Martin EuserBosweg
33922 GJ Elst (Utrecht) The Netherlands To
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However it may be, I focused on the factor root-(2N-1) and thought it would be interesting to consider this factor as a geometrical mean. Hence I sought two terms that would yield this factor. Proceed as follows: a : root-(2N-1) = root-(2N-1):
b hence: I dissolved 2 root-(2N) + 1and root-(2,N) -1which can be taken to be Some interesting features arise from these factors: When you elaborate these values for 2 The number 2 is prevalent here. The terms a: 3,5,7,9, ..c: 1,3,5,7,..which are of the form 2 One hypothesis could be that these values are the input in
the term root-(2
p/q) where p = 2d^{2} and
q = 2e^{2}.This formula gives rise to musical proportions: d = 2 and e = 1 it yields: 2/1 =
2 , the octaved = 3 and e = 2 it yields: 3/2 ,
the third or fifthd = 4 and e = 3 it yields: 4/3,
the fourthetc. The formula 2N Now, for some intuitive reason, I started to combine the variables
d
and e: d^{2} + e^{2} = r;d and e: d^{2}
- e^{2} = p;d and e : 2de
= q.for r = ; d^{2} + e^{2}
= 5 ; p= d^{2} - e^{2}
=3,q = 2de = 4which reminded me of the famous Pythagorean triangle with sides 3, 4 and 5. This triangle is common in the physical shapes/forms of many fishes, as Doczi shows in his book on the power of limits.[3] For ; q = 2de = 12 ; r =
d^{2} + e^{2} = 13,p
= d^{2} - e^{2} =5and, yes, this is another famous Pythagorean triangle with sides 5, 12, 13. And so on for higher values for d and e. So, a correlation is established between Pythagorean triangles and musical proportions, as I have proven here mathematically. These triangles seem to be connected to the root-number scheme too. I would not be surprised when they would play a vital role in the structure-function-order of nature (e.g. in atomic shell structure). After all, Pythagoras was a very wise man, who taught the mysteries in his Mystery-school and it is very likely that he knew a lot of the workings of nature, being a (high degree) initiate himself. Very simple properties for the Pythagorean triangles can be derived from what I have deduced already: r - q = d^{2} + e^{2}
- 2de = (d - e)^{2} = 1because ,r - p = d^{2} + e^{2}
- ( e^{2} - d^{2}) = 2d^{2}which is the famous formula for periodicity of electronic
shell-structure. This means that the following relations between
; p = d + e = 2(e - 1) ;q = 2de = 2(e^{2} -e)r = q + 1 = 2de + 1 = 2(e^{2}
- e) + 1(Figure 1). Thus, the Pythagorean triangles can be characterized by one
variable, Now, N^{2} + N [root-(2N-1)]and .[4]
N^{2} - N [root-(2N-1)]This is an interesting result, the ramifications of which are not yet clear. There may be a relation to a very simple generative set of number-pairs: N + root-Nand N - root-N.These pairs have the following means: A= ,N ; G = root-[N*(N-1)];
H = N -1which means are connected to the previously mentioned pairs: ,N^{2} +/- N [root-(2N-1)]because in that case A = One hypothesis is that A = Note that A + H = 2 After considering the terms A=N and H= So, this is another way of viewing the Pythagorean triangles, and it may be interesting to connect the former description with this one. Mathematically, this points to some transformation of variables, mapping one description onto the other. Indeed, I had a look at these two ways of describing Pythagorean triangles and discovered some new facets: - embeddedness of triangles into a larger set (hierarchies, classes/subclasses, sets and subsets or whatever is an appropriate designation for this feature);
- a kind of repetition of these sets of triangles (fractality?);
- appearance of the famous phi proportion as sides of a Pythagorean prototypical triangle;
- connections between the small side of a triangle and a corresponding one in the next set of triangles, as well as within the same set.
In order to present my findings in a visual way (and not making this article too boring :)), I'll include a couple of images that show the features meantioned above (Figure 3). Comment: a simple squaring function will transform To make this abstract picture a bit more understandable we
must fill in some values for In the tables I've indicated the connection between triangles
within and across tables by the transformation One thing is certain: there remains a lot to discover regarding the field of Sacred Geometry as it pertains to nature.
[2] Theon of Smyrna
: [3]Doczi, Gyorgy.
[4] See my previous
article, Sacred
Geometry , music and a possible correlation with quantum-mechanics
for a discussion of this pair.
in statistics, methods of research and as a courseware developer. Currently he is employed in the ICT business as a network specialist and programmer. He is the author and publisher of numerous articles on contemporary spirituality, varying from Gnosis, Theosophy and Kabbalah to Sacred Geometry.
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