Well, today I wrote something of a super-equation, that defines 7 different 3D shapes. After playing with three different shapes in 7D and 8D, I came to this one, which seemed to be the most efficient. I call it the Cubsphonindrone, defined as IOO>II> in 7D. Didn't animate it, but here's a script for CalcPlot3D. It's the rotate function at the bottom:
And, how I derived the function:
IOO>II> - 7D Cubsphonindrone, pyramid of (sphone,square) prism
I - 1D Line : Start with line in 1-plane X
IO - 2D Circle : bisecting rotate Line around origin, along X into Y
IOO - 3D Sphere : bisecting rotate Circle around X, along Y into Z
IOO> - 4D Sphone : scale Sphere to a point along W
|√(x²+y²+z²) + 2w| + √(x²+y²+z²) = a
IOO>I - 5D Sphoninder : extend Sphone along V, sphone-xyzw times line-v
||√(x²+y²+z²)+2w|+√(x²+y²+z²) - 2v| + ||√(x²+y²+z²)+2w|+√(x²+y²+z²) + 2v| = a
IOO>II - 6D Cubsphoninder : extend Sphoninder along U, sphone-xyzw times square-vu
||√(x²+y²+z²)+2w|+√(x²+y²+z²) - |v-u|-|v+u|| + ||√(x²+y²+z²)+2w|+√(x²+y²+z²) + |v-u|+|v+u|| = a
IOO>II> - 7D Cubsphonindrone : scale Cubsphoninder to a point along T
|||√(x²+y²+z²)+2w|+√(x²+y²+z²)-|v-u|-|v+u||+||√(x²+y²+z²)+2w|+√(x²+y²+z²)+|v-u|+|v+u|| + 4t| + ||√(x²+y²+z²)+2w|+√(x²+y²+z²)-|v-u|-|v+u||+||√(x²+y²+z²)+2w|+√(x²+y²+z²)+|v-u|+|v+u|| = a
Full 7D Implicit Surface Equation:
f(x,y,z,w,v,u,t) = |||√(x²+y²+z²)+2w|+√(x²+y²+z²)-|v-u|-|v+u||+||√(x²+y²+z²)+2w|+√(x²+y²+z²)+|v-u|+|v+u|| + 4t| + ||√(x²+y²+z²)+2w|+√(x²+y²+z²)-|v-u|-|v+u||+||√(x²+y²+z²)+2w|+√(x²+y²+z²)+|v-u|+|v+u|| - a
abs(abs(abs(sqrt(x^2+y^2+z^2) +2w)+sqrt(x^2+y^2+z^2) -abs(v-u)-abs(v+u)) + abs(abs(sqrt(x^2+y^2+z^2) +2w)+sqrt(x^2+y^2+z^2) +abs(v-u)+abs(v+u)) + 4t) + abs(abs(sqrt(x^2+y^2+z^2) +2w)+sqrt(x^2+y^2+z^2) -abs(v-u)-abs(v+u)) + abs(abs(sqrt(x^2+y^2+z^2) +2w)+sqrt(x^2+y^2+z^2) +abs(v-u)+abs(v+u)) = a
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3D Hyperplane Intersections of IOO>II> along XYZWVUT, i = dimension set to zero
IOOiiii - IOO sphere, WVUT=0
abs(abs(abs(sqrt(x^2+y^2+z^2) +2*0)+sqrt(x^2+y^2+z^2) -abs(0-0)-abs(0+0)) + abs(abs(sqrt(x^2+y^2+z^2) +2*0)+sqrt(x^2+y^2+z^2) +abs(0-0)+abs(0+0)) + 4*0) + abs(abs(sqrt(x^2+y^2+z^2) +2*0)+sqrt(x^2+y^2+z^2) -abs(0-0)-abs(0+0)) + abs(abs(sqrt(x^2+y^2+z^2) +2*0)+sqrt(x^2+y^2+z^2) +abs(0-0)+abs(0+0)) = 20
IOi>iii - IO> cone, ZVUT=0
abs(abs(abs(sqrt(x^2+y^2+0^2) +2z)+sqrt(x^2+y^2+0^2) -abs(0-0)-abs(0+0)) + abs(abs(sqrt(x^2+y^2+0^2) +2z)+sqrt(x^2+y^2+0^2) +abs(0-0)+abs(0+0)) + 4*0) + abs(abs(sqrt(x^2+y^2+0^2) +2z)+sqrt(x^2+y^2+0^2) -abs(0-0)-abs(0+0)) + abs(abs(sqrt(x^2+y^2+0^2) +2z)+sqrt(x^2+y^2+0^2) +abs(0-0)+abs(0+0)) = 20
IOiiIii - IOI cylinder, ZWUT=0
abs(abs(abs(sqrt(x^2+y^2+0^2) +2*0)+sqrt(x^2+y^2+0^2) -abs(z-0)-abs(z+0)) + abs(abs(sqrt(x^2+y^2+0^2) +2*0)+sqrt(x^2+y^2+0^2) +abs(z-0)+abs(z+0)) + 4*0) + abs(abs(sqrt(x^2+y^2+0^2) +2*0)+sqrt(x^2+y^2+0^2) -abs(z-0)-abs(z+0)) + abs(abs(sqrt(x^2+y^2+0^2) +2*0)+sqrt(x^2+y^2+0^2) +abs(z-0)+abs(z+0)) = 20
IiiiIIi - III cube, YZWT=0
abs(abs(abs(sqrt(x^2+0^2+0^2) +2*0)+sqrt(x^2+0^2+0^2) -abs(y-z)-abs(y+z)) + abs(abs(sqrt(x^2+0^2+0^2) +2*0)+sqrt(x^2+0^2+0^2) +abs(y-z)+abs(y+z)) + 4*0) + abs(abs(sqrt(x^2+0^2+0^2) +2*0)+sqrt(x^2+0^2+0^2) -abs(y-z)-abs(y+z)) + abs(abs(sqrt(x^2+0^2+0^2) +2*0)+sqrt(x^2+0^2+0^2) +abs(y-z)+abs(y+z)) = 20
IiiiiI> - II> sq pyramid, YZWV=0
abs(abs(abs(sqrt(x^2+0^2+0^2) +2*0)+sqrt(x^2+0^2+0^2) -abs(0-y)-abs(0+y)) + abs(abs(sqrt(x^2+0^2+0^2) +2*0)+sqrt(x^2+0^2+0^2) +abs(0-y)+abs(0+y)) + 4z) + abs(abs(sqrt(x^2+0^2+0^2) +2*0)+sqrt(x^2+0^2+0^2) -abs(0-y)-abs(0+y)) + abs(abs(sqrt(x^2+0^2+0^2) +2*0)+sqrt(x^2+0^2+0^2) +abs(0-y)+abs(0+y)) = 20
Iii>Iii - I>I triangle prism, YZUT=0
abs(abs(abs(sqrt(x^2+0^2+0^2) +2y)+sqrt(x^2+0^2+0^2) -abs(z-0)-abs(z+0)) + abs(abs(sqrt(x^2+0^2+0^2) +2y)+sqrt(x^2+0^2+0^2) +abs(z-0)+abs(z+0)) + 4*0) + abs(abs(sqrt(x^2+0^2+0^2) +2y)+sqrt(x^2+0^2+0^2) -abs(z-0)-abs(z+0)) + abs(abs(sqrt(x^2+0^2+0^2) +2y)+sqrt(x^2+0^2+0^2) +abs(z-0)+abs(z+0)) = 20
Iii>ii> - I>> tetrahedron, YZVU=0
abs(abs(abs(sqrt(x^2+0^2+0^2) +2y)+sqrt(x^2+0^2+0^2) -abs(0-0)-abs(0+0)) + abs(abs(sqrt(x^2+0^2+0^2) +2y)+sqrt(x^2+0^2+0^2) +abs(0-0)+abs(0+0)) + 4z) + abs(abs(sqrt(x^2+0^2+0^2) +2y)+sqrt(x^2+0^2+0^2) -abs(0-0)-abs(0+0)) + abs(abs(sqrt(x^2+0^2+0^2) +2y)+sqrt(x^2+0^2+0^2) +abs(0-0)+abs(0+0)) = 20
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Building the rotate function for turning the 3-plane to all 7 distinct intersections in 3D:
Start with 7D equation,
abs(abs(abs(sqrt(x^2+y^2+z^2) +2w)+sqrt(x^2+y^2+z^2) -abs(v-u)-abs(v+u)) + abs(abs(sqrt(x^2+y^2+z^2) +2w)+sqrt(x^2+y^2+z^2) +abs(v-u)+abs(v+u)) + 4t) + abs(abs(sqrt(x^2+y^2+z^2) +2w)+sqrt(x^2+y^2+z^2) -abs(v-u)-abs(v+u)) + abs(abs(sqrt(x^2+y^2+z^2) +2w)+sqrt(x^2+y^2+z^2) +abs(v-u)+abs(v+u)) = 20
Define rotation parameters,
Y -> W
Z -> V
(Y at W) -> U
(Z at V) -> T
y = (y*sin(a))
z = (z*sin(b))
w = ((y*cos(a))*sin(c))
v = ((z*cos(b))*sin(d))
u = ((y*cos(a))*cos(c))
t = ((z*cos(b))*cos(d))
abs(abs(abs(sqrt(x^2+(y*sin(a))^2+(z*sin(b))^2) +2((y*cos(a))*sin(c)))+sqrt(x^2+(y*sin(a))^2+(z*sin(b))^2) -abs(((z*cos(b))*sin(d))-((y*cos(a))*cos(c)))-abs(((z*cos(b))*sin(d))+((y*cos(a))*cos(c)))) + abs(abs(sqrt(x^2+(y*sin(a))^2+(z*sin(b))^2) +2((y*cos(a))*sin(c)))+sqrt(x^2+(y*sin(a))^2+(z*sin(b))^2) +abs(((z*cos(b))*sin(d))-((y*cos(a))*cos(c)))+abs(((z*cos(b))*sin(d))+((y*cos(a))*cos(c)))) + 4((z*cos(b))*cos(d))) + abs(abs(sqrt(x^2+(y*sin(a))^2+(z*sin(b))^2) +2((y*cos(a))*sin(c)))+sqrt(x^2+(y*sin(a))^2+(z*sin(b))^2) -abs(((z*cos(b))*sin(d))-((y*cos(a))*cos(c)))-abs(((z*cos(b))*sin(d))+((y*cos(a))*cos(c)))) + abs(abs(sqrt(x^2+(y*sin(a))^2+(z*sin(b))^2) +2((y*cos(a))*sin(c)))+sqrt(x^2+(y*sin(a))^2+(z*sin(b))^2) +abs(((z*cos(b))*sin(d))-((y*cos(a))*cos(c)))+abs(((z*cos(b))*sin(d))+((y*cos(a))*cos(c)))) = 20
Set Ranges to:
0 < a,b,c,d < π/2
XYZbox = -10,10
Plot in 37 Cubes Resolution
Navigating around the shape:[a,b,c,d] - Adjustable Parameter Values for 3D Midsection Positions
IOO Sphere = [π/2,π/2,π/2,π/2], [π/2,π/2,0,π/2], [π/2,π/2,0,0]
IOI Cylinder = [π/2,0,π/2,π/2], [π/2,0,0,π/2], [0,π/2,0,π/2], [0,π/2,0,0]
IO> Cone = [0,π/2,π/2,π/2], [π/2,0,0,0], [0,π/2,π/2,0], [π/2,0,π/2,0]
III Cube = [0,0,0,π/2]
II> Square Pyramid = [0,0,0,0]
I>I Triangle Prism = [0,0,π/2,π/2]
I>> Tetrahedron = [0,0,π/2,0]
When a=1.57 or b=1.57, adjusting c or d, respectively, will have no effect. Only when they are equal to zero, will c and d rotate to another 3-plane.
Good Morph Cycles through all 7 intersections:
• Set a,b,c,d to 1.57
• On the first occurrence of the parameter, slide it to 0
• On the second occurrence, slide it back to 1.57
a,c,b,d,c,d,b,a
a,c,b,c,d,a,d,b