Special case of the three-dimensional Pythagorean gear

Luis Teia University of Lund, Sweden luisteia@sapo .pt

Introduction

n mathematics, three integer numbers or triples have been shown to Igovern a specific geometrical balance between triangles and squares. The first to study triples were probably the Babylonians, followed by Pythagoras some 1500 years later (Friberg, 1981). This geometrical balance relates parent triples to child triples via the central square method (Teia, 2015). The great family of triples forms a tree—the Pythagorean tree—that grows its branches from a fundamental seed (3, 4, 5) making use of one single very specific motion. The diversity of its branches rises from different starting points (i.e., triples) along the tree (Teia, 2016). All this organic geometric growth sprouts from the dynamics of the Pythagorean geometric gear (Figure 1). This gear interrelates geometrically two fundamental alterations in the Pythagorean theorem back to its origins—the fundamental process of summation x + y = z (Teia, 2018). But reality has a three dimensional nature to it, and hence the next important question to ask is: what does the Pythagorean gear look like in three dimensions?

Figure 1. Overall geometric pattern or ‘geometric gear’ connecting the three equations (Teia, 2018). Australian Senior Mathematics Journal vol. 32 no. 2

36 Special case of the three-dimensional Pythagorean gear Australian Senior Mathematics Journal vol. 32 no. 2 37 = 1 and (Flannery, (Flannery, = y 2 of the three … = y 6 9 2 6 9 2 6 9 2 1 2 4 1 4 . 1 = 3 0 0 1 6 + 2 1 0 5 6 + 0 4 6 2 + 1 ≈ 2 Figure 2. Special case x = y: (a) the Pythagorean gear and 2. Special case x = y: (a) the Pythagorean Figure , contemporarily termed in mathematics as the Pythagoras’ , contemporarily termed in mathematics 2 (b) Babylonian clay tablet YBC 7289 (courtesy of the Yale Babylonian Collection). (b) Babylonian clay tablet YBC 7289 (courtesy of the Yale Indeed, ancient Greek philosophers such as Aristotle and Euclid studied Indeed, ancient Greek philosophers A good starting point for studying any theorystudying for point starting good A the is any dimension in several of the two dimensional versions aligned along the three orthogonal several of the two dimensional versions aligned along the three and mathematical expression of this special case, where the later holds an and mathematical expression of Swetz, 2018). It was found that this table contained in itself the geometrical Swetz, 2018). It was found that this planes. dimensional version of the Pythagorean geometric gear is constructed from geometric Pythagorean version of the dimensional impressive approximation to the triangle’s hypotenuse length triangle’s impressive approximation to the importance in ancient times (as indicated by the clay tablet of the Babylon’s clay tablet of the Babylon’s times (as indicated by the importance in ancient discoveries such as Einstein’s theory was speed where relativity special of Einstein’s as such discoveries considered constant (the exceptional case) that then evolved to the theory (the exceptional case) that considered constant for (the general case) where acceleration was accounted of general relativity examination of the exceptional cases. This approach was used in eminent used in eminent approach was cases. This of the exceptional examination constant (Finsch, 2003), and presented proofs of its irrational nature (Euclid, constant (Finsch, 2003), and presented the Pythagorean theorem this exceptional condition occurs when the right theorem this exceptional condition the Pythagorean same length x that is, when both legs have the triangle is isosceles, this number named YBC 7289, estimated to date back to circa 1800–1600 BCE; Beery estimated to date back to circa named YBC 7289, & Hypothesis hold the same angles (Figure 2a). This special case is seen to play particular play to seen is case special This 2a). (Figure angles same the hold 2006), given as:

300 BC). The hypothesis of this paper is that the special case x (Einstein, 1961). Following on his approach, and in particular for the case of Following on his approach, and (Einstein, 1961). 38 Australian Senior Mathematics Journal vol. 32 no. 2 Teia Theory (Figure 4b).Truncated meansthattheedgeofelementwasclippedata Three-dimensional Pythagorean gear Consider theisoscelesrighttriangle(twolegsx hexagonal faceisformedbythethreehypotenuseofeachtrianglealignedin hexagon. Oneisinside the other, andthetwo dimensionaltheoremsinall to theprevioustriangle,givesFigure3b.RotatingnowintoYZplane, the three orthogonalplanes drivealltogetherinagroupmovementthethree third ofitslength.Therefore,ifinthetransition from onetotwodimensions, between themthethreedimensionalequivalent—the truncatedtetrahedron between thematriangle (Figure 4a), three perpendiculartrianglesform in maintaining thepoint0atcentreofhypotenuse,andperpendicular each orthogonalplanehavingasacommonpointthecentre0. other, fromtwotothreedimensions,thenew‘hypotenuse’ isformedbythe perpendicular totheprevioustrianglegivesFigure3c.Thefinaloutcome diagonal linesofthree triangles perpendiculartoeachother, whichisthe gives theconstructinFigure3dholdingahexagonalfaceatcentre.This again whilemaintainingthepoint0atcentreofhypotenuseand angles equal)andpoint0atthecentreofhypotenusealignedalong a hypotenuseoftriangleisformedbytwolines perpendiculartoeach This approach suggests that just like two perpendicular lines form in (b) rotation intoXYplane,(c) rotation intoYZplaneand(d)joiningtheedgesgivesahexagon. XZ plane(Figure3a).RotatingthetriangleintoXYplane,while Figure ofthethree 3.Formation dimensional diagonal:(a)triangleinXZplane, =y =1andcorresponding Special case of the three-dimensional Pythagorean gear Australian Senior Mathematics Journal vol. 32 no. 2 39 YZ or Figure 4. Diagonals of: (a) a square is the hypotenuse and (b) cube is a hexagon. is the hypotenuse and (b) cube Diagonals of: (a) a square 4. Figure sketch) the square in a perpendicular direction by 1 (Figure 5a). selecting the midpoint and making it perpendicular to the diagonal selecting the midpoint and making it perpendicular to the diagonal and draw a line from one opposing corner of the cube to the other and draw a line from one opposing corner of the cube to the face of the cube (Figure 5d). This completes the proof and the exercise. face of the cube (Figure 5d). This completes midpoint, as often certain perspectives can cause optical illusions and illusions optical cause can perspectives certain often as midpoint, technical term used in CAD that means to add material based on a technical term used in CAD that means to add material based the model to make sure that indeed the point is on the line and is the the model to make sure that indeed the point is on the line and by the point may actually be floating somewhere else). Add a midplane line (Figure 5c). Create a 3-D sketch (that is, a sketch that is not bound by a plane) a sketch Create a 3-D sketch (that is, Intersecting the plane with the cube, results in a hexagon as the diagonal To create a cube of side 1, first select an orthogonal plane XY, To (Figure 5b). Add a midpoint to this diagonal line (tip: always rotate and sketch a square of side length 1. Then extrude boss (this is a XZ and sketch a square of side length 1. Then extrude boss (this The proof that the diagonal face of a cube is a hexagon is an exercise The proof that the diagonal face students can also use other equivalent software existing in their school/ students can also use other equivalent and three dimensional theorems, all in one. While two isosceles right triangles right isosceles two While one. in all theorems, dimensional three and described in this exercise (e.g., sketches, extrusion, planes, etc). The solution described in this exercise (e.g., sketches, extrusion, planes, etc). The in geometry teachers alike to do as an assessed assignment. and for students dimensional theorem. The Pythagorean gear is an ensemble of all one, two an ensemble of gear is The Pythagorean theorem. dimensional of these programs may be different, but they all have the same functions of these programs may be different, but they all have the same computer-aided design (CAD) programs in the context of the Pythagorean in the context of programs (CAD) design computer-aided opposing each other form a square, two truncated tetrahedra opposing each opposing each truncated tetrahedra a square, two each other form opposing diagonal of a square, while Hence, the hypotenuse is the other form a cube. to this exercise is described in the following steps: university (like Catia, Ansys DesignModeler, Siemens NX, etc). The layout Siemens NX, etc). The layout university (like Catia, Ansys DesignModeler, the hexagon (composed of three hypotenuses) is the diagonal of a cube. of three hypotenuses) is the hexagon (composed 2. 3. Whilst this example was created with the free academic version of Solidworks, version of Solidworks, academic the free created with was this example Whilst Theorem which is relevant to the Australian Curriculum (ACARA, n.d.). Theorem which is relevant to This exercise develops skills in how to solve pure geometrical problems using This exercise develops skills in how 1. 40 Australian Senior Mathematics Journal vol. 32 no. 2 Teia three axisgivesatruncatedoctahedron(Figure6b).A octahedron withtheouter cubeside2,andtherelationshipisdefinedlater case ofthethree-dimensional Pythagoreangearrelatestheinnertruncated in equation(6). is composedof6squarefacesand8hexagonalfaces. inner square side square insideasquare,wherethehassidelengthhypotenuse 2 The specialcasex Revolving the isosceles right triangle givesthe now familiarshape of a (Figure6a).Similarly, revolvingthetruncatedtetrahedronaroundall Figure 5.Exercise onhowtocreate thediagonalofacube:(a–d)stepsinvolvedinsolution. Figure 6.Innerelementsside 2 =y with the outer square side 2, as 2( ofthetwo-dimensionalPythagoreangearrelates the 2 : (a)square and(b)octahedron. 2 ) 2 = 2 2 . Thespecial Special case of the three-dimensional Pythagorean gear Australian Senior Mathematics Journal vol. 32 no. 2 41 ) to its dual (the outer square of ) to its dual (the 2 Figure 7. Outer (dual) elements side 2: (a) square and (b) cube. Outer (dual) elements side 2: (a) square 7. Figure ) to its dual (the outer cube of side 2) is achieved by adding 8 outer cube of side 2) is achieved ) to its dual (the 2 A square is the dual of itself, and it is well known since the times of Kepler times of Kepler known since the and it is well is the dual of itself, A square Figure 8. (a) A simplified view of the Pythagorean Gear and (b) its extrusion into three dimensions. Gear and (b) its extrusion into three 8. (a) A simplified view of the Pythagorean Figure side 2) is achieved by adding four additional isosceles right triangles (Figure by adding four additional isosceles side 2) is achieved arrangement of the terms of that expression. additional truncated tetrahedrons (Figure 7b). additional truncated Start with the simplified special case of the Pythagorean gear in Figure 8a. Start with the simplified special original equation by 1 gives the same equation. The subtlety lies in the re- original equation by 1 gives the same equation. The subtlety lies context of the special case of the Pythagorean Gear, it is the special version of it is the special Gear, of the Pythagorean of the special case context theorem to three dimensions (Figure 8b). Mathematically, multiplying the theorem to three dimensions (Figure 8b). Mathematically, the octahedron—the truncated octahedron—that is the dual of the cube. A truncated octahedron—that the octahedron—the inner square (side length transition of the that the octahedron is the dual of the cube (Kepler, 1619). However, in the in However, 1619). (Kepler, cube the of dual the is octahedron the that length half Extruding all surfaces the by 0.5 in both directions (up and down) transits Construction volumes using 7a). Similarly, the transition from the inner truncated octahedron (side octahedron truncated inner the from transition the Similarly, 7a). 42 Australian Senior Mathematics Journal vol. 32 no. 2 Teia which isthetruncatedtetrahedron.NotethatFigure10alsoholdsinitself width version ofsumlengths. larger squareABCD(ofside the sumofvolumestwocubesside1islinkedtovolumehalf three piecesabouttheedgesofmidplanesquaregivesFigure 9b. both thetwodimensional version ofsumsquares, and theonedimensional in Figure 9a. Sectioning the volume along these lines, and rotating each of the its edgestotheprojectedlargersquareA gives thevolumetricvariantofgear. Extendinglinesfromthenodesof Since novolumewasaddedorsubtracted,thegeometricassemblystatesthat a truncatedoctahedron(orhalf It isthereorganisationofthisvolume(noneaddedorsubtracted)that Figure 9isrotatedandredrawnin10tobettervisualiseitsshape. 2 , andcentredatthetopofparallelepiped),extendinginturn Figure 9. (a) Partition oftheparallelepipedand(b)rotationFigure ofthefourpieces. 9.(a)Partition Figure 10. The geometry oftheone,two,andthree dimensionalequations. Figure 10.Thegeometry 2 ) tothesmallersquareEFGH(withhalf 2 ( 2 'B ( 1 'C ) ) 'D 3 ) viaanintermediateelement, ' givesthewireframepresented Special case of the three-dimensional Pythagorean gear Australian Senior Mathematics Journal vol. 32 no. 2 43 , termed by the y ≠

. The general case where x . The general case = y XZ plane and (d) the volume superimposed becoming the 3D Pythagorean gear. XZ plane and (d) the volume superimposed becoming the 3D Pythagorean plane (Figure 11c), and finally superimposing the volumetric content and finally superimposing the XZ plane (Figure 11c), However, the two dimensional Pythagorean gear needs to be satisfied in needs to be satisfied Pythagorean gear dimensional the two However, The same cube side 2 can be split into two principal ways: as a sum of 8 The same cube side 2 can be split into two principal ways: as a Figure 11. Integration of the 2-D Pythagorean gear: (a) along XY plane, (b) along YZ plane, (c) along 11. Integration of the 2-D Pythagorean Figure shown previously in Figure 3, the three orthogonal Pythagorean theorems take theorems Pythagorean orthogonal the three Figure 3, in previously shown plane (Figure 11b), along plane (Figure 11b), then along the YZ along the XY plane (Figure 11a), author the ‘universal gear’, will be presented in a future publication. author the ‘universal gear’, will be all three orthogonal planes (Figure 10 shows only one plane) in order to be in order to only one plane) (Figure 10 shows orthogonal planes all three gives Figure 11d. By inspection, we can see that Figure 11d is the expression By inspection, we can see that Figure gives Figure 11d. for the special case when x for the special case of the three dimensional Pythagorean gear as a function of three orthogonal Pythagorean gear as a function of the three dimensional explained later in the section that refers to grids. cubes of side 1 (as shown in Figure 12a, this is the orthogonal way), or as a sum or as the orthogonal way), this is 12a, in Figure shown 1 (as of side cubes Figure 12b, of an inner and outer central truncated octahedron (as shown in two-dimensional Pythagorean gears (shown initially in Figure 2a). This is valid Pythagorean gears (shown initially two-dimensional the the disposition in Figure 11 as follows. Aligning the two dimensional gear first Figure 11 as follows. Aligning the the disposition in this is the diagonal way). The meaning of orthogonal and diagonal way will be this is the diagonal way). The meaning of orthogonal and diagonal volumetrically correct. Therefore, following the orientation of the triangles of the triangles the orientation following correct. Therefore, volumetrically 44 Australian Senior Mathematics Journal vol. 32 no. 2 Teia (redrawn fromFigure7b).Inonedimension,thebalancewithingearis And inbeingmathematicallymoreaccurate,theexponent1canbeaddedto This onedimensionalbalancerepeatsitselfallaroundthecube. Consider theshadedthree-dimensionalversionofgearinFigure13 1

Mathematics ofthePythagorean gear that ofthesumlengths.Thatis,additiontwolineslength1gives denote anequationthatvariesalongonedimension,resultingin (2): another linelength2,resultinginequation(1): Figure 12. Grid lines: (a) the orthogonal wayand(b)thediagonalway.Figure 12.Gridlines:(a)theorthogonal Figure 13.Theonedimensionalbalanceinthesumoflengths. 1 +=2 1 +1 1 =2 1 (2) (1) Special case of the three-dimensional Pythagorean gear Australian Senior Mathematics Journal vol. 32 no. 2 45 . 2 (6) (4) (3) (5) 2

3 2

2 = = 2, and equation (3) can be = 2, and equation 2 3 ) 2 = ) 1 2 ( ) 2 2 ) 1 ( ( 2 2 ( + + 2 2 + 2 = 4 4 + 4 = 8 3 ) ) ) ) 1 1 ( ( 2 2 ( ( 2 Figure 14. The two dimensional balance in the sum of areas. 14. Figure In three dimensions, the balance within the gear is that of the sum of volumes. of sum the of that is gear the within the balance dimensions, three In In essence, this 2 is in fact the product In essence, this 2 In two dimensions, the balance within the gear is that of the sum of areas. the sum of areas. gear is that of within the the balance In two dimensions, 1) lines (equation from a sum of 14. The transition gear in Figure seen in the gives another volume of size 8 (the cube). This is again highlighted on the gives another volume of size 8 (the cube). This is again highlighted to a sum of gear in Figure 15. The transition from a sum of areas (equation 3) to a sum of squares (equation 3) is denoted by a multiplication factor of 2. (equation 3) is denoted by a to a sum of squares reorganised to express its two dimensionality as equation (4): its two dimensionality as equation reorganised to express volumes (equation 5) is denoted again by a multiplication factor of 2 = volumes (equation 5) is denoted again by a multiplication factor of

That is, the addition of two volumes of size 4 (the truncated octahedrons) truncated (the 4 size of volumes two of addition the is, That That is, the addition of two areas size 1 give another area size 2. This is again area size 2. 1 give another of two areas size the addition That is, This two dimensional balance also repeats itself all around the cube. balance also repeats itself all This two dimensional Therefore, equation 5 can be reorganised into equation 6: 46 Australian Senior Mathematics Journal vol. 32 no. 2 Teia A gearcanbeseenasanindividualbalance,oranetworkbalanceofmany A typicalcombinationpackageofsoftwareused foranalysisisSolidworks The one-,two-andthree-dimensional grid CAD, ICEMformeshing,Fluentsolvingand Tecplot forpost-processing. Each cellcommunicateswithitsneighbourinanetworkofinformation.In 16b shows an example of the computed surface pressuredistribution Figure 16b showsanexampleofthecomputedsurface world). Eachcellholdsinitselfasetofequationsthatexpressesthestate that cancollectandplotinameaningfulwaythe fieldofthegenerateddata. the computationalfluiddynamics(CFD)world,a‘converged’solutionto computer-aided was split into design (CAD) YF-17 fighter jet geometry conversion ofinformationbetweentheorthogonalanddiagonalgrids. of physics(e.g.,pressure,temperature,density, etc.)inthatparticularcell. cells andequations has reacheda sufficientlystablestate.Thereissoftware of applications,suchasredesigningtheaircraft. on anF-16XLaircraftinflight.Suchinformation isthenusedinamultitude directional pathtothediagonal,andviceversa,thenvariousmultipliers field referstothesituationwhennumericalbalancebetweenallthose gears. Influiddynamics,anetworkofgeometricalelements(tetrahedral a variablesizenetwork of elements (often called ‘cells’ in theaerospace a body(AndersonJr, 1995). Figure16ashowshowthevolumearounda and prismatic)areusedtostudythebehaviourofafluidwithinoraround 2 Note thatif appearinginequations(4)and(6)expressaseriesofmovements Figure 15.Thethree dimensionalbalanceinthesumofvolumes. 2 isanoperatorthatconvertsinformationfromoneorthogonal Special case of the three-dimensional Pythagorean gear Australian Senior Mathematics Journal vol. 32 no. 2 47 (b) pressure distribution around a F-16 XL (Tomac & Eller, 2014). & Eller, a F-16 XL (Tomac distribution around (b) pressure Figure 16. Computational fluid dynamics: (a) grid mesh around a YF-17 fighter jet and a YF-17 fighter jet and dynamics: (a) grid mesh around 16. Computational fluid Figure Figure 17. Network of Pythagorean Gears: (a) in two dimensions and (b) in three dimensions. Gears: (a) in two dimensions and (b) in three 17. Network of Pythagorean Figure Figure 18a shows the various one dimensional grids (sequence of unit line Figure 18a shows the various one dimensional grids (sequence of While CFD is a very which complex topic, the fundamental principle on seen as coupled to its neighbour in a network of information. This network seen as coupled to its neighbour segments connected in chain) disposed both orthogonally and diagonally, diagonally, segments connected in chain) disposed both orthogonally and area 1 transit into the diagonal grid as two squares of area 2, and return to area 1 transit into the diagonal grid composed of many three dimensional Pythagorean gears (Figure 15). grid composed of many three dimensional Pythagorean gears grid via the intermediate nodes, and back again. This process is fractal, that grid via the intermediate nodes, and back again. This process is forming the two dimensional grid of the Pythagorean gears (as shown forming the two dimensional grid of the Pythagorean gears paper, volume information (as shown in Figure 17b). Similarly, eight cubes information (as shown in Figure 17b). Similarly, volume paper, it operates is the same as that for the Pythagorean gear. One gear can be One gear can the Pythagorean gear. it operates is the same as that for is, it happens at different scales of the grid simultaneously. Aligning many is, it happens at different scales of the grid simultaneously. previously in Figure 2a). Information flows from the orthogonal to the diagonal the orthogonal to flows from Information 2a). in Figure previously of volume 1 in the orthogonal grid transit into two truncated octahedrons of of volume 1 in the orthogonal grid of triples shown in Figure 17a show). These arrows imply that four squares of of triples shown in Figure 17a show). of these two dimensional grids as per Figure 11d gives a three dimensional of these two dimensional grids as per Figure 11d gives a three dimensional the orthogonal grid as one square of area 4, and also, as shown by the present the orthogonal grid as one square transmits conservation Pythagoras tree of length and area (as the arrows in the volume 4 in the diagonal grid, and return to the diagonal grid as one cube of cube one as grid diagonal the to return and grid, diagonal the in 4 volume volume 8. 48 Australian Senior Mathematics Journal vol. 32 no. 2 Teia ‘hypotenuse’ thatisahexagon(andoverall, theyformatruncated The natureofspatialrealityisatleastthreedimensional.Followingonthe The particularityofthistruncatedoctahedronwherex Therefore, thetruncatedtetrahedronisthree dimensionalequivalentof The hypothesisisthatthethreedimensionalPythagoreangearbuiltfrom Pythagorean geartoitsthreedimensionality. Agoodstartingpointforthe Each truncatedoctahedroniscoupled to itsneighbour. Ultimately, the the two-dimensional isosceles right triangle. Ultimately, thePythagoreangear tetrahedron). Justasthehexagonisformedby three diagonalhypotenuses, that justastwolinesperpendiculartoeachother defineahypotenuse(and examination of any theory istheinspectionofitsexceptions.Therefore, examination ofanytheory case is beyond the scope of the present article and will be published in a clock governed by the jewel movement of the inner orthogonal and diagonal octahedrons, thus forming an infinite grid occupyingallspace(Figure18b). of thegeardecomposesinto orthogonalanddiagonaldirectionsforming two overall a right-angled triangle), three triangles aligned along the orthogonal orthogonal versionsofitstwodimensionalexpression. Asaresult,itisshown defining a perfect balance between lengths, areas and volumes.This balance defining a perfect is an ensemble of all one-, two- and three-dimensional theorems, all in one, planes havingasacommoncentrepointdefine thenewthree-dimensional footsteps ofapreviouspublication,thisarticleextendsthedynamics future article. grids composedofvariablesizeandshaperight-angledtriangles.Thisgeneral general casex Conclusion and hexagonal surfaces are a perfect matchto otherneighbouringtruncated areaperfect and hexagonalsurfaces special casex so isthetruncatedtetrahedronformedbythree isoscelesrighttriangles. Figure 18. The geometrical grids: (a) orthogonal anddiagonalin2-D, Figure 18.Thegeometricalgrids: (a)orthogonal =y

≠ y , andtheroleofPythagoras’ constant ofthePythagoreangear—the‘universalgear’—islikea (b) cubicandoctahedralin3-D. =y 2 isthatitssquare , isinvestigated. Special case of the three-dimensional Pythagorean gear Australian Senior Mathematics Journal vol. 32 no. 2 49 , the = y was termed by the author the ‘universal gear’ and will be the was termed by the author the ‘universal y ≠

can be seen as an operator that converts information (i.e., length, areas length, (i.e., information converts that an operator seen as can be australiancurriculum.edu.au 54–64. org/press/periodicals/convergence/the-best-known-old-babylonian-tablet. Planck, 1619). 227–318. Mathematical Association of America. Retrieved 21 October 2018 from https://www.maa. Mathematical Association of America. Retrieved McGraw-Hill Higher Education. Copernicus. 38–47. Trade Paperbacks. Trade Procedia Journal, 82, 377–389. Procedia curriculum: Mathematics F–10. Retrieved 21 October 2018 from http://www. Journal, 29(1), 7–15. 2 subject matter of research of this article—applies to the special case x subject matter of research of this subject of discussion of a future publication. a network of geometrical elements (tetrahedral and prismatic) termed ‘cells’ termed ‘cells’ and prismatic) elements (tetrahedral of geometrical a network behaviour of a fluid within grids that allow the study of the are used to form and volumes) back and forth between the orthogonal and diagonal grids in grids diagonal and orthogonal the between forth and back volumes) and gear—the three dimensional Pythagorean Since the present a fractal way. geared onto each other, form a network that fills space. Pythagoras’ constant form a network that fills space. other, geared onto each general case x in a similar manner. A gear can be seen as holding an individual balance, gear can be seen as holding an A in a similar manner. distinct grids. This topic is practical in nature, and its implications are veryand its implications in nature, is practical grids. This topic distinct real. or a network balance of many gears. Neighbouring truncated octahedrons, truncated Neighbouring gears. of many balance network or a or around a body (like an aircraft). The Pythagorean gear can be regarded (like an aircraft). The Pythagorean or around a body . New York: A dialogue concerning of 2: a number and a sequence. New York: root D. (2006). The square Flannery, . New York: Cambridge University Press. Cambridge University . New York: Finch, S. R. (2003). Mathematical constants of Babylonian mathematics. Historia Mathematica, 8, Friberg, J. (1981). Methods and traditions References Kepler, J. (1619). Harmonices [The harmony of the world] (Linz, (Austria): Johann mundi Kepler, (2nd ed.). New York: Crown Crown Einstein, A. (1961). Relativity: The special and the general theory (2nd ed.). New York: Euclid (300 BC). Elements. Book X. Beery, J. L. & Swetz, F. J. (2018). The best known old Babylonian tablet? Convergence. J. (2018). J. L. & Swetz, F. Beery, Teia, L. (2016). Anatomy of the Pythagoras’ tree. Australian Senior Mathematics Journal, L. (2016). Anatomy of the Pythagoras’ tree. 30 (2), Teia, Australian Senior Mathematics Journal, 32 (1), L. (2018). The Pythagorean geometric gear. Teia, hybrid-prismatic mesh generation. Elsevier automated D. (2014). Towards M. & Eller, Tomac, Teia, L. (2015). Pythagoras’ triples explained via central squares. Australian Senior Mathematics L. (2015). Pythagoras’ triples explained via Teia, In modern aerospace industry, in particular in computational fluid dynamics, fluid particular in computational in aerospace industry, In modern . New York: applications. New York: J. D. (1995). Computational fluid dynamics: The basics with Anderson Jr, (n.d.). Australian Reporting Authority [ACARA]. Australian Curriculum, Assessment and