem Biol Drug Des, 0\u003c/em\u003e, 1\u0026ndash;13. doi:10.1111/cbdd.13988\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eBanavar, J. R., Moses, M. E., Brown, J. H., Damuth, J., Rinaldo, A., Sibly, R. M., \u0026amp; Maritan, A. (2010). A general basis for quarter-power scaling in animals. \u003cem\u003eProc Nat Acad Sci, 107\u003c/em\u003e, 15816\u0026ndash;15820. doi:10.1073/pnas.1009974107\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eBrody, S. (1945). \u003cem\u003eBioenergetics and Growth.\u003c/em\u003e New York: Reinhold Publishing Corporation.\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eHuang, W., Geng, L., Deng, R., Lu, S., Ma, G., Yu, J.,. .. Lu, X. (2015). Prediction of Human Clearance Based on Animal Data and Molecular Properties. \u003cem\u003eChem. Biol. Drug Des., 86\u003c/em\u003e, 990\u0026ndash;997.\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eKempen, L., \u0026amp; Biehler, R. (2020). Using figurate numbers in elementary number theory - discussing a \u0026apos;useful\u0026apos; herustic from the perspectives of semiotics and cignitive psychology. \u003cem\u003eFront Psychol, 11\u003c/em\u003e, 1180. doi:10.3389/fpsyg.2020.01180\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eKleiber, M. (1932). Body size and metabolism. \u003cem\u003eHilgardia, 6\u003c/em\u003e, 315\u0026ndash;353.\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eKleiber, M. (1961). \u003cem\u003eThe Fire of Life.\u003c/em\u003e New York, London: John Wiley and Sons, Inc.\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eKozlowski, J., Konarzewski, M., \u0026amp; Gawelczyk, A. T. (2003). Cell size as a link between noncoding DNA and metabolic rate scaling. \u003cem\u003eProc. Natl. Acad. Sci. U.S.A., 100\u003c/em\u003e(24), 14080\u0026ndash;14085.\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eRazavi, M. S., Shirani, E., \u0026amp; Kassa, G. S. (2018). Scaling laws of flow rate, vessel blood volume, lengths, and transit times with number of capillaries. \u003cem\u003eFront Physiol, 9\u003c/em\u003e, 581. doi:10.3389/fphys.2018.00581\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eWeaver, C. (1974). Figurate numbers. \u003cem\u003eMath Teach, 67\u003c/em\u003e, 661\u0026ndash;666.\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eWest, G. B., Brown, J. H., \u0026amp; Enquist, B. J. (1997). A general model for the origin of allometric scaling laws in biology. \u003cem\u003eScience, 276\u003c/em\u003e, 122\u0026ndash;126.\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eWest, G. B., Brown, J. H., \u0026amp; Enquist, B. J. (2001). A general model for ontogenetic growth. \u003cem\u003eNature, 413\u003c/em\u003e, 628\u0026ndash;631.\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eWest, G. B., Enquist, J. H., \u0026amp; Enquist, B. J. (1999). The fourth dimension of life: fractal geometry and allometric scaling of organisms. \u003cem\u003eScience, 284\u003c/em\u003e, 1677\u0026ndash;1679.\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eWhite, C. R., \u0026amp; Seymour, R. S. (2005). Allometric scaling of mammalian metabolism. \u003cem\u003eThe Journal of Experimental Biology, 208\u003c/em\u003e, 1611\u0026ndash;1619. doi:10.1242/jeb.01501\u003c/span\u003e\u003c/li\u003e\n \u003cli\u003e\u003cspan\u003eWhite, J. F., \u0026amp; Gould, S. J. (1965). Interpretatıon of the coefficient in the allometric equation. \u003cem\u003eThe American Naturalist, 99\u003c/em\u003e, 5\u0026ndash;18.\u003c/span\u003e\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Allometric relation, triangular numbers, square numbers, cube numbers, pentagonal numbers, hexagonal numbers","lastPublishedDoi":"10.21203/rs.3.rs-3050595/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3050595/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eAllometric scaling is a widely-studied, and discussed phenomena in biology that relates the parameters with a power function. There is still space for further investigation. Accordingly, the relationship is aimed to be explained for the first time in this work, through the figurate numbers, specifically the triangular, square, pentagonal, and hexagonal numbers. These mathematical series have not been considered in terms of a possible relation with allometric scaling, which is a mathematical relation in essence. We formulated the possible means of the relation of these figurate numbers with allometric scaling and suggested that their utilization can end up in different exponents and constant multipliers at varying scaling ranges. We also discussed further implications and exemplified the possible association with the blood flow-related metabolic rate adaptation. It is revealed that figurate numbers are having an inherent relation with the allometric scaling.\u003c/p\u003e","manuscriptTitle":"Allometric Scaling Inspection Through Figurate Numbers","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2023-06-24 01:07:22","doi":"10.21203/rs.3.rs-3050595/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"bf8a31b6-e45b-4170-9cbe-c42be4754063","owner":[],"postedDate":"June 24th, 2023","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2023-06-24T01:07:22+00:00","versionOfRecord":[],"versionCreatedAt":"2023-06-24 01:07:22","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3050595","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3050595","identity":"rs-3050595","version":["v1"]},"buildId":"cBFmMYwuxLRRLfASyISRj","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}