~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~ ~ ~ ~ ~ ~ QSO and the FRACTAL HEXAHEDRON by J. Snuszka ~ ~ ~ ~ ~ ~
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

A 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])),C_3=vector(sin([2*pi*t]),cos([2*pi*t]),0),D_1=vector(1,0,0),D_2=vector(s,p,0),D_3=vector(0,s,p),D_4=vector(p,0,s)

D_5=vector(s,0,p),D_6=vector(p,s,0),D_7=vector(0,p,s),E_1=vector(0,2,0),E_2=vector(s,-1,s),D_8=vector(s,p,q),D_9=vector(q,s,p),D_10=vector(p,q,s)

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=0.9,r_2=0.03,s=plusorminus(1)

S_1=set(0,pi/4),S_2=set(0,pi/4,ldots*pi),S_3=set(-1/2,1/2),S_4=set(-2/3,2/3)

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 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])

vector(x,y,z)=r_1*A

vector(x,y,z)=r_1*B,'radius'='tubeR','color'=0.62

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ The FRACTAL HEXAHEDRON ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

vector(x,y,z)=r_2*A+I*E_2,in(g,S_1),in(h,S_2),in(d,S_2)

vector(x,y,z)=t*I*E_1+I*E_2,in(g,S_1),in(h,S_2),in(d,S_2),'radius'='tubeR'

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

vector(x,y,z)=I*C_1+I*D_1,in(g,S_1),in(h,S_1),in(d,S_2),'radius'='tubeR'

vector(x,y,z)=1/2*C_1+D_2,in(p,S_3),'radius'='tubeR'

vector(x,y,z)=1/2*C_2+D_3,in(p,S_3),'radius'='tubeR'

vector(x,y,z)=1/2*C_3+D_4,in(p,S_3),'radius'='tubeR'

vector(x,y,z)=1/3*C_1+D_5,in(p,S_4),'radius'='tubeR'

vector(x,y,z)=1/3*C_2+D_6,in(p,S_4),'radius'='tubeR'

vector(x,y,z)=1/3*C_3+D_7,in(p,S_4),'radius'='tubeR'

vector(x,y,z)=1/6*C_1+D_8,in(p,S_3),in(q,S_4),'radius'='tubeR'

vector(x,y,z)=1/6*C_2+D_9,in(p,S_3),in(q,S_4),'radius'='tubeR'

vector(x,y,z)=1/6*C_3+D_10,in(p,S_3),in(q,S_4),'radius'='tubeR'

Graph of the formula

This file was created by Graphing Calculator 4.0.
Visit Pacific Tech to download the helper application to view and edit these equations live.