Octahedral Graph Scaling

Transcription

1 Octahedral Graph Scalig Peter Russell Jauary 1, 2015 Abstract There is presetly o strog iterpretatio for the otio of -vertex graph scalig. This paper presets a ew defiitio for the term i the cotext of Octahedral Scalig or 4-vertex graph scalig, that may be cosidered for other values of, ad presets two differet calculatio methods based o two differet iterpretatios of scale; oe from fractal geometry ad the other from the geometry of platoic solids. A cojecture is preseted as to the uiqueess of the Pythagorea series to form atural graphs ad its relevace to mathematical cosmology. 1 Itroductio -vertex graphs such as the 3-vertex cubical ad dodecahedral graphs, ad the 4-vertex octahedral graph have o iheret scalig property ad so a method for determiig a scalig factor for them seems ot to exist i curret literature. Scalig factors are foud i fractal geometry, ad a umber of differet methods exist to determie scale give differet fractal geometries. My origial goal was to fid a suitable series o the coutig umbers that could be formed ito a graph where its scale could be determied. This is a compoet step i the evetual costructio of a causal set based o the series. Eve though there are may graph like structure i various braches or mathematics they are i the mai etirely ma-made ad artificial represetatios that flow from artificial axiomatic spaces. I was lookig for a atural graph that had its roots close to some iate atural aspect of the coutig umbers. I the aalysis of the Pythagorea series of uique hypoteuse based o a 2 + b 2 = c 2 the Simple Tree of Pythagorea Triples seems to form a graph suitable for study. Locally each ode i the graph is 4-vertex, ad the scale of the graph could be determied by examiig the ratio of the desities of members betwee geeratios of the graph. A stepwise umerical calculatio i a spread sheet based o the three matrix Barig method [2] showed that the ratio appears to asymptote to with each geeratio of the graph. I the searched usig Google for this umber to see if it had ay sigificace ad foud Iside a cube - solutios [1] which 1

2 presets the stadard result to the questio of platoic solids iside the uit cube. This fractal scalig method derived decimal is very close to 3(3 2 2) which represets the diameter of each of six spheres (octahedral close packed), iside the uit cube. Give that I was dealig with a 4-vertex graph i the Simple Tree of Pythagorea Triples, a result that idicated a relatioship with 4-vertex platoic solid scalig seemed iterestig. If it could be prove that the scale of the tree of primitive Pythagorea triples was equal to 3(3 2 2), by solvig the ifiite matrix sequece, the this would idicate that the otio of scale from the uit cube packig example for a octahedro, ad the otio of scale from the tree were mutually reiforcig. I posted this questio o StackExchage Mathematics ad was delighted by a succict proof. 2 Tree of Primitive Pythagorea Triples The questio Tree of Primitive Pythagorea Triples graph scale ifiite series was posted by me o StackExchage Mathematics[3]. The proof is provided by a solutio from Ricardo Burig (as Commuity Wiki). 2.1 Questio Cosider the tree of primitive Pythagorea triples as see here: of primitive Pythagorea triples Cosider the values for c i the triples (a, b, c) ad their desity i each geeratio layer of the tree. Takig the umber of c s i ay geeratio ad dividig by the differece betwee the maximum ad miimum values for c i that geeratio gives the desity of that geeratio. The scale of the graph is give by the ratio of desities as the geeratios go o to ifiity. I have calculated the maximum, miimum ad umber of c s i icreasig geeratios of the tree ad the scale asymptotes to ? by calculatio. Note: This is a graph scalig questio ad is ot the same questio as the simple desity of primitive triples as solved by Lehmer (1900). I postulate that this graph scalig value is i fact 3(3 2 2). Ca this be proved? 2

3 2.2 Aswer If (a, b, c) is a primitive Pythagorea triple, the the c values of its descedats (by the matrix expressio) are 2a 2b + 3c, 2a + 2b + 3c, 2a + 2b + 3c Clearly the middle oe is the largest. Hece the middle oe is the largest i ay geeratio. With the startig value (3, 4, 5), the first is the smallest. Hece the first is the smallest i ay geeratio. I the otatio of Wikipedia ad by diagoalizatio we have A = Hece the miimum c value i the th geeratio is c mi = Similarly but more horrifically (diagoalizig B), the maximum c value i the th geeratio is = 1 ( 5 7 ) (3 2 2) + 1 ( ) ( ) The desity of the th geeratio is d = 3 c mi The quotiet of two cosecutive desities is q = d +1 d = 3 cmax c mi +1 cmi +1 Fially, we have q = lim q = 3 lim c mi +1 cmi +1 The last limit was computed usig a CAS = 3(3 2 2) 3

4 3 Octahedral Explosio of the Uit Sphere i the Uit Cube The optimal packig of spheres i the uit cube is give by cosiderig the positioig of the three orthogoal squares that are at the heart of the octahedro. The diameters of each of the resultig six spheres is 3(3 2 2). This is a stadard geometric result [1]. The ratio of the edig diameter to the startig diameter (1 i the uit cube) gives the scale factor. I this case s = 6spheres 1sphere = 3(3 2 2) It is a coveiet otio that platoic solids scale by virtue of their explosio i the uit cube ad that this scale is give by the edig diameters of the spheres so packed at the vertices. 4 Coclusio A relatioship betwee the primitive Tree of Pythagorea Triples ad the octahedral graph has bee established. These two graphs share the followig properties: 1. 4-vertex 2. scale factor s = 3(3 2 2) The Simple Tree of Pythagorea Triples is a form of ope graph ad the octahedral graph is a form of closed graph. The scale factor of the Simple Tree Pythagorea of Triples was calculated usig a fractal scale determiatio method via the limit of the ifiite series of ratios of the desity of members i each successive geeratio of the graph. The scale factor of the octahedral graph was calculated usig the sphere explosio i uit cube method (beig postulated i this paper) via traditioal 3D geometry. The scales where foud to be idetical. 5 Further Work The meaig of the result to future researchers is that traditioal Euclidia results ca also be obtaied by cosiderig solids i their alterative graph (ope or closed) formulatios where the limitig values of some graph based features ca be show to be represetative of the properties of 3D projected solids. 4

5 It is of iterest that there is aother uderlyig commo property of both calculatio methods. The fractal scalig method ivolves a projectio to ifiity by the takig of a limit i order to establish the value of scale. The uit cube method also ivolves takig a projectio from a two dimesioal octahedral graph to a three dimesioal octahedro solid i order to recover the cocept of scale. Scale is ot apparet as a property of either the ope (tree) or closed (graph) util a form of dimesioal (+1) projectio is take. The result was obtaied exclusively for the 4-vertex graph i the octahedral cotext oly. Further study is eeded to exted the result where possible to other -vertex graphs ad their associated 3 space solid projectios. Due to this result the properties of octahedral graphs ad octahedros may be cosidered more viable as the basis of atural causal set costructio ad dyamics. I cosideratio of the varied cadidate series o the coutig umbers, I would postulate that a filter based upo the ability of the series to form a fixed vertex cout graph that has both a ope ad closed form ad a calculable scale is a importat discrimiat. I this case the series is based o the tree of primitive Pythagorea triples ad the graph is octahedral, both ope ad closed with calculable scale. Further work is required o the ature ad properties of atural series ad their ability to form graphs to uderstad whether this example is the oly oe (the cojecture), or simply a member of a class of series that have this behaviour. Should this cojecture prove true the this would be a importat result to mathematical cosmology as it illumiates a uique pathway from the coutig umbers to higher order mathematical costructs. Refereces [1] Philippe Chevae. Iside a cube - solutios. [2] Wikipedia Commuity. Tree of primitive pythagorea triples. [3] Peter Russell. Tree of primitive pythagorea triples graph scale ifite series. 5