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GEOMETRY OF THE PARABOLA ACCORDING TO THE GOLDEN NUMBER

 
 
-FROM THE HARMONY OF NATURE TO THAT OF ARCHITECTURE-

CARLOS CALVIMONTES ROJAS

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To cover spaces is the main architectural problem faces man. He, in their best achievements for solve it, finds solutions that join conditions of beauty, constructive easiness, saving of materials and good structural quality, repeating perfect forms of nature. This has occurred with the parabolic and pointed arches, whose geometry is present in the section of a body existing in the nature that has a system of great stability due to the harmony between its parts.

The author found the parabola in the elongated part of the longitudinal section of the hen's egg. He considered that this should have its form configured by the Golden Number, whose presence is over all in all that has life or relation with life. He checked this ruling manifestation that transcends the knowledge of the egg. The result, in addition to benefiting the design and construction of the parabolic and pointed arches, contributes to the better knowledge of the parabola.

 
THE GOLDEN NUMBER IN THE HARMONY OF WHAT IS CREATED

The Golden Number (F =1.6180339...) is the measure of the perfect proportion between two unequal parts. Known since a very remote antiquity, in the plane geometry Vitruvio proposed that "so that a space divided into unequal parts results pleasant and aesthetic, the relation between the smallest and the largest part must be the same as between the largest and the whole". Regarding linear systems Euclides defined that such proportion is the "division of one length in medium and extreme ratio" or Golden Section.

As explained by Ghyka, that code of nature rather than being fully described it exhibits in the unstable concourse of all the parts of the living beings and on the matter with some form of dynamic structures, with the expression of the summarized numeric pattern that is F, with the basis of the number 5. Rather, in the forms of the matter not organized for life, in the exponents of stable symmetries and in the structures of crystalline balance its numeric scheme resorts in general to the number 6.

THE PROPORTIONS IN ARCHITECTURE

In the architectural creation the best harmony of the composing parts of a work incorporate F connatural to man due to being in its same proportions. The human being has applied criteria developed on that harmony in spontaneous or voluntary form in the best of its creation since the antiquity, in the cultures of Sumer, Egypt and Tiwanaku; and, in Occident, starting with the studies of Policleto, Euclides and Vitruvio, those of Alberti, Leonardo da Vinci, Durero, Miguel Angel, Zeising and Le Corbusier, among many others.

The architect expresses by intuition the concept of the ‘desired design’ and, according to Le Corbusier, "automatically makes the necessary optic corrections", in addition to what was addressed by Leibniz regarding the application of the pre-existing harmony, as "we own much knowledge of what we do not have clear conscience even when we apply it "assuming that "all the arithmetic and all the geometry exist in an innate and virtual in our soul", although "the innate principles only appear when the attention is fixed on them".
 


FIN THE CONFIGURATION OF THE HEN'S EGG


NON-PARABOLIC GEOMETRY IN THE OVAL SECTION

The drawing with arcs of four circles is the one that better configure the oval inscribed in the vesica piscis. Though in this drawing there is no parabola, it has with exactness the proportions of its chord and sagitta. The tiny difference between the parabola and the combination of circular arcs in non significant parts (see graphic of the parabola) makes that the product of the analysis of the oval figure serves for the fast design of the parabolic and pointed arcs and for a higher knowledge of the parabola.
 
 


 
 




GEOMETRY OF THE POINTED AND PARABOLIC ARCHES


GEOMETRY OF THE PARABOLA ACCORDING TO F

All parabolas are similar: while the size varies, the constants of their configuration are the same for all them. Consequently, having demonstrated that the parabola is on the oval section configured by F, the relative proportion of the main dimensions of every parabola are established: parameter, sagitta and chord, with the subsequent implications. Also the harmonic coherence of the joint assembly that associates the oval figure with the parabola permits to confirm the location of the focus of it.


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