QSO and the FRACTAL OCTAHEDRON by J. SnuszkaA Quasi-Spherical Orbit (QSO) is the path of a particle in orbit simultaneously about two or more axes
with a common centre. Defined by QSO (a:b), the spin rates can be changed by changing the slider values.
For more information on QSOs refer to:
"QSO - The Mathematics and Physics of Quasi-Spherical Orbits" by Robert G. Chester @ Amazon.com.

I=matrix(3,3,c*X^2+M,c*X*Y-(N*Z),c*X*Z+N*Y,c*X*Y+N*Z,c*Y^2+M,c*Y*Z-(N*X),c*X*Z-(N*Y),c*Y*Z+N*X,c*Z^2+M),A=vector(cos([2*pi*u])*sin([pi*v]),sin([2*pi*u])*sin([pi*v]),cos([pi*v])),B=vector(cos([a*2*pi*t])*sin([b*2*pi*t]),s*sin([a*2*pi*t])*sin([b*2*pi*t]),cos([b*2*pi*t]))

C_1=vector(0,sin([2*pi*t]),cos([2*pi*t])),C_2=vector(sin([2*pi*t]),0,cos([2*pi*t])),D_1=vector(0,0,s_2-s_1),D_2=vector(0,p,s_7*s_2-s_1),D_3=vector(p,0,s_7*s_2-s_1)

D_4=vector(0,p,s_8*s_2-s_1),D_5=vector(p,0,s_8*s_2-s_1),E_1=vector(0,0,s_5),E_2=vector(0,2,0),E_3=vector(1,-1,0)

c=1-cos([2*d]),X=cos([2*g])*sin([2*h]),Y=sin([2*g])*sin([2*h]),Z=cos([2*h]),M=cos([2*d]),N=sin([2*d])

'tubeR'=0.015,r_1=0.8,r_2=0.03,s=plusorminus(1),s_1=tan([pi/6])^(-1),s_2=tan([pi/6]),s_3=asin([tan(pi/6)])

s_4=asin([1/sqrt(2)]),s_5=sqrt(s_1^2-1),s_6=(s_1*s_5-(2*s_5^2))/s_1,s_7=(s_1-(2*s_2))/s_1,s_8=s_7*[(s_1-(2*s_2))/s_1]

S_1=set(pi/8,3*pi/8,5*pi/8,7*pi/8),S_2=set(pi/4),S_3=set(s_4/2),S_4=set(0),S_5=set(0,pi/4,ldots*pi),S_6=set(0,pi/4,2*pi/4)

S_7=set(s*s_3/2),S_8=set(s_3/2),S_9=set(pi/32),S_10=set(1-s_7,s_7-1),S_11=set(1-s_8,s_8-1)

The QSO (a:b) and the sphere
Change slider values to change QSO spin ratios

a=slider([1,20,19])

b=slider([1,20,19])

r_1*A

r_1*B,'radius'='tubeR'

The FRACTAL OCTAHEDRON

r_2*A+s*I*E_1,in(g,S_1),in(h,S_2),in(d,S_6),'color'=0.123

t*s*I*E_2+I*s*E_3,in(g,S_1),in(h,S_2),in(d,S_6),'radius'='tubeR','color'=0.123

s_2*I*C_1+plusorminus(I*D_1)+plusorminus(E_1),in(g,S_2),in(h,S_2),in(d,S_7),'radius'='tubeR','color'=0.123

s_2*I*C_2+plusorminus(I*D_1)+plusorminus(E_1),in(g,S_4),in(h,S_2),in(d,S_7),'radius'='tubeR','color'=0.123

s_2*s_7*I*C_1+plusorminus(s_7*I*D_1)+plusorminus(E_1),in(g,S_2),in(h,S_2),in(d,S_7),'radius'='tubeR','color'=0.123

s_2*s_7*I*C_1+plusorminus(I*D_2)+plusorminus(E_1),in(g,S_2),in(h,S_2),in(d,S_7),in(p,S_10),'radius'='tubeR','color'=0.123

s_2*s_7*I*C_2+plusorminus(s_7*I*D_1)+plusorminus(E_1),in(g,S_4),in(h,S_2),in(d,S_7),'radius'='tubeR','color'=0.123

s_2*s_7*I*C_2+plusorminus(I*D_3)+plusorminus(E_1),in(g,S_4),in(h,S_2),in(d,S_7),in(p,S_10),'radius'='tubeR','color'=0.123

s_2*s_8*I*C_1+plusorminus(s_8*I*D_1)+plusorminus(E_1),in(g,S_2),in(h,S_2),in(d,S_7),'radius'='tubeR','color'=0.123

s_2*s_8*I*C_1+plusorminus(I*D_4)+plusorminus(E_1),in(g,S_2),in(h,S_2),in(d,S_7),in(p,S_11),'radius'='tubeR','color'=0.123

s_2*s_8*I*C_2+plusorminus(s_8*I*D_1)+plusorminus(E_1),in(g,S_4),in(h,S_2),in(d,S_7),'radius'='tubeR','color'=0.123

s_2*s_8*I*C_2+plusorminus(I*D_5)+plusorminus(E_1),in(g,S_4),in(h,S_2),in(d,S_7),in(p,S_11),'radius'='tubeR','color'=0.123

Graph of the formula

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