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Either in English or Spanish I will try to publish in what follows any new contribution dealing with math. related to sunflower pattern; even well documented images describing the motion of sunflowers at night and during the day (in hot climates from Mexico to Venezuela, say).
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Sea en Espanyol o Ingles tratare de publicar aqui las nuevas contribuciones relacionadas con matemat. de espirales en girasoles; y tambien algunas documentadas imagenes del movimiento de girasoles en la noche y durante el dia (en climas calientes desde Mejico a Venezuela, digamos).
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Contact // Contacto:
Angel Garcia, secr. "Universitas Americae (UNIAM)"
. Tel.: 4164657779, Toronto; Canada.
E-mali//Correo electr.: bp887@freenet.carleton.ca.
(Sea breve. Refiera 'girasol' // Be brief. Refer 'sunflower')
URLs:
http://www.interlog.com/~uniam
http://www.interlog.com/~uniam/books.txt
http://www.ncf.ca/~bp887
http://eltasico.tripod.com
http://www.geocities.com/Eureka/Concourse/9460/index.html
http://www.geocities.com/Eureka/Network/1679/face.html
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(8-I-2002). "Golden Section in transition-radii"; by Angel Garcia:
As first new contribution (and to set a short example) let me say that the 6 radii where 'change of Phyllotaxis' takes place in real sunflowers have the following new property (compare 'pag. 117' and 'pag.: 6 radi.'):
"The Ratio of consecutive radii of 'Phyllotaxis transition' in ideal sunflowers is the 'Golden Section' = 0.61803..; with excellent approximation for practical cases and exactly in the limit of infinitely large (ideal) radii". In fact: From Lahoz's relation (bottom of pag. 117):
R_k = 2R_(k-1) - R_(k-3) .....(1)
it follows that:
R_k = R_(k-1) + R_(k-2) + C .....(2)
where C is a small number near 0.66; dividing (2) by R_(k-1) and neglecting C / R_(k-1) for large k and R_k we have:
F = [R_(k-2) / R_(k-1)] = [R_k / R_(k-1)] - 1 .....(3)
Thus, since in the limit of k==>infinity we have F = R_(k-1) / R_k, it follows (large k):
F^2 + F - 1 = 0 ; with F = [sqrt(5)-1]/2 = 0.61803... .....(4) ......Q.E.D. 
Prof. Dr. D.G. Lahoz has conscientiously refined his original progr. B) (pag. 114) still using QBASIC for DOS 5 and 6 (Microsoft). It is a very simple programming method for Math. which does not require specific Computer-programming skills... although a very good background in general Mathematics is still required. Microsoft did excellent job in refining QBASIC up to present level as here used. Nor Lahoz nor myself have explored further refinements by Microsoft in its new platforms of Windows-95 or beyond that. Clearly QBASIC cannot calculate my ratio F for radii larger than R_23 because more than 16 digit calculations would be needed and QBASIC cannot do that (as far as we know). With above corrected program we tested my little theorem-conjecture up to 6-DIGIT approximation which is achieved at R_22.
It remains as interesting exercise for computer experts in math. programing to test it up to ANY desired number of digits... perhaps computations with 100 digit precision (and 100-digit approx. for Pi, of course) will suffice to reach ratios F with more than a dozen digits approximation to the golden section G. Go ahead, you experts !.
Angel Garcia, secretary of UNIAM, (19-jan-2002). Presentation, debate: conjecture C) and UBASIC.
(http://www.geocities.com/Eureka/Concourse/9460/jan02.html) Learning UBASIC to test conjectures C) and SC). Transition-radii up to k=100.
(http://www.geocities.com/Eureka/Concourse/9460/feb02.html) |
