~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~~~~~~~~~~ Interlocking Spirals by J. Snuszka ~~~~~~~~~~~~
~~~~~~~~~~~~ inspired by Oskar van Deventer ~~~~~~~~~~~~~
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

A_1=vector(p_3*cos(2*pi*u)*sin(pi*v),p_3*sin(2*pi*u)*sin(pi*v),p_3*cos(pi*v)),A_2=vector(p_3*e*cos(2*pi*u)*sin(pi*v),p_3*e*sin(2*pi*u)*sin(pi*v),p_3*e*cos(pi*v)),M=matrix(3,3,a_1*X_1^2+M_1,a_1*X_1*Y_1-(N_1*Z_1),a_1*X_1*Z_1+N_1*Y_1,a_1*X_1*Y_1+N_1*Z_1,a_1*Y_1^2+M_1,a_1*Y_1*Z_1-(N_1*X_1),a_1*X_1*Z_1-(N_1*Y_1),a_1*Y_1*Z_1+N_1*X_1,a_1*Z_1^2+M_1)

B_1=vector(p_2*cos([2*pi*n]),-(p_2*sin([2*pi*n])),0),B_2=vector(p_2*e*cos([3*pi-(2*pi*n)]),p_2*e*sin([3*pi-(2*pi*n)]),0),B_3=vector(-(p_2*cos([2*pi*n])),-(p_2*sin([2*pi*n])),0),B_4=vector(-(p_2*e*cos([3*pi-(2*pi*n)])),p_2*e*sin([3*pi-(2*pi*n)]),0)

B_5=vector(p_2*cos([3*pi/2-(2*pi*n)]),-(p_2*sin([3*pi/2-(2*pi*n)])),0),B_6=vector(p_2*e*cos([3*pi-(2*pi*n)+3*pi/2]),-(p_2*e*sin([3*pi-(2*pi*n)+3*pi/2])),0),B_7=vector(p_2*cos([3*pi/2-(2*pi*n)]),p_2*sin([3*pi/2-(2*pi*n)]),0),B_8=vector(p_2*e*cos([3*pi-(2*pi*n)+3*pi/2]),p_2*e*sin([3*pi-(2*pi*n)+3*pi/2]),0)

C_1=vector(p_1*e^v*cos([3*pi*v-(2*pi*n)])*[p_4+sin(2*pi*u)],p_1*e^v*sin([3*pi*v-(2*pi*n)])*[p_4+sin(2*pi*u)],p_1*e^v*cos(2*pi*u)),C_2=vector(p_1*e^t*cos([3*pi*t-(2*pi*n)])*[p_4+sin(2*pi*p)],p_1*e^t*sin([3*pi*t-(2*pi*n)])*[p_4+sin(2*pi*p)],p_1*e^t*cos(2*pi*p)),C_3=vector(p_1*e^s*cos([3*pi*s-(2*pi*n)])*[p_4+sin(2*pi*t)],p_1*e^s*sin([3*pi*s-(2*pi*n)])*[p_4+sin(2*pi*t)],p_1*e^s*cos(2*pi*t))

C_4=vector(-(p_1*e^v*cos([3*pi*v-(2*pi*n)])*[p_4+sin(2*pi*u)]),p_1*e^v*sin([3*pi*v-(2*pi*n)])*[p_4+sin(2*pi*u)],-(p_1*e^v*cos(2*pi*u))),C_5=vector(-(p_1*e^t*cos([3*pi*t-(2*pi*n)])*[p_4+sin(2*pi*p)]),p_1*e^t*sin([3*pi*t-(2*pi*n)])*[p_4+sin(2*pi*p)],-(p_1*e^t*cos(2*pi*p))),C_6=vector(-(p_1*e^s*cos([3*pi*s-(2*pi*n)])*[p_4+sin(2*pi*t)]),p_1*e^s*sin([3*pi*s-(2*pi*n)])*[p_4+sin(2*pi*t)],-(p_1*e^s*cos(2*pi*t)))

C_7=vector(p_1*e^v*cos([3*pi*v-(2*pi*n)+3*pi/2])*[p_4+sin(2*pi*u)],-(p_1*e^v*sin([3*pi*v-(2*pi*n)+3*pi/2])*[p_4+sin(2*pi*u)]),-(p_1*e^v*cos(2*pi*u))),C_8=vector(p_1*e^t*cos([3*pi*t-(2*pi*n)+3*pi/2])*[p_4+sin(2*pi*p)],-(p_1*e^t*sin([3*pi*t-(2*pi*n)+3*pi/2])*[p_4+sin(2*pi*p)]),-(p_1*e^t*cos(2*pi*p))),C_9=vector(p_1*e^s*cos([3*pi*s-(2*pi*n)+3*pi/2])*[p_4+sin(2*pi*t)],-(p_1*e^s*sin([3*pi*s-(2*pi*n)+3*pi/2])*[p_4+sin(2*pi*t)]),-(p_1*e^s*cos(2*pi*t)))

C_10=vector(p_1*e^v*cos([3*pi*v-(2*pi*n)+3*pi/2])*[p_4+sin(2*pi*u)],p_1*e^v*sin([3*pi*v-(2*pi*n)+3*pi/2])*[p_4+sin(2*pi*u)],-(p_1*e^v*cos(2*pi*u))),C_11=vector(p_1*e^t*cos([3*pi*t-(2*pi*n)+3*pi/2])*[p_4+sin(2*pi*p)],p_1*e^t*sin([3*pi*t-(2*pi*n)+3*pi/2])*[p_4+sin(2*pi*p)],-(p_1*e^t*cos(2*pi*p))),C_12=vector(p_1*e^s*cos([3*pi*s-(2*pi*n)+3*pi/2])*[p_4+sin(2*pi*t)],p_1*e^s*sin([3*pi*s-(2*pi*n)+3*pi/2])*[p_4+sin(2*pi*t)],-(p_1*e^s*cos(2*pi*t)))

D_1=vector(p_3*sin(2*pi*p)*sin(2*pi*t),p_3*cos(2*pi*t),p_3*cos(2*pi*p)*sin(2*pi*t)),D_2=vector(p_3*e*sin(2*pi*p)*sin(2*pi*t),p_3*e*cos(2*pi*t),p_3*e*cos(2*pi*p)*sin(2*pi*t)),D_3=vector(p_3*cos(2*pi*t),p_3*cos(2*pi*p)*sin(2*pi*t),p_3*sin(2*pi*p)*sin(2*pi*t)),D_4=vector(p_3*e*cos(2*pi*t),p_3*e*cos(2*pi*p)*sin(2*pi*t),p_3*e*sin(2*pi*p)*sin(2*pi*t))

E_1=matrix(3,3,cos(2*pi*n),sin(2*pi*n),0,-sin(2*pi*n),cos(2*pi*n),0,0,0,1),E_2=matrix(3,3,cos(2*pi*n),-sin(2*pi*n),0,sin(2*pi*n),cos(2*pi*n),0,0,0,1)

S_1=set(0),S_2=set(1/4),S_3=set(1/8),S_4=set(3/8),S_5=set(0,1/4,ldots*(3/4)),S_6=set(0,1/48,ldots*(48/48)),S_7=set(0,1/4)

a_1=1-cos([2*pi*b_1]),X_1=cos([2*pi*g_1])*sin([2*pi*h_1]),Y_1=sin([2*pi*g_1])*sin([2*pi*h_1]),Z_1=cos([2*pi*h_1]),M_1=cos([2*pi*b_1]),N_1=sin([2*pi*b_1])

p_1=1/6,p_2=6.1/6,p_3=0.987/6,p_4=6.1

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

A_1+M*B_1,in(g_1,S_1),in(h_1,S_2),in(b_1,S_3),vector(h,s,v)=vector(0.12,0.9,1)

A_2+M*B_2,in(g_1,S_1),in(h_1,S_2),in(b_1,S_3),vector(h,s,v)=vector(0.12,0.9,1)

M*C_1,in(g_1,S_1),in(h_1,S_2),in(b_1,S_3),vector(h,s,v)=vector(0.12,0.9,1)

M*C_2,in(g_1,S_1),in(h_1,S_2),in(b_1,S_3),in(p,S_5)

M*C_3,in(g_1,S_1),in(h_1,S_2),in(b_1,S_3),in(s,S_6)

M*E_1*D_1+M*B_1,in(g_1,S_1),in(h_1,S_2),in(b_1,S_3),in(p,S_7)

M*E_1*D_2+M*B_2,in(g_1,S_1),in(h_1,S_2),in(b_1,S_3),in(p,S_7)

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

A_1+M*B_3,in(g_1,S_1),in(h_1,S_2),in(b_1,S_4),vector(h,s,v)=vector(0.06,0.9,1)

A_2+M*B_4,in(g_1,S_1),in(h_1,S_2),in(b_1,S_4),vector(h,s,v)=vector(0.06,0.9,1)

M*C_4,in(g_1,S_1),in(h_1,S_2),in(b_1,S_4),vector(h,s,v)=vector(0.06,0.9,1)

M*C_5,in(g_1,S_1),in(h_1,S_2),in(b_1,S_4),in(p,S_5)

M*C_6,in(g_1,S_1),in(h_1,S_2),in(b_1,S_4),in(s,S_6)

M*E_2*D_1+M*B_3,in(g_1,S_1),in(h_1,S_2),in(b_1,S_4),in(p,S_7)

M*E_2*D_2+M*B_4,in(g_1,S_1),in(h_1,S_2),in(b_1,S_4),in(p,S_7)

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

A_1+M*B_5,in(g_1,S_2),in(h_1,S_2),in(b_1,S_3),vector(h,s,v)=vector(0.18,0.9,0.8)

A_2+M*B_6,in(g_1,S_2),in(h_1,S_2),in(b_1,S_3),vector(h,s,v)=vector(0.18,0.9,0.8)

M*C_7,in(g_1,S_2),in(h_1,S_2),in(b_1,S_3),vector(h,s,v)=vector(0.18,0.9,0.8)

M*C_8,in(g_1,S_2),in(h_1,S_2),in(b_1,S_3),in(p,S_5)

M*C_9,in(g_1,S_2),in(h_1,S_2),in(b_1,S_3),in(s,S_6)

M*E_2*D_3+M*B_5,in(g_1,S_2),in(h_1,S_2),in(b_1,S_3),in(p,S_7)

M*E_2*D_4+M*B_6,in(g_1,S_2),in(h_1,S_2),in(b_1,S_3),in(p,S_7)

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

A_1+M*B_7,in(g_1,S_2),in(h_1,S_2),in(b_1,S_4),vector(h,s,v)=vector(0,0.9,0.8)

A_2+M*B_8,in(g_1,S_2),in(h_1,S_2),in(b_1,S_4),vector(h,s,v)=vector(0,0.9,0.8)

M*C_10,in(g_1,S_2),in(h_1,S_2),in(b_1,S_4),vector(h,s,v)=vector(0,0.9,0.8)

M*C_11,in(g_1,S_2),in(h_1,S_2),in(b_1,S_4),in(p,S_5)

M*C_12,in(g_1,S_2),in(h_1,S_2),in(b_1,S_4),in(s,S_6)

M*E_1*D_3+M*B_7,in(g_1,S_2),in(h_1,S_2),in(b_1,S_4),in(p,S_7)

M*E_1*D_4+M*B_8,in(g_1,S_2),in(h_1,S_2),in(b_1,S_4),in(p,S_7)

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~`

Graph of the formula

This file was created by Graphing Calculator 4.0.5.
Visit Pacific Tech to download the helper application to view and edit these equations live.