Rambler MATHEMATICS OF TAOISM-ONLY HERE!-HOW TO DRAW WITH A COMPUTER THE ONLY TRUE MAGICAL YIN-YANG SYMBOL. Now this beautiful HARMONY belongs to all the people! DOBRO POZHALOVAT'Note, that such an invention of Yin-Yang Symbol is similar to the invention of wheel, bicycle, fire,baseball etc. and nobody know the first inventor of the wheel or of the bicycle or of fire! Scientific Approach to the Yin-Yang Geometry by Sergey Yu. Shishkov Here is given (below) the most generalized definition of the astroid-like hypocycloid as the trajecory of a point P of a rotating with angular velocity "omega1"=1 circle of radius "radius1"=a, with centre of which also being rotating around the origin by the circle of radius "radius2"=1-a , and angular velocity "omega2"=-3, so that"radius1" +"radius2"=1, and "omega2"/"omega1" =-3. Then for coordinates X[t], Y[t] of this point P we have: X[t]=(a)*cos(t)+(1-a)*cos(3*t); Y[t]=(a)*sin(t)-(1-a)*sin(3*t); 1-X[t]^2-Y[t]^2=factor(simplify(expand(1-((a)*cos(t)+(1-a)*cos(3 *t))^2-((a)*sin(t)-(1-a)*sin(3*t))^2)))=16*a*cos(t)^2*(cos(t)-1) *(cos(t)+1)*(-1+a)=16*a*cos(t)^2*(cos(t)^2-1)*(-1+a)=16*a*cos(t) ^2*(sin(t)^2)*(1-a)=FULL SQUARE!=> If Z[t]=4*cos(t)*sin(t)*(a*(1-a))^(1/2), then X[t]^2+Y[t]^2+z[t]^2=1 ;(i.e., For every time t {X[t],Y[t],Z[t]} is on the unit SPHERE!!!). With different values of the parameter a we obtain the whole class of astroid-like hypocycloids with FOUR PARTS. Below is given the Maple 5.4 Text programm for plotting of these trajectories.; > a=0.6339;plot([(a)*cos(t)+(1-a)*cos(3*t),(a)*sin(t)-(1-a)*sin(3*t) ,t=0..2*Pi]); plot([(a)*cos(t)+(1-a)*cos(3*t),4*cos(t)*sin(t)*(a*(1-a))^(1/2), t=0..2*Pi]); plot([(a)*sin(t)-(1-a)*sin(3*t),4*cos(t)*sin(t)*(a*(1-a))^(1/2), t=0..2*Pi]); Look also in Maple 5: > factor(simplify(expand(1-((b)*cos(t)+(1-b)*cos(3*t))^2-((b)*sin( t)-(1-b)*sin(3*t))^2))); > The Optimal Value for the parametr a is a=(1/2)*(3-3^(1/2))=0.6339, as will be shown elsewher;-). It corresponds to the most "BEAUTIFUL" 3D-hypo-astroid. Such a configuration may serve also as the Yin-Yang MAGNETIC TRAP for adiabatic freezing of Bose condensate in modern Atomic Beam Lasers and for hot plazma in thermonuclear fusion systems. However, this is beyond the scope of this site. Let us call this unique value of the parameter a as "THE YIN-YANG PLATINUM SECTION", analogous to the famous "GOLDEN SECTION GS" (i.e.,GS=1/2*5^(1/2)-1/2=0.6180339887), suggested by Leonardo da Vinci! > [>a:=0.6339;plot([(a)*cos(t)+(1-a)*cos(3*t),(a)*sin(t)-(1-a)*sin (3*t),t=0..2*Pi]); plot([(a)*cos(t)+(1-a)*cos(3*t),4*cos(t)*sin(t)*(a*(1-a))^(1/2), t=0..2*Pi]); plot([(a)*sin(t)-(1-a)*sin(3*t),4*cos(t)*sin(t)*(a*(1-a))^(1/2), t=0..2*Pi]);

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