************************************************************************* nD Notation - Name , Fiber Bundle Sequence , Diameter Size Hierarchy ** -----------------------------------------------------------------------** Surface Equation ** • Translation Equations {a,b,c,d} ** • Rotation Equations {X,x} {Y,y} {Z,z} ** • Trans + Rot Equations {A,a} {C,c} ** • MultiPosition Rotate Eq {X,a,b} {Y,a,b} {Z,a,b} ** • MultiDimension Rotate Eq {X,Y,[xy]} {X,Z,[xz]} {Y,Z,[yz]} ** --- Notes ** ** ************************************************************************* ************************************************************************* * Explore Function Library for Hypertoric Varieties * ***************************************************** 3D ((II)I) - Torus , S1xS1 , ((maj)min) ----------------------------------------------------------------- (sqrt(x^2+y^2) - R1)^2 + z^2 - R2^2 = 0 • ((II)I) (sqrt(x^2+y^2) - 2)^2 + z^2 - 0.5^2 = 0 • ((I)I) (sqrt(x^2+a^2) - 2)^2 + y^2 - 0.5^2 = 0 • ((II)) (sqrt(x^2+y^2) - 2)^2 + a^2 - 0.5^2 = 0 • ((Iy)Y) (sqrt(x^2+(y*cos(a))^2) - 2)^2 + (y*sin(a))^2 - 0.5^2 = 0 • ((II)I) 3D rotate (sqrt((x*sin(a) + z*cos(a))^2+y^2) - 2)^2 + (x*cos(a) - z*sin(a))^2 - 0.5^2 = 0 • Rotate + Translate (sqrt((x*sin(b) + (z+a)*cos(b))^2+y^2) - 2)^2 + (x*cos(b) - (z+a)*sin(b))^2 = 1 4D (IIII) - Glome , S3 --------------------------------- x^2 + y^2 + z^2 + w^2 - R^2 = 0 • (IIIi) x^2 + y^2 + z^2 + a^2 - 2^2 = 0 • (IYIy) x^2 + (y*sin(a))^2 + z^2 + (y*cos(a))^2 - 2^2 = 0 4D ((III)I) - Toripshere , S1xS2 , ((maj)min) -------------------------------------------------------------------- (sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2 - R2^2 = 0 • ((IIi)I) - torus (sqrt(x^2 + y^2 + a^2) - 4)^2 + z^2 -0.75^2 = 0 • ((III)) - concentric spheres (sqrt(x^2 + y^2 + z^2) - 4)^2 + a^2 -0.75^2 = 0 Ymin/max to -3 / 5 • ((IIz)Z) - rotation (sqrt(x^2 + y^2 + (z*cos(a))^2) - 4)^2 + (z*sin(a))^2 -0.75^2 = 0 0 < a < 1.57 z= -2 / 5 • ((IIa)A) - Trans+Rotate (sqrt(x^2 + y^2 + (z*cos(b/57.3)-a*sin(b/57.3))^2) - 3)^2 + (z*sin(b/57.3)+a*cos(b/57.3))^2 = 1 4D ((II)II) - Spheritorus , S2xS1 , ((maj)min) --------------------------------------------------------------------- (sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2 - R2^2 = 0 • ((I)II) - displaced spheres (sqrt(x^2 + a^2) - 2.5)^2 + y^2 + z^2 - 1^2 = 0 -4 < a < 4 • ((II)Ii) - torus (sqrt(x^2 + y^2) - 2.5)^2 + z^2 + a^2 - 1^2 = 0 -0.7 < a < 0.7 • ((IY)Iy) - rotation (sqrt(x^2 + (y*sin(a))^2) - 2.5)^2 + z^2 + (y*cos(a))^2 - 1^2 = 0 • ((IA)Ia) - Trans+Rotate (sqrt(x^2 + (y*sin(b/57.3)+a*cos(b/57.3))^2) - 3)^2 + z^2 + (y*cos(b/57.3)-a*sin(b/57.3))^2 = 1 4D (((II)I)I) - Ditorus , T3 , (((maj)med)min) ----------------------------------------------------------------- (sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + w^2 - R3^2 = 0 • (((I)I)I) - displaced toruses (sqrt((sqrt(x^2 + a^2) - 2.5)^2 + y^2) - 1)^2 + z^2 -0.5^2 = 0 -4.2 < a < 4.2 • (((II))I) - concentric toruses (sqrt((sqrt(x^2 + y^2) - 2.5)^2 + a^2) - 1)^2 + z^2 -0.5^2 = 0 -1.4 < a < 1.4 • (((II)I)) - cocircular toruses (sqrt((sqrt(x^2 + y^2) - 2.5)^2 + z^2) - 1)^2 + a^2 -0.5^2 = 0 -0.5 < a < 0.5 Zmin/max to -1 / 5 Ymin/max to -3 / 5 • (((IY)y)I) (sqrt((sqrt(x^2 + (y*sin(a))^2) - 9)^2 + (y*cos(a))^2) - 3)^2 + z^2 = 1 • (((IY)I)y) (sqrt((sqrt(x^2 + (y*sin(a))^2) - 9)^2 + z^2) - 3)^2 + (y*cos(a))^2 = 1 • (((II)Z)z) (sqrt((sqrt(x^2 + y^2) - 9)^2 + (z*sin(a))^2) - 3)^2 + (z*cos(a))^2 = 1 Trans+Rotate: • (((IA)a)I) (sqrt((sqrt(x^2 + (y*sin(b/57.3)+a*cos(b/57.3))^2) - 6)^2 + (y*cos(b/57.3)-a*sin(b/57.3))^2) - 3)^2 + z^2 = 1 • (((IA)I)a) (sqrt((sqrt(x^2 + (y*sin(b/57.3)+a*cos(b/57.3))^2) - 6)^2 + z^2) - 3)^2 + (y*cos(b/57.3)-a*sin(b/57.3))^2 = 1 • (((II)A)a) (sqrt((sqrt(x^2 + y^2) - 6)^2 + (z*sin(b/57.3)+a*cos(b/57.3))^2) - 3)^2 + (z*cos(b/57.3)-a*sin(b/57.3))^2 = 1 (((II)I)I) Exploring in 2D Slices ------------------------------------ • Step 1: (((Ac)C)a) - Rotate + Translate (sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*cos(d) - c*sin(d))^2) - 2)^2 + (y*sin(d) + c*cos(d))^2) - 1)^2 + (x*cos(b) - a*sin(b))^2 -0.5^2 = 0 - A,C Translates - B,D Rotates - Set X,Y to -5,+5 - Set Z to -0.01 , +0.01 for 2D • Step 2: (((Ac)a)C) - Rotate + Translate (sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*cos(d) - c*sin(d))^2) - 2)^2 + (x*cos(b) - a*sin(b))^2) - 1)^2 + (y*sin(d) + c*cos(d))^2 -0.5^2 = 0 • Step 3: (((I))I) - 4x1 row of circles (sqrt((sqrt(x^2 + a^2) - 2)^2 + b^2) - 1)^2 + y^2 -0.5^2 = 0 - A,B Translates away from center • Step 4: (((II))) - 4x concentric circles (sqrt((sqrt(x^2 + y^2) - 2)^2 + a^2) - 1)^2 + b^2 -0.5^2 = 0 • Step 5: (((I)I)) - 2x pair in 2x1 row of circles (sqrt((sqrt(x^2 + a^2) - 2)^2 + y^2) - 1)^2 + b^2 -0.5^2 = 0 4D ((II)(II)) - Tiger , S1xC2 , ((maj1)(maj2)min) --------------------------------------------------------------------------- (sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 - R2^2= 0 • ((II)(Ii)) - vertical stack of torii (sqrt(x^2 + y^2) - 2.5)^2 + (sqrt(z^2 + a^2) - 2.5)^2 -0.5^2 = 0 -2.5 < a < 2.5 • ((IY)(Iy)) (sqrt(x^2 + (y*sin(a))^2) - 2)^2 + (sqrt(z^2 + (y*cos(a))^2) - 2)^2 -0.5^2 = 0 • ((IA)(Ia)) - Rotation + Translation (sqrt(x^2 + (y*sin(b/57.3) + a*cos(b/57.3))^2) - 3)^2 + (sqrt(z^2 + (y*cos(b/57.3) - a*sin(b/57.3))^2) - 3)^2 = 1 5D (((II)I)(II)) - Tiger Torus , S1xC2xS1 = S1x[T2*S1] , (((maj1)med1)(med2)min) --------------------------------------------------------------------------------------------------- (sqrt((sqrt(x^2 + y^2) -R1a)^2 + z^2) -R2)^2 + (sqrt(w^2 + v^2) -R1b)^2 -R3^2 = 0 • (((I))(II)) - 4 torii in 1x1x4 column (Sqrt((Sqrt(x^2 + a^2) -2)^2 + b^2) -1)^2 + (Sqrt(y^2 + z^2) -2)^2 -0.5^2 = 0 • (((II))(I)) - 4 torii in 2 concentric along 1x1x2 column (Sqrt((Sqrt(x^2 + y^2) - 2)^2 + a^2) - 1)^2 + (Sqrt(z^2 + b^2) - 2)^2 - 0.5^2 = 0 • (((I)I)(I)) - 4 torii in 2x1x2 vertical square (Sqrt((Sqrt(x^2 + a^2) -2.5)^2 + y^2) -1.2)^2 + (Sqrt(z^2 + b^2) -1.5)^2 -0.4^2 = 0 • (((IY)z)(Zy)) Rotation (Sqrt((Sqrt(x^2 + (y*sin(a))^2) - 2.5)^2 + (z*cos(b))^2) - 1.2)^2 + (Sqrt((z*sin(b))^2 + (y*cos(a))^2) - 1.5)^2 - 0.5^2 = 0 (Sqrt((Sqrt(x^2 + a^2) -2.5)^2 + b^2) -1.2)^2 + (Sqrt(y^2 + z^2) - 1.5)^2 - 0.5^2 = 0 • (((Ic)a)(AC)) (sqrt((sqrt(x^2 + (z*cos(d) - c*sin(d))^2) -2)^2 + (y*cos(b) - a*sin(b))^2) -1)^2 + (sqrt((y*sin(b) + a*cos(b))^2 + (z*sin(d) + c*cos(d))^2) -2)^2 -0.5^2 = 0 • (((Ac)a)(CI)) (sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*cos(d) - c*sin(d))^2) -2)^2 + (x*cos(b) - a*sin(b))^2) -1)^2 + (sqrt((y*sin(d) + c*cos(d))^2 + z^2) -2)^2 -0.5^2 = 0 • (((I))(II)) - rotates whole cut, no trans or slice rotate (sqrt((sqrt((x*sin(b) + y*cos(a))^2 + a^2) -2)^2 + b^2) -1)^2 + (sqrt((x*cos(b) - y*sin(a))^2 + z^2) -2)^2 -0.5^2 = 0 • (((Ia)b)(YI)) from (((I))(II)) (Sqrt((Sqrt(x^2 + (y*cos(a))^2) -2)^2 + (y*cos(b))^2) -1)^2 + (Sqrt((y*((sin(a))*(sin(b))))^2 + z^2) -2)^2 -0.5^2 = 0 -- Start [A,B] at [1.57,1.57], rotate to either A or B = 0 • (((IY)a)(Ib)) from (((II))(I)) (Sqrt((Sqrt(x^2 + (y*((sin(a))*(sin(b))))^2) - 2)^2 + (y*cos(a))^2) - 1)^2 + (Sqrt(z^2 + (y*cos(b))^2) - 2)^2 - 0.5^2 = 0 • (Sqrt((Sqrt(x^2 + ((y*sin(c))*((sin(a))*(sin(b))))^2) - 2)^2 + (y*cos(a))^2) - 1)^2 + (Sqrt(z^2 + ((y*cos(c))*cos(b))^2) - 2)^2 - 0.5^2 = 0 • (((Ia)c)(AC)) - Dual Translate+Rotate , a/b = slide/rotate in 4D ; c/d = slide/rotate in 5D (sqrt((sqrt(x^2 + (y*cos(b) - a*sin(b))^2) -6)^2 + (z*cos(d) - c*sin(d))^2) -3)^2 + (sqrt((y*sin(b) + a*cos(b))^2 + (z*sin(d) + c*cos(d))^2) -3)^2 = 1 • (((AC)c)(Ia)) - Dual Translate+Rotate , a/b = slide/rotate in 4D ; c/d = slide/rotate in 5D (sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*sin(d) + c*cos(d))^2) -6)^2 + (y*cos(d) - c*sin(d))^2) -3)^2 + (sqrt(z^2 + (x*cos(b) - a*sin(b))^2) -3)^2 = 1 XYZbox = -12 / +12 -12 < a,c < 12 0 < b,d < 1.57 5D ((((II)I)I)I) - Tritorus , T4 , ((((maj)sec)tert)min) -------------------------------------------------------------------------------------- (sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 ) - R2)^2 + w^2) - R3)^2 + v^2 - R4^2= 0 • ((((I)I))I) (sqrt((sqrt((sqrt(x^2 + a^2) - 9)^2 + y^2 ) - 4)^2 + b^2) - 2)^2 + z^2 - 1.25^2= 0 • ((((Iy)Y)z)Z) (sqrt((sqrt((sqrt(x^2 + (y*cos(a))^2) - 9)^2 + (y*sin(a))^2 ) - 4)^2 + (z*cos(b))^2) - 2)^2 + (z*sin(b))^2 - 1.25^2= 0 • ((((Iz)Y)y)Z) (sqrt((sqrt((sqrt(x^2 + (z*cos(a))^2) - 9)^2 + (y*sin(b))^2 ) - 4)^2 + (y*cos(b))^2) - 2)^2 + (z*sin(a))^2 - 1.25^2= 0 • ((((IY)z)y)Z) (sqrt((sqrt((sqrt(x^2 + (y*sin(b))^2) - 9)^2 + (z*cos(a))^2 ) - 4)^2 + (y*cos(b))^2) - 2)^2 + (z*sin(a))^2 - 1.25^2= 0 • ((((XY)x)I)y) (sqrt((sqrt((sqrt((x*sin(a))^2 + (y*sin(b))^2) - 8)^2 + (x*cos(a))^2 ) - 4)^2 + z^2) - 2)^2 + (y*cos(b))^2 - 1.25^2= 0 • ((((XI)I)a)b) (sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2 + y^2) - 9)^2 + z^2 ) - 4)^2 + (x*cos(a))^2) - 2)^2 + (x*cos(b))^2 - 1= 0 • ((((XI)Z)x)z) (sqrt((sqrt((sqrt((x*sin(a))^2 + y^2) - 9)^2 + (z*sin(b))^2 ) - 4)^2 + (x*cos(a))^2) - 2)^2 + (z*cos(b))^2 - 1= 0 • ((((XI)Z)z)x) (sqrt((sqrt((sqrt((x*sin(a))^2 + y^2) - 9)^2 + (z*sin(b))^2 ) - 4)^2 + (z*cos(b))^2) - 2)^2 + (x*cos(a))^2 - 1= 0 • ((((XI)x)z)Z) (sqrt((sqrt((sqrt((x*sin(a))^2 + y^2) - 9)^2 + (x*cos(a))^2 ) - 4)^2 + (z*cos(b))^2) - 2)^2 + (z*sin(b))^2 - 1= 0 R1 Void Rotations • ((((XY)x)y)I) (sqrt((sqrt((sqrt((x*sin(a))^2 + (y*sin(b))^2) - 9)^2 + (x*cos(a))^2 ) - 4)^2 + (y*cos(b))^2) - 2)^2 + z^2 - 1= 0 • ((((XY)I)x)y) (sqrt((sqrt((sqrt((x*sin(a))^2 + (y*sin(b))^2) - 9)^2 + z^2 ) - 4)^2 + (x*cos(a))^2) - 2)^2 + (y*cos(b))^2 - 1= 0 • ((((XY)x)I)y) (sqrt((sqrt((sqrt((x*sin(a))^2 + (y*sin(b))^2) - 9)^2 + (x*cos(a))^2 ) - 4)^2 + z^2) - 2)^2 + (y*cos(b))^2 - 1= 0 • (sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2 + y^2) - 9)^2 + (z*sin(c) + (x*cos(a))*cos(c))^2 ) - 4)^2 + (z*cos(c) - (x*cos(a))*sin(c))^2) - 2)^2 + (x*cos(b))^2 - 1= 0 --- [A,C,a,B,c,b] • ((((XY)Z)[a,z])[y,b]) - [x->a,b] [z->c] [y->d] (sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2 + (y*sin(d) + (x*cos(b))*cos(d))^2) - 9)^2 + (z*sin(c) + (x*cos(a))*cos(c))^2 ) - 4)^2 + (z*cos(c) - (x*cos(a))*sin(c))^2) - 2)^2 + (y*cos(d) - (x*cos(b))*sin(d))^2 - 1 = 0 —— good morph sequence : set a,b,c,d to 1.57; slider sequence for A=0 -> a=1.57 : [D,C,d,A,c,a] and {C,A,c,a,D,C,d,c] 5D (((II)(II))I) - Toratiger , T2xC2 , (((maj1)(maj2)med)min) ----------------------------------------------------------------------------------- (((II)(II))I) ((II)(II))I ( (II) (II)) I ( (xy) (zw)) v ( (x+y) + (z+w)) + v ( (x+y -R1a) + (z+w -R1b) -R2) + v = Rminor ( (x+y -R1a)² + (z+w -R1b)² -R2)² + v = Rminor² ( (√(x+y) -R1a)² + (√(z+w) -R1b)² -R2)² + v = Rminor² ( √((√(x+y) -R1a)² + (√(z+w) -R1b)²) -R2)² + v = Rminor² (√((√(x²+y²) -R1a)² + (√(z²+w²) -R1b)²) -R2)² + v² = Rminor² (sqrt((sqrt(x^2+y^2) -R1a)^2 + (sqrt(z^2+w^2) -R1b)^2) -R2)^2 + v^2 = Rminor^2 • (((II)(II))I) (sqrt((sqrt(x^2+y^2) -6)^2 + (sqrt(z^2+w^2) -6)^2) -3)^2 + v^2 = 1 • (((II)(I))) : 1x1x2x[Rminor pair] (sqrt((sqrt(x^2+y^2) -6)^2 + (sqrt(z^2+0^2) -6)^2) -3)^2 + 0^2 = 1 • (((I)(I))I) : 2x2x1 square (sqrt((sqrt(x^2+0^2) -6)^2 + (sqrt(y^2+0^2) -6)^2) -3)^2 + z^2 = 1 • (((AC)(Ia))c) : Dual Translate + Rotate (sqrt((sqrt((x*sin(b)+a*cos(b))^2+(y*sin(d)+c*cos(d))^2)-6)^2+(sqrt(z^2+(x*cos(b)-a*sin(b))^2)-6)^2)-3)^2+(y*cos(d)-c*sin(d))^2 = 1 5D ((III)II) Spherisphere , S2xS2 , (maj)min) ----------------------------------------------- • ((III)II) (sqrt(x^2 + y^2 + z^2) -R1)^2 + w^2 + v^2 -R2^2 = 0 R1 = 3 / R2 = 1 (sqrt(x^2 + y^2 + z^2) -3)^2 + w^2 + v^2 -1 = 0 • ((I)II) - 2x Spheres (III) in 2x1x1 row (sqrt(x^2 + 0^2 + 0^2) -3)^2 + w^2 + v^2 -1 = 0 • ((II)I) - 1x Torus (sqrt(x^2 + y^2 + 0^2) -3)^2 + w^2 + 0^2 -1 = 0 • ((III)) - 2x Spheres (III) in concentric pair (sqrt(x^2 + y^2 + z^2) -3)^2 + 0^2 + 0^2 -1 = 0 • ((IYZ)yz) (sqrt(x^2 + (y*sin(a))^2 + (z*sin(b))^2) -3)^2 + (y*cos(a))^2 + (z*cos(b))^2 -1 = 0 5D (((II)I)II) - Spheriditorus , S2xT2 , (((maj)med)min) --------------------------------------------------------------------------- (sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2 ) - R2)^2 + w^2 + v^2 - R3^2 = 0 • (((I))II) (sqrt((sqrt(x^2 + a^2) - 3.5)^2 + b^2 ) - 2)^2 + y^2 + z^2 = 1^2 (sqrt((sqrt(x^2 + (y*cos(a))^2) - 4)^2 + (z*cos(b))^2 ) - 2)^2 + (y*sin(a))^2 + (z*sin(b))^2 - 1^2 = 0 5D (((III)I)I) - Ditorisphere , T2xS2 , (((maj)med)min) -------------------------------------------------------------------------------- (sqrt((sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2) - R2)^2 + v^2))^2 - R3^2 = 0 major - R1 / medium - R2 / minor - R3 • (((IYZ)y)z) (sqrt((sqrt(x^2 + (y*sin(a))^2 + (z*sin(b))^2) - 2.5)^2 + (y*cos(a))^2) - 1)^2 + (z*cos(b))^2 - 0.5^2 = 0 5D (((II)II)I) - Torispheritorus , S1xS2xS1 , (((r1)r2)r3) ----------------------------------------------------------- (sqrt((sqrt(x^2 + y^2) -r1)^2 + z^2 + w^2) -r2)^2 + v^2 -r3^2 = 0 r1= 4 / r2= 2 / r3= 1 (sqrt((sqrt(x^2 + y^2) -4)^2 + z^2 + w^2) -2)^2 + v^2 -1 = 0 • ((()II)I) - empty, moving out makes 1x torisphere evolution of ((IIi)I) (sqrt((sqrt(0^2 + 0^2) -4)^2 + x^2 + y^2) -2)^2 + z^2 -1 = 0 • (((I)I)I) - 2x Tori ((II)I) in 2x1x1 row (sqrt((sqrt(x^2 + 0^2) -4)^2 + y^2 + 0^2) -2)^2 + z^2 -1 = 0 • (((I)II)) - 4x Spheres (III) as 2x1x1 row of R1 pairs (sqrt((sqrt(x^2 + 0^2) -4)^2 + y^2 + z^2) -2)^2 + 0^2 -1 = 0 • (((II))I) - 2x Tori ((II)I) as R1 pair (sqrt((sqrt(x^2 + y^2) -4)^2 + 0^2 + 0^2) -2)^2 + z^2 -1 = 0 • (((II)I)) - 2X Tori ((II)I) as R2 pair (sqrt((sqrt(x^2 + y^2) -4)^2 + z^2 + 0^2) -2)^2 + 0^2 -1 = 0 • (((IY)yz)Z) (sqrt((sqrt(x^2 + (y*sin(a))^2) -4)^2 + (y*cos(a))^2 + (z*cos(b))^2) -2)^2 + (z*sin(b))^2 -1 = 0 • (((IA)ac)C) (sqrt((sqrt(x^2 + (y*sin(b) + a*cos(b))^2) -4)^2 + (y*cos(b) - a*sin(b))^2 + (z*cos(d) - c*sin(d))^2) -2)^2 + (z*sin(d) + c*cos(d))^2 -1 = 0 5D ((III)(II)) - Cylspherintigroid , S1x[S2*S1] ----------------------------------------------------- (√(x²+y²+z²) -R1a)² + (√(w²+v²) -R1b)² = Rminor² (sqrt(x^2+y^2+z^2) -R1a)^2 + (sqrt(w^2+v^2) -R1b)^2 = Rminor^2 -------------------------------------- ((3)(3)1) - Ring-Torus Diameter Values (sqrt(x^2+y^2+z^2) -3)^2 + (sqrt(w^2+v^2) -3)^2 = 1 3D Midsections along ((xyz)(wv)) -------------------------------- XYZboz = -6 , +6 • ((xy)(w)) - zv=0 , 1x1x2 column of tori (sqrt(x^2+y^2+0^2) -3)^2 + (sqrt(z^2+0^2) -3)^2 = 1 • ((x)(wv)) - yz=0 , 1x1x2 column of tori (sqrt(x^2+0^2+0^2) -3)^2 + (sqrt(y^2+z^2) -3)^2 = 1 • ((xyz)()) - wv=0 , Void R1b , ring intercepts at [±3] are 2 places of ((III)) , [R pair] of 2 spheres (sqrt(x^2+y^2+z^2) -3)^2 + (sqrt(0^2+0^2) -3)^2 = 1 Dual Translate + Rotate Function --------------------------------- • ((Iac)(AC)) : dual trans+rotate (sqrt(x^2+(y*cos(b)-a*sin(b))^2+(z*cos(d)-c*sin(d))^2)-3)^2+(sqrt((y*sin(b)+a*cos(b))^2+(z*sin(d)+c*cos(d))^2)-3)^2 = 1 -6 < a,c < 6 0 < b,d < 1.57 6D (((II)I)((II)I)) - Tiger Duotorus , S1xC2xC2 (((maj1)med1)((maj2)med2)min) ------------------------------------------------------------------------------------------------------------- (sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) -R2a)^2 + (sqrt((sqrt(w^2 + v^2) - R1b)^2 + u^2) - R2b)^2 - R3^2 = 0 • (((I)I)((I))) (sqrt((sqrt(x^2 + a^2) - 2)^2 + y^2) -1)^2 + (sqrt((sqrt(z^2 + b^2) - 2)^2 + c^2) -1)^2 = 0.4^2 • (((Xz)Y)((Zx)y)) (sqrt((sqrt((x*sin(b))^2 + (z*cos(a))^2) - 2)^2 + (y*sin(c))^2) -1)^2 + (sqrt((sqrt((z*sin(a))^2 + (x*cos(b))^2) - 2)^2 + (y*cos(c))^2) -1)^2 = 0.4^2 • (((Xz)Y)((Zy)x)) (sqrt((sqrt((x*sin(c))^2 + (z*cos(a))^2) - 2)^2 + (y*sin(b))^2) -1)^2 + (sqrt((sqrt((z*sin(a))^2 + (y*cos(b))^2) - 2)^2 + (x*cos(c))^2) -1)^2 = 0.4^2 • (((II))((I))) (sqrt((sqrt(x^2 + y^2) - 2)^2 + a^2) -1)^2 + (sqrt((sqrt(z^2 + b^2) - 2)^2 + c^2) -1)^2 = 0.4^2 • (((XY)z)((Zx)y)) (sqrt((sqrt((x*sin(b))^2 + (y*sin(c))^2) - 2)^2 + (z*cos(a))^2) -1)^2 + (sqrt((sqrt((z*sin(a))^2 + (x*cos(b))^2) - 2)^2 + (y*cos(c))^2) -1)^2 = 0.4^2 -- A,C=1.57 / B=0.785 is (((xI))((xI))), quadruple tiger cage // A,C=0 / B=0.785 is (((x)I)((x)I)) 2x oblique tiger scan • (((IA))((Ia))) translate A , rotate B (sqrt((sqrt(x^2 + (y*sin(b) + a*cos(b))^2) - 2)^2) -1)^2 + (sqrt((sqrt(z^2 + (y*cos(b) - a*sin(b))^2) - 2)^2) -1)^2 - 0.4^2 = 0 -- B=0.785 , Adjust A for flythrough of (((OI))((OI))) oblique structure, di-duoring structural scan with R3=0.2 -- XYZ = -5/+5 -- A = -4.5~4.5 -- A = 1.22 , adj B for neat topo change of six tori • (((A)I)((Ca)c)) translate A,C rotate B,D (sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 2)^2 + y^2) -1)^2 + (sqrt((sqrt((z*sin(d) + c*cos(d))^2 + (x*cos(b) - a*sin(b))^2) - 2)^2 + (z*cos(d) - c*sin(d))^2) -1)^2 = 0.4^2 -- B=0.785 ; C,D=0 , Adjust A for fantastic diagonal translate along 2x1x2x1 square of tigers (((I)I)((I)I)) !! • (sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*sin(d) + c*cos(d))^2) - 2)^2) -1)^2 + (sqrt((sqrt(z^2 + (y*cos(d) - c*sin(d))^2) - 2)^2 + (x*cos(b) - a*sin(b))^2) -1)^2- 0.3^2 = 0 -- c=1.16667 , adj [B,D] through 4x 90deg turns • (sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*sin(d) + c*cos(d))^2) - 2)^2 + (x*cos(b) - a*sin(b))^2) -1)^2 + (sqrt((sqrt(z^2 + (y*cos(d) - c*sin(d))^2) - 2)^2) -1)^2 - 0.3^2 = 0 • (Sqrt((Sqrt(x^2 + (y*((sin(a))*(sin(b))))^2) - 2)^2 + (y*cos(a))^2) - 1)^2 + (Sqrt(z^2 + (y*cos(b))^2) - 2)^2 - 0.5^2 = 0 • sqrt((sqrt(x^2 + a^2) - 2)^2 + y^2) -1)^2 + (sqrt((sqrt(z^2 + b^2) - 2)^2 + c^2) -1)^2 = 0.4^2 • (sqrt((sqrt(x^2 + (y*cos(a))^2) - 2)^2 + (y*((sin(a))*(sin(b))*(sin(c))))^2) -1)^2 + (sqrt((sqrt(z^2 + (y*cos(b))^2) - 2)^2 + (y*cos(c))^2) -1)^2 = 0.4^2 --- Interesting, works too. At least two out of three have to be 1.57, setting one to 0 makes morphs • (sqrt((sqrt(x^2 + ((y*cos(a))*sin(d))^2) - 2)^2 + (y*((sin(a))*(sin(b))*(sin(c))))^2) -1)^2 + (sqrt((sqrt(z^2 + ((y*cos(b))*cos(d))^2) - 2)^2 + (y*cos(c))^2) -1)^2= 0.4^2 -- Works with interesting results! 6D ((((II)I)I)(II)) - Tiger Ditorus , S1xC2xT2 / S1x[T3*S1] / T3xC2 , ((((maj)sec)tert1)(tert2)min) ------------------------------------------------------------------------------------------------------------- (sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + w^2) - R3a)^2 + (sqrt(v^2 + u^2) - R3b)^2 - R4^2 = 0 • ((((I))I)(I)) - ((((X))Y)(Z)) (sqrt((sqrt((sqrt(x^2 + a^2) - 4.5)^2 + b^2) - 2.2)^2 + y^2) - 1.1)^2 + (sqrt(z^2 + c^2) - 2)^2 - 0.7^2 = 0 • ((((II)))(I)) - ((((XY)))(Z)) (sqrt((sqrt((sqrt(x^2 + y^2) - 4.5)^2 + a^2) - 2.2)^2 + b^2) - 1.1)^2 + (sqrt(z^2 + c^2) - 2)^2 - 0.7^2 = 0 • ((((Xz)x)Y)(Zy)) (sqrt((sqrt((sqrt((x*sin(b))^2 + (z*cos(a))^2) - 4.5)^2 + (x*cos(b))^2) - 2.2)^2 + (y*sin(c))^2) - 1.1)^2 + (sqrt((z*sin(a))^2 + (y*cos(c))^2) - 2)^2 - 0.7^2 = 0 • ((((Xy)z)Y)(Zx)) (sqrt((sqrt((sqrt((x*sin(c))^2 + (y*cos(a))^2) - 4.5)^2 + (z*cos(b))^2) - 2.2)^2 + (y*sin(a))^2) - 1.1)^2 + (sqrt((z*sin(b))^2 + (x*cos(c))^2) - 2)^2 - 0.7^2 = 0 • ((((Xz)y)Y)(Zx)) (sqrt((sqrt((sqrt((x*sin(c))^2 + (z*cos(a))^2) - 4.5)^2 + (y*cos(b))^2) - 2.2)^2 + (y*sin(b))^2) - 1.1)^2 + (sqrt((z*sin(a))^2 + (x*cos(c))^2) - 2)^2 - 0.7^2 = 0 • ((((XY)y)z)(Zx)) same as ((((Xz)x)Y)(Zy)) (sqrt((sqrt((sqrt((x*sin(c))^2 + (y*sin(a))^2) - 4.5)^2 + (y*cos(a))^2) - 2.2)^2 + (z*cos(b))^2) - 1.1)^2 + (sqrt((z*sin(b))^2 + (x*cos(c))^2) - 2)^2 - 0.7^2 = 0 • ((((Ac)a)I)(C)) - ((((I))I)(I)) (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (z*cos(d) - c*sin(d))^2) - 5.75)^2 + (x*cos(b) - a*sin(b))^2) - 3)^2 + y^2) - 1.75)^2 + (sqrt((z*sin(d) + c*cos(d))^2 + 0^2) - 2.85)^2 - 0.75^2 = 0 • ((((A)c)a)(CI)) - ((((I)))(II)) (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 5.75)^2 + (y*cos(d) - c*sin(d))^2) - 3)^2 + (x*cos(b) - a*sin(b))^2) - 1.5)^2 + (sqrt((y*sin(d) + c*cos(d))^2 + z^2) - 2.85)^2 - 0.75^2 = 0 • ((((Ia))c)(AC)) - ((((I)))(II)) (sqrt((sqrt((sqrt(x^2 + (y*cos(b) - a*sin(b))^2) - 5.75)^2 + 0^2) - 3)^2 + (z*cos(d) - c*sin(d))^2) - 1.5)^2 + (sqrt((y*sin(b) + a*cos(b))^2 + (z*sin(d) + c*cos(d))^2) - 2.85)^2 - 0.75^2 = 0 • ((((I)a)c)(AC)) - ((((I)))(II)) (sqrt((sqrt((sqrt(x^2 + 0^2) - 6)^2 + (y*cos(b) - a*sin(b))^2) - 3)^2 + (z*cos(d) - c*sin(d))^2) - 1.5)^2 + (sqrt((y*sin(b) + a*cos(b))^2 + (z*sin(d) + c*cos(d))^2)- 3)^2 - 0.6^2 = 0 --- a=3 , c=0 / [b,d] go thru [0,0]>[1.57,0]>[1.57,1.57]>[0,1.57]>[0,0] Very interesting topology change!!! • ((((Ia)c))(AC)) - ((((I)))(II)) (sqrt((sqrt((sqrt(x^2 + (y*cos(b) - a*sin(b))^2) - 6)^2 + 0^2) - 3)^2 + (z*cos(d) - c*sin(d))^2) - 1.5)^2 + (sqrt((y*sin(b) + a*cos(b))^2 + (z*sin(d) + c*cos(d))^2) - 2.85)^2 - 0.75^2 = 0 • ((((A)a)C)(Ic)) - ((((I))I)(I)) (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 5.75)^2 + (x*cos(b) - a*sin(b))^2) - 3)^2 + (y*sin(d) + c*cos(d))^2) - 1.5)^2 + (sqrt(z^2 + (y*cos(d) - c*sin(d))^2) - 2.85)^2 - 0.75^2 = 0 --- a=5.75 , trans out to single tigritorus, slide b for neat rotations • ((((Ac))C)(Ia)) - ((((I))I)(I)) (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*cos(d) - c*sin(d))^2) - 5.75)^2 + 0^2) - 3)^2 + (y*sin(d) + c*cos(d))^2) - 1.5)^2 + (sqrt(z^2 + (x*cos(b) - a*sin(b))^2) - 2.85)^2 - 0.75^2 = 0 • ((((AC)a))(Ic)) - ((((II)))(I)) (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*sin(d) + c*cos(d))^2) - 5.75)^2 + (x*cos(b) - a*sin(b))^2) - 3)^2 + 0^2) - 1.5)^2 + (sqrt(z^2 + (y*cos(d) - c*sin(d))^2) - 2.85)^2 - 0.75^2 = 0 • ((((AC)a)c)(I)) - ((((II)))(I)) (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*sin(d) + c*cos(d))^2) - 5.75)^2 + (x*cos(b) - a*sin(b))^2) - 3)^2 + (y*cos(d) - c*sin(d))^2) - 1.5)^2 + (sqrt(z^2 + 0^2) - 2.85)^2 - 0.75^2 = 0 • ((((Ac)I)a)(C)) - ((((I)I))(I)) (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (z*cos(d) - c*sin(d))^2) - 5.75)^2 + y^2) - 3)^2 + (x*cos(b) - a*sin(b))^2) - 1.5)^2 + (sqrt((z*sin(d) + c*cos(d))^2 + 0^2) - 2.85)^2 - 0.75^2 = 0 • ((((Ac)I))(Ca)) - ((((I)I))(I)) (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (z*cos(d) - c*sin(d))^2) - 5.75)^2 + y^2) - 3)^2 + 0^2) - 1.5)^2 + (sqrt((z*sin(d) + c*cos(d))^2 + (x*cos(b) - a*sin(b))^2) - 2.85)^2 - 0.6^2 = 0 • ((((a))I)(AI)) - (((())I)(II)) (sqrt((sqrt((sqrt((y*cos(b) - a*sin(b))^2 + 0^2) - 5.8125)^2 + 0^2) - 3)^2 + x^2) - 2)^2 + (sqrt((y*sin(b) + a*cos(b))^2 + z^2) - 2)^2 - 0.6^2 = 0 --- set b=0.785 , adj A for 4x OBLQ tiger scan along line • ((((a))A)(II)) - (((())I)(II)) (sqrt((sqrt((sqrt((x*cos(b) - a*sin(b))^2 + 0^2) - 5.8125)^2 + 0^2) - 3)^2 + (x*sin(b) + a*cos(b))^2) - 2)^2 + (sqrt(y^2 + z^2) - 2)^2 - 0.6^2 = 0 --- set B=0.785 , adj A for 4x tiger dance along line • ((((a)c)C)(AI)) - (((())I)(II)) (sqrt((sqrt((sqrt((y*cos(d) - c*sin(d))^2 + 0^2) - 5.75)^2 + (x*cos(b) - a*sin(b))^2) - 3)^2 + (x*sin(b) + a*cos(b))^2) - 1.5)^2 + (sqrt((y*sin(d) + c*cos(d))^2 + z^2) - 2.85)^2 - 0.75^2 = 0 --- Very cool exploration!!!! 6D (((II)(II))(II)) - Double Tiger , T2xC3 , (((maj1)(maj2)med)(maj3)min) ----------------------------------------------------------------------------------------------------------- (sqrt((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2) - R2)^2 + (sqrt(v^2 + u^2) - R1c)^2 - R3^2 = 0 • (((I)(I))(I)) (sqrt((sqrt(x^2 + a^2) - R1)^2 + (sqrt(y^2 + b^2) - R2)^2) - R3)^2 + (sqrt(z^2 + c^2) - R4)^2 - R5^2 = 0 • (((I)(I))(I)) (sqrt((sqrt(x^2 + a^2) - 3)^2 + (sqrt(y^2 + b^2) - 3)^2) - 1.5)^2 + (sqrt(z^2 + c^2) - 3)^2 - 0.5^2 = 0 • (((Xz)(Yx))(Zy)) (sqrt((sqrt((x*sin(b))^2 + (z*cos(a))^2) - 3)^2 + (sqrt((y*sin(c))^2 + (x*cos(b))^2) - 3)^2) - 1.5)^2 + (sqrt((z*sin(a))^2 + (y*cos(c))^2) - 2.5)^2 - 0.5^2 = 0 Octatangent Cut at a = 0.36497 b = 0.3492 c = 0.91242 •(sqrt((sqrt((x*sin(c))^2 + (y*cos(a))^2) - 3)^2 + (sqrt((y*sin(a))^2 + (z*cos(b))^2) - 3)^2) - 1.5)^2 + (sqrt((z*sin(b))^2 + (x*cos(c))^2) - 2.5)^2 - 0.5^2 = 0 • (((A)(Ic))(Ca)) (sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 3)^2 + (sqrt(y^2 + (z*cos(d) - c*sin(d))^2) - 3)^2) - 1.5)^2 + (sqrt((z*sin(d) + c*cos(d))^2 + (x*cos(b) - a*sin(b))^2) - 3)^2 - 0.5^2 = 0 --- Set a=3 , rotate B,D for tiger torus morphing • (((A)(Ca))(Ic)) (sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 3)^2 + (sqrt((y*sin(d) + c*cos(d))^2 + (x*cos(b) - a*sin(b))^2) - 3)^2) - 1.5)^2 + (sqrt(z^2 + (y*cos(d) - c*sin(d))^2) - 3)^2 - 0.5^2 = 0 --- XYZ = -7,+7 --- Set a=-3 , c=1.85 / [b,d] go through [0,0]>[1.57,0]>[1.57,1.57]>[0,1.57]>[0,0] for interesting topology morphing 6D ((II)(II)(II)) - Tritiger , S2xC3 , ((maj1)(maj2)(maj3)min) ---------------------------------------------------------------------------------------------- (sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 + (sqrt(v^2 + u^2) - R1c)^2 - R2^2 = 0 • ((I)(I)(I)) - 2x2x2 array of 8 spheres (sqrt(x^2 + a^2) - 2.5)^2 + (sqrt(y^2 + b^2) - 2.5)^2 + (sqrt(z^2 + c^2) - 2.5)^2 - 1^2 = 0 • ((Xy)(Yz)(Zx)) (sqrt((x*sin(c))^2 + (y*cos(a))^2) - 2.5)^2 + (sqrt((y*sin(a))^2 + (z*cos(b))^2) - 2.5)^2 + (sqrt((z*sin(b))^2 + (x*cos(c))^2) - 2.5)^2 - 1^2 = 0 XYZmin/max = -5,+5 • ((Xz)(Yx)(Zy)) (sqrt((x*sin(b))^2 + (z*cos(a))^2) - 3)^2 + (sqrt((y*sin(c))^2 + (x*cos(b))^2) - 3)^2 + (sqrt((z*sin(a))^2 + (y*cos(c))^2) - 3)^2 - 0.8805^2 = 0 -- Special diameter values for 45x45x45 cut, dodecatangent • ((A)(Ca)(Ic)) : Dual R+T for Triger (sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 2.5)^2 + (sqrt((y*sin(d) + c*cos(d))^2 + (x*cos(b) - a*sin(b))^2) - 2.5)^2 + (sqrt(z^2 + (y*cos(d) - c*sin(d))^2) - 2.5)^2 - 1^2 = 0 6D ((((II)(II))I)I) - Ditoratiger , T3xC2 , ((((maj1)(maj2)sec)tert)min) ----------------------------------------------------------------------------------------------------- (sqrt((sqrt((sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2) - R2)^2 + v^2) - R3)^2 + u^2 - R4^2 = 0 • Diameter Adjustment Function (sqrt((sqrt((sqrt(x^2 + 0^2) - a)^2 + (sqrt(y^2 + 0^2) - a)^2) - b)^2 + 0^2) - c)^2 + z^2 = 1 XYZbox = -20 / +20 - a = 8 - b = 4 - c = 2 • ((((I)(I)))I) (sqrt((sqrt((sqrt(x^2 + a^2) - 8)^2 + (sqrt(y^2 + b^2) - 8)^2) - 4)^2 + c^2) - 2)^2 + z^2 = 1 • ((((Xz)(Yx))y)Z) (sqrt((sqrt((sqrt((x*sin(b))^2 + (z*cos(a))^2) - 8)^2 + (sqrt((y*sin(c))^2 + (x*cos(b))^2) - 8)^2) - 4)^2 + (y*cos(c))^2) - 2)^2 + (z*sin(a))^2 = 1 • ((((Xy)(Yz))x)Z) (sqrt((sqrt((sqrt((x*sin(c))^2 + (y*cos(a))^2) - 8)^2 + (sqrt((y*sin(a))^2 + (z*cos(b))^2) - 8)^2) - 4)^2 + (x*cos(c))^2) - 2)^2 + (z*sin(b))^2 = 1 6D (((II)I)(II)I) - Spheritiger Torus , S2xC2xS1 = S2x[T2*S1] , (((maj)med1)(med2)min) ------------------------------------------------------------------------------------------------------ (sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2a)^2 + (sqrt(w^2 + v^2) - R2b)^2 + u^2 - R3^2 = 0 • (((Ii)i)(Ii)I) = (((I))(I)I) - 4x2 array of 8 spheres (sqrt((sqrt(x^2 + a^2) -2)^2 + b^2) -1)^2 + (Sqrt(y^2 + c^2) -2)^2 + z^2 -0.75^2 = 0 • (((II)i)(Ii)i) = (((II))(I)) - 2 conc stacked 2 high of 4 torii (sqrt((sqrt(x^2 + y^2) -2)^2 + a^2) -1)^2 + (Sqrt(z^2 + b^2) -2)^2 + c^2 -0.75^2 = 0 • (((Ii)i)(II)i) = (((I))(II)) - vert column of 4 torii (sqrt((sqrt(x^2 + a^2) -2)^2 + b^2) -1)^2 + (Sqrt(y^2 + c^2) -2)^2 + z^2 -0.75^2 = 0 • (((Xz)y)(Yx)Z) (sqrt((sqrt((x*sin(c))^2 + (z*cos(a))^2) -2)^2 + (y*cos(b))^2) -1)^2 + (Sqrt((y*sin(b))^2 + (x*cos(c))^2) -2)^2 + (z*sin(a))^2 -0.75^2 = 0 • (((Xy)z)(yx)Z) (sqrt((sqrt((x*sin(c))^2 + (y*cos(b))^2) -2)^2 + (z*cos(a))^2) -1)^2 + (Sqrt((y*sin(b))^2 + (x*cos(c))^2) -2)^2 + (z*sin(a))^2 -0.75^2 = 0 • (((Xy)x)(Yz)Z) - (((I))(I)I) (sqrt((sqrt((x*sin(b))^2 + (y*cos(a))^2) -4)^2 + (x*cos(b))^2) -1.5)^2 + (sqrt((y*sin(a))^2 + (z*cos(c))^2) -4)^2 + (z*sin(c))^2 -0.75^2 = 0 6D ((((II)I)(II))I) - Toratiger Torus , T2xC2xS1 , ((((maj)sec)(maj2)tert)min) ---------------------------------------------------------------------------------------------------------------- (√((√((√(x²+y²) -R1a)² +z²) -R2)² + (√(w²+v²) -R1b)²) -R3)² +u² = Rminor² (sqrt((sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2)^2 + (sqrt(w^2 + v^2) - R1b)^2) - R3)^2 + u^2 - R4^2 = 0 ((((8)4)(4)2)1) • ((((I))(I))I) - 4x2 array of 8 torii (sqrt((sqrt((sqrt(x^2 + a^2) - 8)^2 + b^2) - 4)^2 + (sqrt(y^2 + c^2) - 4)^2) - 2)^2 + z^2 = 1 • ((((II))(I))) - 2 cocirc by 2 conc stacked 2 high of 8 torii (sqrt((sqrt((sqrt(x^2 + y^2) - 8)^2 + a^2) - 4)^2 + (sqrt(z^2 + b^2) - 4)^2) - 2)^2 + c^2 = 1 • ((((I))(II))) - 2 cocirc stacked 4 high of 8 torii (sqrt((sqrt((sqrt(x^2 + a^2) - 8)^2 + b^2) - 4)^2 + (sqrt(y^2 + z^2) - 4)^2) - 2)^2 + c^2 = 1 • ((((I)I)(I))) - 2 cocirc in 2x1x2 vert square of 8 torii (sqrt((sqrt((sqrt(x^2 + a^2) - 8)^2 + y^2) - 4)^2 + (sqrt(z^2 + b^2) - 4)^2) - 2)^2 + c^2 = 1 • ((((Xy)z)(Yx))Z) (sqrt((sqrt((sqrt((x*sin(c))^2 + (y*cos(a))^2) - 8)^2 + (z*cos(b))^2) - 4)^2 + (sqrt((y*sin(a))^2 + (x*cos(c))^2) - 4)^2) - 2)^2 + (z*sin(b))^2 = 1 • ((((Xz)y)(Yx))Z) (sqrt((sqrt((sqrt((x*sin(c))^2 + (z*cos(a))^2) - 8)^2 + (y*cos(b))^2) - 4)^2 + (sqrt((y*sin(b))^2 + (x*cos(c))^2) - 4)^2) - 2)^2 + (z*sin(a))^2 = 1 • ((((Xy)x)(Yz))Z) (sqrt((sqrt((sqrt((x*sin(b))^2 + (y*cos(a))^2) - 8)^2 + (x*cos(b))^2) - 4)^2 + (sqrt((y*sin(a))^2 + (z*cos(c))^2) - 4)^2) - 2)^2 + (z*sin(c))^2 = 1 • ((((Ac)I)(a))C) (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (z*cos(d) - c*sin(d))^2) - 8)^2 + y^2) - 4)^2 + (sqrt(0^2 + (x*cos(b) - a*sin(b))^2) - 4)^2) - 2)^2 + (z*sin(d) + c*cos(d))^2 = 1 -- Very good exploration of ((((I)I)(I))I) cut, though Z rotations make major-->minor concentric morph • ((((A)C)(ac))I) (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) - 8)^2 + (y*sin(d) + c*cos(d))^2) - 4)^2 + (sqrt((x*cos(b) - a*sin(b))^2 + (y*cos(d) - c*sin(d))^2) - 4)^2) -2)^2 + z^2 = 1 -- A= -1.29 , C= 0.9 : Adjust B,D angles to all four combinations, very cool stuff! • ((((A)c)(Ia))C) (sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2)-8)^2+(z*cos(d)-c*sin(d))^2)-4)^2+(sqrt(y^2+(x*cos(b)-a*sin(b))^2)-4)^2)-2)^2+(z*sin(d)+c*cos(d))^2 = 1 • ((((Ac))(Ia))C) - 4x2 array of 8 torii (sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2 + (z*cos(d)-c*sin(d))^2) - 8)^2 + 0^2) - 4)^2 + (sqrt(y^2 + (x*cos(b)-a*sin(b))^2) - 4)^2) - 2)^2 + (z*sin(d)+c*cos(d))^2 = 1 7D ((((II)(II))I)(II)) - Tigritiger , T3xC3 = S1xC2xS1xC2 = S1x[(T2xC2)*S1] , ((((maj1)(maj2)sec)tert)(maj3)min) ------------------------------------------------------------------------------------------------------------------------------------- (sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 + (sqrt(z^2+w^2) -R1b)^2) -R2)^2 +v^2) -R3)^2 + (sqrt(u^2+t^2) -R1c)^2 = Rminor^2 • ((((I)(I)))(I)) : 2x2x2x[R1 pair] array of 16 tori (sqrt((sqrt((sqrt(x^2+0^2) -7.5)^2 + (sqrt(y^2+0^2) -7.5)^2) -3.5)^2 +0^2) -1.5)^2 + (sqrt(z^2+0^2) -3.5)^2 = 1 ———— XYZbox = -17/+17 • ((((A)(Ic))a)(C)) (sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2+0^2) -7.5)^2 + (sqrt(y^2+(z*cos(d)-c*sin(d))^2) -7.5)^2) -3.5)^2 +(x*cos(b)-a*sin(b))^2) -1.5)^2 + (sqrt((z*sin(d)+c*cos(d))^2+0^2) -3.5)^2 = 1 ——— -15 < a,c < 15 ——— 0 < b,d < 1.5707 • ((((Xc)(I))a)(Cb)) (sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2+(z*cos(d) - c*sin(d))^2) -7.5)^2 + (sqrt(y^2+0^2) -7.5)^2) -3.5)^2 +(x*cos(a))^2) -1.5)^2 + (sqrt((z*sin(d) + c*cos(d))^2+(x*cos(b))^2) -3.5)^2 = 1 — -15 < c < 15 — 0 < a,b,d < 1.5707 • ((((X)(Ia))c)(Cb)) (sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2+0^2) -7.5)^2 + (sqrt(y^2+(x*cos(a))^2) -7.5)^2) -3.5)^2 +(z*cos(d) - c*sin(d))^2) -1.5)^2 + (sqrt((z*sin(d) + c*cos(d))^2+(x*cos(a))^2) -3.5)^2 = 1 • ((((A)(Ca))c)(I)) (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+0^2) -7.5)^2 + (sqrt((y*sin(d) + c*cos(d))^2+(x*cos(b) - a*sin(b))^2) -7.5)^2) -3.5)^2 +(y*cos(d) - c*sin(d))^2) -1.5)^2 + (sqrt(z^2+0^2) -3.5)^2 = 1 ——— -17 < a,c < 17 ——— 0 < b,d < 1.5707 • ((((A)(Ca)))(Ic)) ** good function ** (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+0^2) -7.5)^2 + (sqrt((y*sin(d) + c*cos(d))^2+(x*cos(b) - a*sin(b))^2) -7.5)^2) -3.5)^2 +0^2) -1.5)^2 + (sqrt(z^2+(y*cos(d) - c*sin(d))^2) -3.5)^2 = 1 ——— -17 < a,c < 17 ——— 0 < b,d < 1.5707 • ((((A)(C))a)(Ic)) (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+0^2) -7.5)^2 + (sqrt((y*sin(d) + c*cos(d))^2+0^2) -7.5)^2) -3.5)^2 +(x*cos(b) - a*sin(b))^2) -1.5)^2 + (sqrt(z^2+(y*cos(d) - c*sin(d))^2) -3.5)^2 = 1 ——- a=7.5,c=7.5±3.5 for 1x1x2 column of tori in cut (((()())I)(II)) 7D (((II)I)((II)I)I) - Spheritiger Duotorus , S2xC2xC2 = S2x[T2*T2] , (((maj1)med1)((maj2)med2)min) ------------------------------------------------------------------------------------------------------------------------------------- (sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) -R2a)^2 + (sqrt((sqrt(w^2 + v^2) - R1b)^2 + u^2) - R2b)^2 + t^2 - R3^2 = 0 • (((I))((I))I) (sqrt((sqrt(x^2 + a^2) - 2.35)^2 + b^2) -1.15)^2 + (sqrt((sqrt(y^2 + c^2) - 2.6)^2 + d^2) -1.25)^2 + z^2 - 0.5^2 = 0 • (((Xz)y)((Yx)d)Z) (sqrt((sqrt((x*sin(c))^2 + (z*cos(a))^2) - 2.35)^2 + (y*cos(b))^2) -1.15)^2 + (sqrt((sqrt((y*sin(b))^2 + (x*cos(c))^2) - 2.6)^2 + d^2) -1.25)^2 + (z*sin(a))^2 - 0.5^2 = 0 • (((Xy)z)((Yc)x)Z) (sqrt((sqrt((x*sin(d))^2 + (y*cos(a))^2) - 2.35)^2 + (z*cos(b))^2) -1.15)^2 + (sqrt((sqrt((y*sin(a))^2 + c^2) - 2.6)^2 + (x*cos(d))^2) -1.25)^2 + (z*sin(b))^2 - 0.5^2 = 0 • (((Xa)x)((Yz)y)Z) (sqrt((sqrt((x*sin(b))^2 + a^2) - 2.35)^2 + (x*cos(b))^2) -1.15)^2 + (sqrt((sqrt((y*sin(d))^2 + (z*cos(c))^2) - 2.6)^2 + (y*cos(d))^2) -1.25)^2 + (z*sin(c))^2 - 0.5^2 = 0 7D ((((II)I)(II)I)I) - Torispheric Tigritorus , S1xS2xC2xS1 = S1xS2x[T2*S1] , ((((maj1)sec)(maj2)tert)min) ------------------------------------------------------------------------------------------------------------------------------- (sqrt((sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2)^2 + (sqrt(w^2 + v^2) - R1b)^2 + u^2) - R3)^2 + t^2 - R4^2 = 0 • Diameter Adjustment Function (sqrt((sqrt((sqrt(x^2 + 0^2) - a)^2 + 0^2) - b)^2 + (sqrt(y^2 + 0^2) - b)^2 + 0^2) - c)^2 + z^2 = 1 - XYZbox = -22 / +22 - a = 10 - b = 5 - c = 3 • ((((I))(I))I) - 8x torii ((II)I) stacked in 4x2 major rectangle array (sqrt((sqrt((sqrt(x^2 + a^2) - 10)^2 + b^2) - 5)^2 + (sqrt(y^2 + c^2) - 5)^2 + d^2) - 3)^2 + z^2 = 1 • ((((I))(I)I)) - 16x spheres (III) as concentric pair stacked in 4x2 rectangle array (sqrt((sqrt((sqrt(x^2 + a^2) - 10)^2 + b^2) - 5)^2 + (sqrt(y^2 + c^2) - 5)^2 + z^2) - 3)^2 + d^2 = 1 • ((((I))(II))) - 8x torii ((II)I) as concentric minor pair stacked in 1x1x4 minor column (sqrt((sqrt((sqrt(x^2 + a^2) - 10)^2 + b^2) - 5)^2 + (sqrt(y^2 + z^2) - 5)^2 + c^2) - 3)^2 + d^2 = 1 • ((((I)I)(I))) - 8x torii ((II)I) as concentric minor pair stacked in 2x1x2 maj/min square array (sqrt((sqrt((sqrt(x^2 + a^2) - 10)^2 + y^2) - 5)^2 + (sqrt(z^2 + b^2) - 5)^2 + c^2) - 3)^2 + d^2 = 1 • ((((II))(I))) - 8x torii ((II)I) as concentric major/minor pairs stacked in 1x1x2 minor column (sqrt((sqrt((sqrt(x^2 + y^2) - 10)^2 + a^2) - 5)^2 + (sqrt(z^2 + b^2) - 5)^2 + c^2) - 3)^2 + d^2 = 1 • ((((Xy)z)(Yc)x)Z) - triple rotate a,b,d with sliding c (sqrt((sqrt((sqrt((x*sin(d))^2 + (y*cos(a))^2) - 10)^2 + (z*cos(b))^2) - 5)^2 + (sqrt((y*sin(a))^2 + c^2) - 5)^2 + (x*cos(d))^2) - 3)^2 + (z*sin(b))^2 = 1 • ((((Xz)x)(Yi)y)Z) - triple rotate a,b,d with sliding c (sqrt((sqrt((sqrt((x*sin(b))^2 + (z*cos(a))^2) - 10)^2 + (x*cos(b))^2) - 5)^2 + (sqrt((y*sin(d))^2 + c^2) - 5)^2 + (y*cos(d))^2) - 3)^2 + (z*sin(a))^2 = 1 • ((((Xz)y)(Yx)Z)i) - triple rotate a,b,c with sliding d (sqrt((sqrt((sqrt((x*sin(c))^2 + (z*cos(a))^2) - 10)^2 + (y*cos(b))^2) - 5)^2 + (sqrt((y*sin(b))^2 + (x*cos(c))^2) - 5)^2 + (z*sin(a))^2) - 3)^2 + d^2 = 1 • ((((Ai)c)(Ca)i)I) - dual translate + rotate (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 10)^2 + (y*cos(d) - c*sin(d))^2) - 5)^2 + (sqrt((y*sin(d) + c*cos(d))^2 + (x*cos(b) - a*sin(b))^2) - 5)^2 + 0^2) - 3)^2 + z^2 = 1 • ((((Az)y)(Ya)i)Z) - Trans+Rotate A,a / Dual rotate c,d (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (z*cos(d))^2) - 10)^2 + (y*cos(c))^2) - 5)^2 + (sqrt((y*sin(c))^2 + (x*cos(b) - a*sin(b))^2) - 5)^2 + 0^2) - 3)^2 + (z*sin(d))^2 = 1 • ((((Ai)y)(Ya)z)Z) - , Trans+Rotate A,a / Dual rotate c,d (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 10)^2 + (y*cos(c))^2) - 5)^2 + (sqrt((y*sin(c))^2 + (x*cos(b) - a*sin(b))^2) - 5)^2 + (z*cos(d))^2) - 3)^2 + (z*sin(d))^2 = 1 • ((((Ad)y)(Ya)i)I) - , Trans+Rotate A,a / rotate c / translate d (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + d^2) - 10)^2 + (y*cos(c))^2) - 5)^2 + (sqrt((y*sin(c))^2 + (x*cos(b) - a*sin(b))^2) - 5)^2 + 0^2) - 3)^2 + z^2 = 1 7D ((((II)I)((II)I))I) - Toratiger Duotorus , T2xC2xC2 = T2x[T2*T2] , ((((maj1)sec1)((maj2)sec2)tert)min) --------------------------------------------------------------------------------------------------------------- (sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 +z^2) -R2a)^2 + (sqrt((sqrt(w^2+v^2) -R1b)^2 +u^2) -R2b)^2) -R3)^2 +t^2 -R4^2 = 0 • ((((I))((I)))I) - 16 tori in 4x4x1 maj array - Diameter Adjustment Equation (sqrt((sqrt((sqrt(x^2+0^2) -a)^2 +0^2) -b)^2 + (sqrt((sqrt(y^2+0^2) -a)^2 +0^2) -b)^2) -c)^2 +z^2 -d^2 = 0 XYZbox = -22 / +22 - a = 10 - b = 5 - c = 3 - d = 1 • ((((I))((I)))I) - 16 tori in 4x4x1 maj array (sqrt((sqrt((sqrt(x^2+0^2) -10)^2 +0^2) -5)^2 + (sqrt((sqrt(y^2+0^2) -10)^2 +0^2) -5)^2) -3)^2 +z^2 = 1 (sqrt((sqrt((sqrt(x^2+a^2) -10)^2 +b^2) -5)^2 + (sqrt((sqrt(y^2+c^2) -10)^2 +d^2) -5)^2) -3)^2 +z^2 = 1 --- XYZ = -12,+12 • ((((Xy)z)((Yc)x))Z) - ((((I))((I)))I) (sqrt((sqrt((sqrt((x*sin(a))^2+(y*cos(b))^2) -10)^2 +(z*cos(c))^2) -5)^2 + (sqrt((sqrt((y*sin(b))^2+0^2) -10)^2 + (x*cos(a))^2) -5)^2) -3)^2 +(z*sin(c))^2 = 1 • ((((Ac))((C)a))I) - ((((I))((I)))I) --> Explores ((((II)I)((I)))I) , a 1x1x1x4x1 tertiary column of 4x tritoruses (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+(y*cos(d) - c*sin(d))^2) -10)^2) -5)^2 + (sqrt((sqrt((y*sin(d) + c*cos(d))^2) -10)^2 +(x*cos(b) - a*sin(b))^2) -5)^2) -3)^2 +z^2 = 1 • ((((A)a)((C)c))I) - ((((I))((I)))I) (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) -10)^2 +(x*cos(b) - a*sin(b))^2) -5)^2 + (sqrt((sqrt((y*sin(d) + c*cos(d))^2) -10)^2 +(y*cos(d) - c*sin(d))^2) -5)^2) -3)^2 +z^2 = 1 7D ((((II)I)I)((II)I)) - Tiger Duotoric Torus , S1xC2xC2xS1 , ((((R1)R2a)R3a)((R2b)R3b)R4) ---------------------------------------------------------------------------------------------------------------------------------- • ((((II)I)I)((II)I)) (sqrt((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2a)^2 + w^2) - R3a)^2 + (sqrt((sqrt(v^2 + u^2) - R2b)^2 + t^2) - R3b)^2 - R4^2 = 0 (sqrt((sqrt((sqrt(x^2 + y^2) - 4.25)^2 + z^2) - 2)^2 + w^2) - 1)^2 + (sqrt((sqrt(v^2 + u^2) - 2.5)^2 + t^2) - 1.25)^2 - 0.4^2 = 0 --- XYZ= -9,+9 • ((((I)))((I)I)) - 16x Tori ((II)I) in 2x1x8 vert rectangle array (sqrt((sqrt((sqrt(x^2 + 0^2) - 4.25)^2 + 0^2) - 2)^2 + 0^2) - 1)^2 + (sqrt((sqrt(y^2 + 0^2) - 2.5)^2 + z^2) - 1.25)^2 - 0.4^2 = 0 • ((((I)))((II))) - 16x Tori ((II)I) as R1 conc pair in 1x1x8 column (sqrt((sqrt((sqrt(x^2 + 0^2) - 4.25)^2 + 0^2) - 2)^2 + 0^2) - 1)^2 + (sqrt((sqrt(y^2 + z^2) - 2.5)^2 + 0^2) - 1.25)^2 - 0.4^2 = 0 • ((((I))I)((I))) - 16x Tori ((II)I) in 4x1x4 vert square array (sqrt((sqrt((sqrt(x^2 + 0^2) - 4.25)^2 + 0^2) - 2)^2 + y^2) - 1)^2 + (sqrt((sqrt(z^2 + 0^2) - 2.5)^2 + 0^2) - 1.25)^2 - 0.4^2 = 0 • ((((I)I))((I))) - 16x Tori ((II)I) as R1 conc pair in 2x1x4 vert rectangle array (sqrt((sqrt((sqrt(x^2 + 0^2) - 4.25)^2 + y^2) - 2)^2 + 0^2) - 1)^2 + (sqrt((sqrt(z^2 + 0^2) - 2.5)^2 + 0^2) - 1.25)^2 - 0.4^2 = 0 • ((((II)))((I))) - 16x Tori ((II)I) as R1 conc quartet in 1x1x4 column (sqrt((sqrt((sqrt(x^2 + y^2) - 4.25)^2 + 0^2) - 2)^2 + 0^2) - 1)^2 + (sqrt((sqrt(z^2 + 0^2) - 2.5)^2 + 0^2) - 1.25)^2 - 0.4^2 = 0 • ((((IA)a))((C)c)) (sqrt((sqrt((sqrt(x^2 + (y*sin(b) + a*cos(b))^2) - 4)^2 + (y*cos(b) - a*sin(b))^2) - 2)^2 + 0^2) - 1)^2 + (sqrt((sqrt((z*sin(d) + c*cos(d))^2 + 0^2) - 2.5)^2 + (z*cos(d) - c*sin(d))^2) - 1.25)^2 - 0.5^2 = 0 • ((((AC))a)((I)c)) (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*sin(d) + c*cos(d))^2) - 4)^2 + 0^2) - 2)^2 + (x*cos(b) - a*sin(b))^2) - 1)^2 + (sqrt((sqrt(z^2 + 0^2) - 2.5)^2 + (y*cos(d) - c*sin(d))^2) - 1.25)^2 - 0.5^2 = 0 • ((((XY)z)a)((Zb)y)) - new multi-position rotate with [X -> a,b][Y->c][Z->d] (sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2 + (y*sin(c))^2) - 4.25)^2 + (z*cos(d))^2) - 2)^2 + (x*cos(b))^2) - 1)^2 + (sqrt((sqrt((z*sin(d))^2 + (x*cos(a))^2) - 2.5)^2 + (y*cos(c))^2) - 1.25)^2 - 0.4^2 = 0 --- A=1.0635 , B=1.57 , [C,D] => 4x Rotation Cycle , or [C,D] double rotate 0->1.57 --- A=1.0635 , [B,C,D] => 6x Rotation Cycle, has two different cycles, very amazing! --- A,B,C,D = 1.57 => Rotation sequence to 0 and back to 1.57 : [B,C,B,D,A,C,D,A] • ((((XY)y)b)((Zx)a)) [Z->a,b][X->c][Y->d] (sqrt((sqrt((sqrt((x*sin(c))^2 + (y*sin(d))^2) - 4.25)^2 + (y*cos(d))^2) - 2)^2 + (z*cos(b))^2) - 1)^2 + (sqrt((sqrt((z*((sin(a))*(sin(b))))^2 + (x*cos(c))^2) -2.5)^2 + (z*cos(a))^2) - 1.25)^2 - 0.4^2 = 0 --- Very cool with nice alternating midcut sequences • ((((Xa)z)Y)((Zx)b)) [Y->a,b][X->c][Z->d] (sqrt((sqrt((sqrt((x*sin(c))^2 + (y*cos(a))^2) - 4.25)^2 + (z*cos(d))^2) - 2)^2 + (y*((sin(a))*(sin(b))))^2) - 1)^2 + (sqrt((sqrt((z*sin(d))^2 + (x*cos(c))^2) -2.5)^2 + (y*cos(b))^2) - 1.25)^2 - 0.4^2 = 0 --- A,B,C,D=1.57 => Rotation Loop Slider Sequence: [A,C,D,A,B,C,D,B] and [A,C,D,C,B,A,D,B] --- A=0 , B=1.57 => 4x Rotation Loop Sequence [C,D,C,D]=[0->1.57] • ((((Xy)a)b)((YZ)z)) - [X->a,b][Y->c][Z->d] (sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2 + (y*cos(c))^2) - 4.25)^2 + (x*cos(a))^2) - 2)^2 + (x*cos(b))^2) - 1)^2 + (sqrt((sqrt((y*sin(c))^2 + (z*sin(d))^2) -2.5)^2 + (z*cos(d))^2) - 1.25)^2 - 0.4^2 = 0 • ((((Ic)a)b)((C)Z)) - [Z->a,b] , Y->[C-translate+D-rotate] -----> *****AWESOME EXPLORE FUNCTION , DOES A LOT!!!***** (sqrt((sqrt((sqrt(x^2 + (y*cos(d) - c*sin(d))^2) - 4.25)^2 + (z*cos(a))^2) - 2)^2 + (z*cos(b))^2) - 1)^2 + (sqrt((sqrt((y*sin(d) + c*cos(d))^2 + 0^2) - 2.5)^2 + (z*((sin(a))*(sin(b))))^2) - 1.25)^2 - 0.4^2 = 0 ——- XYZ = -9 , +9 ——- a,b,d = 0 ~ 1.57 / c = -10 ~ +10 ——- [a,b] are multi-position rotate : [1.57,1.57] = ((((I)))((I)I)) / [1.57,0] = ((((I))I)((I))) / [0,1.57] = ((((I)I))((I))) / [0,0] = invalid/no solution ——- [c,d] are translate+rotate --- 6-Step D rotation, animate C from -9 -> +9 for scanning the column of 4x tritoruses ((((II)I)I)((I))) --- [a,b] Gives all three cuts of (((II)I)(II)) intercept while in square array ((((I)I)I)((I)I)) --- c=-1.4 , Adjust a,b,d for awesome morphs of translated empty rotations, has triple rotate parameter on 5D tritorus intercept • ((((C)a)b)((Ic)Z)) - X->[c-trans,d-rot] / Z-start->[a-end1 , b-end2] (sqrt((sqrt((sqrt((x*sin(d) + c*cos(d))^2 + 0^2) - 4.25)^2 + (z*cos(a))^2) - 2)^2 + (z*cos(b))^2) - 1)^2 + (sqrt((sqrt(y^2 + (x*cos(d) - c*sin(d))^2) - 2.5)^2 +(z*((sin(a))*(sin(b))))^2) - 1.25)^2 - 0.4^2 = 0 --- Nice mirror equation for alternate explore scans. Does a few more step rotation empty scans • ((((A)y)z)((Ya)Z)) : X->[a,b] , Y->c , Z->d (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) - 4.0)^2 + (y*cos(c))^2) - 2)^2 + (z*cos(d))^2) - 1)^2 + (sqrt((sqrt((y*sin(c))^2 + (x*cos(b) - a*sin(b))^2) - 2.5)^2 + (z*sin(d))^2) - 1.25)^2 - 0.4^2 = 0 • ((((Xc))I)((Ca)b)) : X->[a,b] , Z->[c+d] (sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2 + (z*cos(d) - c*sin(d))^2) - 4)^2) - 2)^2 + y^2) - 1)^2 + (sqrt((sqrt((z*sin(d) + c*cos(d))^2 + (x*cos(a))^2) - 2.5)^2 +(x*cos(b))^2) - 1.25)^2 - 0.4^2 = 0 - more empty scans not seen before, many empty rotates • ((((Xc)a)b)((CI))) : X->[a,b] , Y->[c+d] (sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2 + (y*cos(d) - c*sin(d))^2) - 4)^2 + (x*cos(a))^2) - 2)^2 + (x*cos(b))^2) - 1)^2 + (sqrt((sqrt((y*sin(d) + c*cos(d))^2 + z^2) - 2.5)^2 + 0^2) - 1.25)^2 - 0.4^2 = 0 • ((((A)c))((Ca)I)) : Dual Trans+Rotate (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2) - 4.0)^2 + (y*cos(d)-c*sin(d))^2) - 2)^2 + 0^2) - 1)^2 + (sqrt((sqrt((y*sin(d)+c*cos(d))^2 + (x*cos(b) - a*sin(b))^2) - 2.5)^2 + z^2) - 1.25)^2 - 0.4^2 = 0 • ((((IY)a)b)((Ic)d)) : Y -> a,b,c,d ; Five Position Rotate (sqrt((sqrt((sqrt(x^2 + (y*((sin(a))*(sin(b))*(sin(c))*(sin(d))))^2) - 4.25)^2 + (y*cos(a))^2) - 2)^2 + (y*cos(b))^2) - 1)^2 + (sqrt((sqrt(z^2 + (y*cos(c))^2) - 2.5)^2 + (y*cos(d))^2) - 1.25)^2 - 0.4^2 = 0 7D ((((II)I)(II))(II)) - Double Tiger R1A-Torus , T3xC3 / T2xC3xS1 --------------------------------------------------------------------------------------------------------------------------- Diameter Size Hierarchy : ((((R1)R2a)(R2b)R3a)(R3b)R4) (sqrt((sqrt((sqrt(x^2 + y^2) -R1)^2 + z^2) -R2a)^2 + (sqrt(w^2 + v^2) -R2b)^2) -R3a)^2 + (sqrt(u^2 + t^2) -R3b)^2 -R4^2 = 0 R1 = 4 / R2 = 2 / R3 = 1.2 / R4 = 0.5 (sqrt((sqrt((sqrt(x^2 + y^2) -4)^2 + z^2) -2)^2 + (sqrt(w^2 + v^2) -2)^2) -1.2)^2 + (sqrt(u^2 + t^2) -1.2)^2 -0.5^2 = 0 • ((((a)b)(b)c)(c)d) - Diamter Adjustment Function (sqrt((sqrt((sqrt(x^2 + 0^2) -a)^2 + 0^2) -b)^2 + (sqrt(y^2 + 0^2) -a)^2) -c)^2 + (sqrt(z^2 + 0^2) -a)^2 -d^2 = 0 • ((((I))(I))(I)) (sqrt((sqrt((sqrt(x^2 + 0^2) -4)^2 + 0^2) -2)^2 + (sqrt(y^2 + 0^2) -2)^2) -1.2)^2 + (sqrt(z^2 + 0^2) -1.2)^2 -0.5^2 = 0 — XYZ = -8 , +8 (sqrt((sqrt((sqrt(x^2 + a^2) -4)^2 + b^2) -2)^2 + (sqrt(y^2 + c^2) -2)^2) -1.2)^2 + (sqrt(z^2 + d^2) -1.2)^2 -0.5^2 = 0 • ((((X)a)(Ib))(Ic)) : X,a,b,c : Four Position Rotate Function from R1a Void (sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))*(sin(c))))^2 + 0^2) -4)^2 + (x*cos(a))^2) -2)^2 + (sqrt(y^2 + (x*cos(b))^2) -2)^2) -1.2)^2 + (sqrt(z^2 + (x*cos(c))^2) -1.2)^2 -0.5^2 = 0 • ((((Ac)d)(I))(Za)): TransRot[A-{b}-a] MPosRot[Z,c,d] (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (z*cos(c))^2) -4)^2 + (z*cos(d))^2) -2)^2 + (sqrt(y^2 + 0^2) -2)^2) -1.2)^2 + (sqrt((z*((sin(c))*(sin(d))))^2 + (x*cos(b) - a*sin(b))^2) -1.2)^2 -0.5^2 = 0 --- Has interesting empty cut scans, new morphs not seen before. Strong correlation to specific void. • ((((Ac))(Ya))(Id)) : X{A,a} Y,c,d (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*cos(c))^2) -4)^2 + 0^2) -2)^2 + (sqrt((y*((sin(c))*(sin(d))))^2 + (x*cos(b) - a*sin(b))^2) -2)^2) -1.2)^2 + (sqrt(z^2 + (y*cos(d))^2) -1.2)^2 -0.5^2 = 0 --- Second type like above ((((Ac)d)(I))(Za)) , scans past 4x tiger toruses as column of 4 tori • ((((XD)a)(Ib))(Ic)) : [X,a,b,c] [D-trans] (sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))*(sin(c))))^2 + d^2) -4)^2 + (x*cos(a))^2) -2)^2 + (sqrt(y^2 + (x*cos(b))^2) -2)^2) -1.2)^2 + (sqrt(z^2 + (x*cos(c))^2) -1.2)^2 -0.5^2 = 0 • ((((X)a)(Cb))(Ic)) : [X,a,b] Y[C,c] (sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2 + 0^2) -4)^2 + (x*cos(a))^2) -2)^2 + (sqrt((y*sin(d) + c*cos(d))^2 + (x*cos(b))^2) -2)^2) -1.2)^2 + (sqrt(z^2 + (y*cos(d) - c*sin(d))^2) -1.2)^2 -0.5^2 = 0 • ((((I)d)(Ac))(Za)) : Y[A,a] [Z,c,d] (sqrt((sqrt((sqrt(x^2 + 0^2) -4)^2 + (z*cos(d))^2) -2)^2 + (sqrt((y*sin(b) + a*cos(b))^2 + (z*cos(c))^2) -2)^2) -1.2)^2 + (sqrt((z*((sin(c))*(sin(d))))^2 + (y*cos(b) - a*sin(b))^2) -1.2)^2 -0.5^2 = 0 • ((((Id)a)(Ac))(Z)) : Y[A,a] [Z,c,d] (sqrt((sqrt((sqrt(x^2 + (z*cos(d))^2) -4)^2 + (y*cos(b) - a*sin(b))^2) -2)^2 + (sqrt((y*sin(b) + a*cos(b))^2 + (z*cos(c))^2) -2)^2) -1.2)^2 + (sqrt((z*((sin(c))*(sin(d))))^2 + 0^2) -1.2)^2 -0.5^2 = 0 • ((((I)A)(ac))(C)) : Y[A,a] Z[C,c] (sqrt((sqrt((sqrt(x^2 + 0^2) -4)^2 + (y*sin(b) + a*cos(b))^2) -2)^2 + (sqrt((y*cos(b) - a*sin(b))^2 + (z*cos(d) - c*sin(d))^2) -2)^2) -1.2)^2 + (sqrt((z*sin(d) + c*cos(d))^2 + 0^2) -1.2)^2 -0.5^2 = 0 --- Set C=1 , Scan B [0~1.57] does 1x Toratiger Torus morph of ((((I)y)(YI))) • ((((AC))(a))(Ic)) : X[A,a] Y[C,c] (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (y*sin(d) + c*cos(d))^2) -4)^2 + 0^2) -2)^2 + (sqrt((x*cos(b) - a*sin(b))^2 + 0^2) -2)^2) -1.2)^2 + (sqrt(z^2 + (y*cos(d) - c*sin(d))^2) -1.2)^2 -0.5^2 = 0 --- Cool Rotation scans, Translate Step-Scans • ((((XC)a)(Ic))(b)) : X,a,b C,c : MPR + T/R (sqrt((sqrt((sqrt((x*((sin(a))*(sin(b))))^2 + (y*sin(d) + c*cos(d))^2) -4)^2 + (x*cos(a))^2) -2)^2 + (sqrt(z^2 + (y*cos(d) - c*sin(d))^2) -2)^2) -1.2)^2 + (sqrt((x*cos(b))^2 + 0^2) -1.2)^2 -0.5^2 = 0 --- Still needs more surveying, unsure of scan morphs • ((((Ac)d)(Ia))(Z)) : TransRot[A-{b}-a] MPosRot[Z,c,d] (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (z*cos(c))^2) -4)^2 + (z*cos(d))^2) -2)^2 + (sqrt(y^2 + (x*cos(b) - a*sin(b))^2) -2)^2) -1.2)^2 + (sqrt((z*((sin(c))*(sin(d))))^2 + 0^2) -1.2)^2 -0.5^2 = 0 --- interesting rotation on parameter b • ((((A)c)(Id))(Za)) ***** Very Good Explore Function ***** (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) -4.8)^2 + (z*cos(c))^2) -2.4)^2 + (sqrt(y^2 + (z*cos(d))^2) -3.25)^2) -1.6)^2 + (sqrt((z*((sin(c))*(sin(d))))^2 + (x*cos(b) - a*sin(b))^2) -1.6)^2 -0.58^2 = 0 ——- XYZ box to -10,+10 --- Good linking with Z rotations and A scans • ((((Ic)d)(A))(Za)) : Y[A,a] [Z,c,d] ***** Very Good Explore Function ***** (sqrt((sqrt((sqrt(x^2 + (z*cos(c))^2) -4.8)^2 + (z*cos(d))^2) -2.4)^2 + (sqrt((y*sin(b) + a*cos(b))^2 + 0^2) -3.25)^2) -1.6)^2 + (sqrt((z*((sin(c))*(sin(d))))^2 + (y*cos(b) - a*sin(b))^2) -1.6)^2 -0.58^2 = 0 --- A,B,C,D = [2.4,0,0,1.57] : rotations to 6D ring intercept of ((((II)I)I)(II)) in empty cut ((((II))())(I)) --- A,B,C,D = [1.6,1.57,0,1.57] : rotations to 6D ring intercept ((((II)I)(II))I) in empty cut ((((II))(I))()) 7D (((II)(II))((II)I)) - Double Tiger 1C-Torus , T2xC3xS1 , S1xC2xS1xC2 ---------------------------------------------------------------------------------------------------- • (((II)(II))((II)I)) - (((R1A)(R1B)R2)((R1C)R3)R4) (sqrt((sqrt(x^2 + y^2) -R1a)^2 + (sqrt(z^2 + w^2) -R1b)^2) -R3)^2 + (sqrt((sqrt(v^2 + u^2) -R1c)^2 + t^2) -R2)^2 -R4^2 = 0 R1 = 4.25 / R2 = 2.125 / R3 = 2.55 / R4 = 0.75 XYZ = -10 , +10 • (((a)(a)c)((a)b)d) - Diameter Adjustment Function (sqrt((sqrt(x^2 + 0^2) -a)^2 + (sqrt(y^2 + 0^2) -a)^2) -c)^2 + (sqrt((sqrt(z^2 + 0^2) -a)^2 + 0^2) -b)^2 -d^2 = 0 --- D,A,B,C seq for neat build up by diameters from 0 • (((I)(I))((I))) - 16x Tori ((II)I) in 2x2x4 vertical tower (sqrt((sqrt(x^2 + 0^2) -4.25)^2 + (sqrt(y^2 + 0^2) -4.25)^2) -2.55)^2 + (sqrt((sqrt(z^2 + 0^2) -4.25)^2 + 0^2) -2.125)^2 -0.75^2 = 0 • (((Ia)(Ib))((Ic)d)) - 4-Axis Translate out of 3D (sqrt((sqrt(x^2 + a^2) -4.25)^2 + (sqrt(y^2 + b^2) -4.25)^2) -2.55)^2 + (sqrt((sqrt(z^2 + c^2) -4.25)^2 + d^2) -2.125)^2 -0.75^2 = 0 • (((A)(Ca))((I)c)) - X[A,a] Y[C,c] (sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) -4.25)^2 + (sqrt((y*sin(d) + c*cos(d))^2 + (x*cos(b) - a*sin(b))^2) -4.25)^2) -2.55)^2 + (sqrt((sqrt(z^2 + 0^2) -4.25)^2 + (y*cos(d) - c*sin(d))^2) -2.125)^2 -0.75^2 = 0 • (((Ac)(I))((Za)d)) : X[A,a] [Z,c,d] (sqrt((sqrt((x*sin(b) + a*cos(b))^2 + (z*cos(c))^2) -4.25)^2 + (sqrt(y^2 + 0^2) -4.25)^2) -2.55)^2 + (sqrt((sqrt((z*((sin(c))*(sin(d))))^2 + (x*cos(b) - a*sin(b))^2) -4.25)^2 + (z*cos(d))^2) -2.125)^2 -0.75^2 = 0 --- Very good explore function, notable scans of empties, plus a 4x2 of 8 tigers ETE scan • (((A)(Ic))((Za)d)) : X[A,a] [Z,c,d] (sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) -4.25)^2 + (sqrt(y^2 + (z*cos(c))^2) -4.25)^2) -2.55)^2 + (sqrt((sqrt((z*((sin(c))*(sin(d))))^2 + (x*cos(b) - a*sin(b))^2) -4.25)^2 + (z*cos(d))^2) -2.125)^2 -0.75^2 = 0 --- Nice alt function to (((Ac)(I))((Za)d)) , has a few new empty scans • (((Xc)(I))((Ca)b)) (sqrt((sqrt((x*((sin(a))*(sin(b))))^2 + (z*cos(d) - c*sin(d))^2) -4.25)^2 + (sqrt(y^2 + 0^2) -4.25)^2) -2.55)^2 + (sqrt((sqrt((z*sin(d) + c*cos(d))^2 + (x*cos(a))^2) -4.25)^2 + (x*cos(b))^2) -2.125)^2 -0.75^2 = 0 --- Use A,B to change scan C of step rotation D --- A,B = [1.57,0] / Step Rotate D 0 to 1.57 with scanning C -12 to +12 : Very nice 2x2x2 cube array of 8 tigers scan, ETE rotation • (((Xc)(Ib))((Ca))) (sqrt((sqrt((x*((sin(a))*(sin(b))))^2 + (z*cos(d) - c*sin(d))^2) -4.25)^2 + (sqrt(y^2 + (x*cos(b))^2) -4.25)^2) -2.55)^2 + (sqrt((sqrt((z*sin(d) + c*cos(d))^2 + (x*cos(a))^2) -4.25)^2 + 0^2) -2.125)^2 -0.75^2 = 0 --- Use A,B to change scan C of step rotation D --- Has some neat 4-cycle sequences 7D (((((II)(II))I)I)I) : Tritoratiger : T4xC2 , (((((R1a)(R1b)R2)R3)R4)minor) ------------------------------------------------------------------------------------------------------------------------ (sqrt((sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 + (sqrt(z^2+w^2) -R1b)^2) -R2)^2 +v^2) -R3)^2 +u^2) -R4)^2 +t^2 = Rminor^2 • Diameter Adjustment Equation (sqrt((sqrt((sqrt((sqrt(x^2+0^2) -a)^2 + (sqrt(y^2+0^2) -a)^2) -b)^2 +0^2) -c)^2 +0^2) -d)^2 +z^2 = 1 • (((((I)(I))))I) - 2x2x[R1quartet] array of 16 tori (sqrt((sqrt((sqrt((sqrt(x^2+0^2) -15)^2 + (sqrt(y^2+0^2) -15)^2) -7)^2 +0^2) -3.2)^2 +0^2) -1.6)^2 +z^2 = 1 —— XYZbox = -35 / +35 —— 55 cubes • (((((A)(Ia))c)d)Z) : X -> A,a / Z -> c,d (sqrt((sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+0^2) -15)^2 + (sqrt(y^2+(x*cos(b) - a*sin(b))^2) -15)^2) -7)^2 +(z*cos(c))^2) -3.2)^2 +(z*cos(d))^2) -1.6)^2 +(z*((sin(c))*(sin(d))))^2 = 1 —— -35 < a < 35 —— 0 < b,c,d < 1.5707 • (((((Xc)(Ix))b)a)Z) : X -> x / Z -> a,b,c (sqrt((sqrt((sqrt((sqrt((x*sin(d))^2+(z*cos(c))^2) -15)^2 + (sqrt(y^2+(x*cos(d))^2) -15)^2) -7)^2 +(z*cos(b))^2) -3.2)^2 +(z*cos(a))^2) -1.6)^2 +(z*((sin(a))*(sin(b))*(sin(c))))^2 = 1 —— 0 < a,b,c,d < 1.5707 7D (((((II)I)(II))I)I) : Ditoratiger Torus, T3xC2xS1 --------------------------------------------------------- (sqrt((sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 +z^2) -R2)^2 + (sqrt(w^2+v^2) -R1b)^2) -R3)^2 +u^2) -R4)^2 +t^2 = Rminor^2 • (((((I))(I)))I) : 16x Tori in 4x2x[R1 quartet] (sqrt((sqrt((sqrt((sqrt(x^2+0^2) -14)^2 +0^2) -7)^2 + (sqrt(y^2+0^2) -7)^2) -3.5)^2 +0^2) -1.75)^2 +z^2 = 1 • (((((I))(I))I)) : 16x Tori in 4x2x[R1 pair]x[Rm pair] (sqrt((sqrt((sqrt((sqrt(x^2+0^2) -14)^2 +0^2) -7)^2 + (sqrt(y^2+0^2) -7)^2) -3.5)^2 +z^2) -1.75)^2 +0^2 = 1 • (((((I))(II)))) : 16x Tori in 1x1x4x[Rm quartet] (sqrt((sqrt((sqrt((sqrt(x^2+0^2) -14)^2 +0^2) -7)^2 + (sqrt(y^2+z^2) -7)^2) -3.5)^2 +0^2) -1.75)^2 +0^2 = 1 • (((((I)I)(I)))) : 16x Tori in 2x1x2x[Rm quartet] (sqrt((sqrt((sqrt((sqrt(x^2+0^2) -14)^2 +y^2) -7)^2 + (sqrt(z^2+0^2) -7)^2) -3.5)^2 +0^2) -1.75)^2 +0^2 = 1 • (((((II))(I)))) : 16x Tori in 1x1x2x[R1 pair]x[Rm quartet] (sqrt((sqrt((sqrt((sqrt(x^2+y^2) -14)^2 +0^2) -7)^2 + (sqrt(z^2+0^2) -7)^2) -3.5)^2 +0^2) -1.75)^2 +0^2 = 1 • Diameter Adjustment Equation of (((((I))(I)))I) (sqrt((sqrt((sqrt((sqrt(x^2+0^2) -a)^2 +0^2) -b)^2 + (sqrt(y^2+0^2) -b)^2) -c)^2 +0^2) -d)^2 +z^2 = 1 a=14 ; b=7 ; c=3.5 ; d=1.75 --- XYZbox = -30,+30 8D ((((II)I)I)(((II)I)I)) : Duoditorus Tiger , S1xC2xC2xC2 ------------------------------------------------------------------------------- (sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 +z^2) -R2a)^2 +w^2) -R3a)^2 + (sqrt((sqrt((sqrt(v^2+u^2) -R1b)^2 +t^2) -R2b)^2 +s^2) -R3b)^2 = Rminor^2 (sqrt((sqrt((sqrt(x^2+y^2) -10)^2 +z^2) -5)^2 +w^2) -2.5)^2 + (sqrt((sqrt((sqrt(v^2+u^2) -10)^2 +t^2) -5)^2 +s^2) -2.5)^2 = 0.5 • ((((II)))(((I)))) (sqrt((sqrt((sqrt(x^2+y^2) -10)^2 +0^2) -5)^2 +0^2) -2.5)^2 + (sqrt((sqrt((sqrt(z^2+0^2) -10)^2 +0^2) -5)^2 +0^2) -2.5)^2 = 0.5 — XYZbox = -23 / +23 -- 40 cubes • ((((IA)))(((Ia)))) : y -> A,a (sqrt((sqrt((sqrt(x^2+(y*sin(b) + a*cos(b))^2) -10)^2 +0^2) -5)^2 +0^2) -2.5)^2 + (sqrt((sqrt((sqrt(z^2+(y*cos(b) - a*sin(b))^2) -10)^2 +0^2) -5)^2 +0^2) -2.5)^2 = 0.5 • ((((AY)c)d)(((Ia)))) (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+(y*((sin(c))*(sin(d))))^2) -10)^2 +(y*cos(c))^2) -5)^2 +(y*cos(d))^2) -2.5)^2 + (sqrt((sqrt((sqrt(z^2+(x*cos(b) - a*sin(b))^2) -10)^2 +0^2) -5)^2 +0^2) -2.5)^2 = 1 • ((((Ac)Y)d)(((Ia)))) (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+(y*cos(c))^2) -10)^2 +(y*((sin(c))*(sin(d))))^2) -5)^2 +(y*cos(d))^2) -2.5)^2 + (sqrt((sqrt((sqrt(z^2+(x*cos(b) - a*sin(b))^2) -10)^2 +0^2) -5)^2 +0^2) -2.5)^2 = 1 • ((((IY)a)b)(((Ic)d))) (sqrt((sqrt((sqrt(x^2+(y*((sin(a))*(sin(b))*(sin(c))*(sin(d))))^2) -10)^2 +(y*cos(a))^2) -5)^2 +(y*cos(b))^2) -2.5)^2 + (sqrt((sqrt((sqrt(z^2+(y*cos(c))^2) -10)^2 +(y*cos(d))^2) -5)^2 +0^2) -2.5)^2 = 1 8D ((II)(II)(II)(II)) - Tetratiger , S3xC4 , ((maj1)(maj2)(maj3)(maj4)min) ------------------------------------------------------------------------------------------------------------------------ (sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 + (sqrt(v^2 + u^2) - R1c)^2 + (sqrt(t^2 + s^2) - R1d)^2- R2^2 = 0 • ((I)(I)(I)(I)) - 2x2x2x2 array of 16 glomes (sqrt(x^2 + a^2) - 2.5)^2 + (sqrt(y^2 + b^2) - 2.5)^2 + (sqrt(z^2 + c^2) - 2.5)^2 + (sqrt(d^2 + 0^2) - 2.5)^2 - 1^2 = 0 • ((Ia)(Ib)(I)(Cd)) (sqrt(x^2 + a^2) - 2.5)^2 + (sqrt(y^2 + b^2) - 2.5)^2 + (sqrt(z^2 + 0^2) - 2.5)^2 + (sqrt(c^2 + d^2) - 2.5)^2 - 1^2 = 0 -- Adjusting C translates away from center of 2x2x2x2 array of glomes, passing by either +W or -W 2x2x2 cube array -- Adjusting D controls merging of +W,-W 2x2x2 arrays in 4-space • ((Xc)(Yz)(Zx)(Cy)) (sqrt((x*sin(0))^2 + (c*cos(a))^2) - 2.5)^2 + (sqrt((y*sin(d))^2 + (z*cos(b))^2) - 2.5)^2 + (sqrt((z*sin(b))^2 + (x*cos(0))^2) - 2.5)^2 + (sqrt((c*sin(a))^2 + (y*cos(d))^2) - 2.5)^2 - 1^2 = 0 -- Set C=3.5 for only visible morphings. • ((Xc)(Yx)(Zy)(Cz)) (sqrt((x*sin(b))^2 + (c*cos(a))^2) - 2.5)^2 + (sqrt((y*sin(0))^2 + (x*cos(b))^2) - 2.5)^2 + (sqrt((z*sin(d))^2 + (y*cos(0))^2) - 2.5)^2 + (sqrt((c*sin(a))^2 + (z*cos(d))^2) - 2.5)^2 - 1^2 = 0 -- Adjusting B scans along diagonal 2x2x2 array, max illum at 0.785/45deg -- Adjusting C translates away fr center of 2x2x2x2 array, max illum at +2.5,-2.5 in 4-space • ((I)(A)(Ca)(c)) - Dual Translate+Rotate of tesseract array of hyperspheres; Explores a 2x2x2x2 array of 16 S^3’s (sqrt(x^2)-3)^2+(sqrt((y*sin(b)+a*cos(b))^2)-3)^2+(sqrt((z*sin(d)+c*cos(d))^2+(y*cos(b)-a*sin(b))^2)-3)^2+(sqrt((z*cos(d)-c*sin(d))^2)-3)^2 = 1 8D ((((II)(II))I)((II)I)) : [Toratiger-Torus] Tiger , S1xC2xS1xS1xC2 , T2xC2xC3 , S1xC2xS1xC3 ------------------------------------------------------------------------------------------------------------------------ (sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 + (sqrt(z^2+w^2) -R1b)^2) -R3)^2 +v^2) -R4)^2 + (sqrt((sqrt(u^2+t^2) -R1c)^2 +s^2) -R2)^2 = Rm^2 • Trace Array Diameter Adjustment Equation (sqrt((sqrt((sqrt(x^2+0^2) -a)^2 + (sqrt(y^2+0^2) -a)^2) -c)^2 +0^2) -d)^2 + (sqrt((sqrt(z^2+0^2) -b)^2 +0^2) -(b/2))^2 = 1 • ((((I)(I)))((I))) : 2x2x4x[R1 pair] array of 32 toruses (sqrt((sqrt((sqrt(x^2+0^2) -9)^2 + (sqrt(y^2+0^2) -9)^2) -4.5)^2 +0^2) -2)^2 + (sqrt((sqrt(z^2+0^2) -6.5)^2 +0^2) -3.25)^2 = 1 —— XYZbox = -22 / +22 • ((((I)(I)))((I))) : standard explore function slate (sqrt((sqrt((sqrt(x^2+0^2) -9)^2 + (sqrt(y^2+0^2) -9)^2) -4.5)^2 +0^2) -2)^2 + (sqrt((sqrt(z^2+0^2) -6.5)^2 +0^2) -3.25)^2 = 1 • ((((A)(Ca)))((Ic))) : X -> A,a / Y -> C,c (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+0^2) -9)^2 + (sqrt((y*sin(d) + c*cos(d))^2+(x*cos(b) - a*sin(b))^2) -9)^2) -4.5)^2 +0^2) -2)^2 + (sqrt((sqrt(z^2+(y*cos(d) - c*sin(d))^2) -6.5)^2 +0^2) -3.25)^2 = 1 —— -22 < a < 22 / 0 < b,d < 1.5707 / -20 < c < 20 • ((((Ic)(A))a)((C))) : Y -> A,a / Z -> C,c (sqrt((sqrt((sqrt(x^2+(z*cos(d) - c*sin(d))^2) -9)^2 + (sqrt((y*sin(b) + a*cos(b))^2+0^2) -9)^2) -4.5)^2 +(y*cos(b) - a*sin(b))^2) -2)^2 + (sqrt((sqrt((z*sin(d) + c*cos(d))^2+0^2) -6.5)^2 +0^2) -3.25)^2 = 1 —— -22 < a < 22 / 0 < b,d < 1.5707 / -20 < c < 20 • ((((A)(I))c)((C)a)) : X -> A,a / Z -> C,c (sqrt((sqrt((sqrt((x*sin(b) + a*cos(b))^2+0^2) -9)^2 + (sqrt(y^2+0^2) -9)^2) -4.5)^2 +(z*cos(d) - c*sin(d))^2) -2)^2 + (sqrt((sqrt((z*sin(d) + c*cos(d))^2+0^2) -6.5)^2 +(x*cos(b) - a*sin(b))^2) -3.25)^2 = 1 8D ((((II)I)(II))((II)I)) - Tigritorus Duotoric Torus , T2xC2xC3 = S1x[(S1xC2xS1)*T2] , ((((maj1)sec1)(sec2)tert)((maj2)sec3)min) --------------------------------------------------------------------------------------------------------------------------------------------------------------- ((sqrt((sqrt(x^2 + y^2) - R1a)^2 + z^2) - R2a)^2 + (sqrt(w^2 + v^2) - R2b)^2 - R3)^2 + (sqrt((sqrt(u^2 + t^2) - R1b)^2 + s^2) - R2c)^2 - R4^2 = 0 • ((((I))(I))((I))) ((sqrt((sqrt(x^2 + a^2) - 5)^2 + 0^2) - 2.25)^2 + (sqrt(y^2 + b^2) - 2.25)^2 - 1.8)^2 + (sqrt((sqrt(z^2 + c^2) - 2)^2 + d^2) - 1)^2 - 0.75^2 = 0 9D (((II)I)((II)I)((II)I)) - Triger Triotorus , S2xC3xC3 = S2x[T2*T2*T2] , (((maj1)med1)((maj2)med2)((maj3)med3)min) ------------------------------------------------------------------------------------------------------------------------------------------------------------ (sqrt((sqrt(x^2+y^2) -R1a)^2 +z^2) -R2a)^2 + (sqrt((sqrt(w^2+v^2) -R1b)^2 +u^2) -R2b)^2 + (sqrt((sqrt(t^2+s^2) -R1c)^2 +r^2) -R2c)^2 = R3^2 • (((I))((I))((I))) (sqrt((sqrt(x^2 + a^2) - 2)^2 + 0^2) -1)^2 + (sqrt((sqrt(y^2 + b^2) - 2)^2 + 0^2) - 1)^2 + (sqrt((sqrt(z^2 + c^2) - 2)^2 + d^2) - 1)^2 - 0.5^2 = 0 • (((A))((Ca)c)((I))) - Dual Trans+Rotate (sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 4)^2 + 0^2) -2)^2 + (sqrt((sqrt((y*sin(d) + c*cos(d))^2 + (x*cos(b) - a*sin(b))^2) - 4)^2 + (y*cos(d) - c*sin(d))^2) - 2)^2 + (sqrt((sqrt(z^2 + 0^2) - 4)^2 + 0^2) - 2)^2 = 1 • (((A)c)((Ia))((c))) - Dual Trans+Rotate (sqrt((sqrt((x*sin(b) + a*cos(b))^2 + 0^2) - 4)^2 + (z*cos(d) - c*sin(d))^2) -2)^2 + (sqrt((sqrt(y^2 + (x*cos(b) - a*sin(b))^2) - 4)^2 + 0^2) - 2)^2 + (sqrt((sqrt((z*sin(d) + c*cos(d))^2 + 0^2) - 4)^2 + 0^2) - 2)^2 = 1 9D (((((II)I)(II))I)((II)I)) - [ToratigerTorus-Torus] Tiger , T3xC2xC3 = S1xC2xT2xC2xS1 --------------------------------------------------------------------------------------------- (sqrt((sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 +z^2) -R2a)^2 + (sqrt(w^2+v^2) -R1b)^2) -R3)^2 +u^2) -R4)^2 + (sqrt((sqrt(t^2+s^2) -R1c)^2 +r^2) -R2b)^2 = Rminor^2 • Diameter Adjustment Equation for Trace Array (sqrt((sqrt((sqrt((sqrt(x^2+0^2) -a)^2 +0^2) -(a/2))^2 + (sqrt(y^2+0^2) -(2a/3))^2) -c)^2 +0^2) -d)^2 + (sqrt((sqrt(z^2+0^2) - b)^2 +0^2) - (b/2))^2 = 1 --- a=15 --- b=7 --- c=4 --- d=1.75 • (((((I))(I)))((I))) : 4x2x4x[R1 pair] of 64 Toruses (sqrt((sqrt((sqrt((sqrt(x^2+0^2) -15)^2 +0^2) -7.5)^2 + (sqrt(y^2+0^2) -7.5)^2) -4)^2 +0^2) -1.75)^2 + (sqrt((sqrt(z^2+0^2) -7)^2 +0^2) -3.5)^2 = 1 —— XYZbox = -32 / +32 —— 55 cubes • (((((I))(I)))((I))) : Exploratory Function Template (sqrt((sqrt((sqrt((sqrt(x^2+0^2) -15)^2 +0^2) -7.5)^2 + (sqrt(y^2+0^2) -7.5)^2) -4)^2 +0^2) -1.75)^2 + (sqrt((sqrt(z^2+0^2) -7)^2 +0^2) -3.5)^2 = 1 • (((((A)c)(Ca)))((I))) : Dual Translate+Rotate (sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2)-15)^2+(y*cos(d)-c*sin(d))^2)-7.5)^2+(sqrt((y*sin(d)+c*cos(d))^2+(x*cos(b)-a*sin(b))^2)-7.5)^2)-4)^2)-1.75)^2+(sqrt((sqrt(z^2)-7)^2)-3.5)^2 = 1 --- -30 < a < 30 ; -20 < c < 20 ; 0 < b,d < 1.57 • (((((A))(C))c)((Ia))) : Dual Translate+Rotate (sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2)-15)^2)-7.5)^2+(sqrt((y*sin(d)+c*cos(d))^2)-7.5)^2)-4)^2+(y*cos(d)-c*sin(d))^2)-1.75)^2+(sqrt((sqrt(z^2+(x*cos(b)-a*sin(b))^2)-7)^2)-3.5)^2 = 1 ((((())(I)))((II))) : [±R1A±R2] Intercepts are 4 places of ((((I)))((II))) - 1x1x8x[R1 pair] of 16 tori • (((((A))(C))a)((I)c)) : Dual Translate+Rotate (sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2)-15)^2)-7.5)^2+(sqrt((y*sin(d)+c*cos(d))^2)-7.5)^2)-4)^2+(x*cos(b)-a*sin(b))^2)-1.75)^2+(sqrt((sqrt(z^2)-7)^2+(y*cos(d)-c*sin(d))^2)-3.5)^2 = 1 • (((((A))(Ca)))((Ic))) : Dual Translate+Rotate (sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2)-15)^2)-7.5)^2+(sqrt((y*sin(d)+c*cos(d))^2+(x*cos(b)-a*sin(b))^2)-7.5)^2)-4)^2)-1.75)^2+(sqrt((sqrt(z^2+(y*cos(d)-c*sin(d))^2)-7)^2)-3.5)^2 = 1 • (((((A))(C)))((Ia)c)) : Dual Translate+Rotate (sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2)-15)^2)-7.5)^2+(sqrt((y*sin(d)+c*cos(d))^2)-7.5)^2)-4)^2)-1.75)^2+(sqrt((sqrt(z^2+(x*cos(b)-a*sin(b))^2)-7)^2+(y*cos(d)-c*sin(d))^2)-3.5)^2 = 1 • (((((A))(C))c)((a)I)) : Dual Translate+Rotate; explores hypervoid regions {±R1A±R2} x {±R1B} x {±R1C} (sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2)-15)^2)-7.5)^2+(sqrt((y*sin(d)+c*cos(d))^2)-7.5)^2)-4)^2+(y*cos(d)-c*sin(d))^2)-1.75)^2+(sqrt((sqrt((x*cos(b)-a*sin(b))^2)-7)^2+z^2)-3.5)^2 = 1 • (((((A))(C))a)((c)I)) : Dual Translate+Rotate; explores hypervoid regions {±R1A±R2} x {±R1B} x {±R1C} (sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2)-15)^2)-7.5)^2+(sqrt((y*sin(d)+c*cos(d))^2)-7.5)^2)-4)^2+(x*cos(b)-a*sin(b))^2)-1.75)^2+(sqrt((sqrt((y*cos(d)-c*sin(d))^2)-7)^2+z^2)-3.5)^2 = 1 9D ((((((II)I)(II))I)I)(II)) : [DitoratigerTorus-Circle] Tiger , S1xC2xT2xC2xS1 ---------------------------------------------------------------------------------------- (sqrt((sqrt((sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 +z^2) -R2)^2 + (sqrt(w^2+v^2) -R1b)^2) -R3)^2 +u^2) -R4)^2 +t^2) -R5)^2 + (sqrt(s^2+r^2) -R1c)^2 = Rminor^2 • ((((((I))(I))))(I)) Diameter Adjustment Function (sqrt((sqrt((sqrt((sqrt((sqrt(x^2+0^2) -a)^2 +0^2) -(a/2))^2 + (sqrt(y^2+0^2) -(2a/3))^2) -b)^2 +0^2) -c)^2 +0^2) -d)^2 + (sqrt(z^2+0^2) -(a/5))^2 = 1 --- a=27 , b=7.1 , c=3.2 , d=1.6 --- XYbox = -55 / +55 , Zbox = -25 / +25 9D (((((II)I)(II))(II))(II)) : Triple Tiger 1A-Torus , T3xC4xS1 ------------------------------------------------------------------ (II) - S1 ((II)I) - T2 (((II)I)I) - T3 ((((II)I)I)I) - T4 ((((II)(II))I)I) - T3xC2 ((((II)(II))(II))I) - T3xC3 ((((II)(II))(II))(II)) - T3xC4 (((((II)I)(II))(II))(II)) - T3xC4xS1 -------------------------------------- (((((II)I)(II))(II))(II)) ((((II)I)(II))(II))(II) ( ( ( (II) I) (II)) (II)) (II) ( ( ( (xy) z) (wv)) (ut)) (sr) ( ( ( (x+y) +z) + (w+v)) + (u+t)) + (s+r) ( ( ( (x+y -R1a) +z -R2) + (w+v -R1b) -R3) + (u+t -R1c) -R4) + (s+r -R1d) = Rminor ( ( ( (x+y -R1a)² +z -R2)² + (w+v -R1b)² -R3)² + (u+t -R1c)² -R4)² + (s+r -R1d)² = Rminor² ( ( ( (√(x+y) -R1a)² +z -R2)² + (√(w+v) -R1b)² -R3)² + (√(u+t) -R1c)² -R4)² + (√(s+r) -R1d)² = Rminor² ( ( ( √((√(x+y) -R1a)² +z) -R2)² + (√(w+v) -R1b)² -R3)² + (√(u+t) -R1c)² -R4)² + (√(s+r) -R1d)² = Rminor² ( ( √((√((√(x+y) -R1a)² +z) -R2)² + (√(w+v) -R1b)²) -R3)² + (√(u+t) -R1c)² -R4)² + (√(s+r) -R1d)² = Rminor² ( √((√((√((√(x+y) -R1a)² +z) -R2)² + (√(w+v) -R1b)²) -R3)² + (√(u+t) -R1c)²) -R4)² + (√(s+r) -R1d)² = Rminor² (√((√((√((√(x²+y²) -R1a)² +z²) -R2)² + (√(w²+v²) -R1b)²) -R3)² + (√(u²+t²) -R1c)²) -R4)² + (√(s²+r²) -R1d)² = Rminor² ---------------------------------------------------------------------------------------------------------------------- • (√((√((√((√(x²+y²)-R1a)²+z²)-R2)²+(√(w²+v²)-R1b)²)-R3)²+(√(u²+t²)-R1c)²)-R4)²+(√(s²+r²)-R1d)² = Rminor² • (sqrt((sqrt((sqrt((sqrt(x^2+y^2) -R1a)^2 +z^2) -R2)^2 + (sqrt(w^2+v^2) -R1b)^2) -R3)^2 + (sqrt(u^2+t^2) -R1c)^2) -R4)^2 + (sqrt(s^2+r^2) -R1d)^2 = Rminor^2 ------------------------------------ (((((R1a)R2)(R1b)R3)(R1c)R4)(R1d)Rm) (((((24)12)(12)6)(8)3)(4)1) - ring torus diameter values • (sqrt((sqrt((sqrt((sqrt(x^2+y^2) -24)^2 +z^2) -12)^2 + (sqrt(w^2+v^2) -12)^2) -6)^2 + (sqrt(u^2+t^2) -12)^2) -3)^2 + (sqrt(s^2+r^2) -6)^2 = 1 ------------------ XYrange = -48,+48 Zrange = -15,+15 ------------------ 3D Scans of the (((((I))(I))(I))(I)) 4x2x2x2 Array, Adjust ‘a’ to move ± in 4-space by [±Rn] value ---------------------------------------------------------------------------------------------------- • ((((())(I))(I))(I)) : [±R1A±R2] intercepts are 4 places of 4x2x2 array of 16 toruses (sqrt((sqrt((sqrt((sqrt(0^2+a^2) -24)^2 +0^2) -12)^2 + (sqrt(x^2+0^2) -12)^2) -6)^2 + (sqrt(y^2+0^2) -8)^2) -3)^2 + (sqrt(z^2+0^2) -4)^2 = 1 --- -48 < a < 48 • (((((I))())(I))(I)) : [±R1B] intercepts are 2 places of 8x2x2 array of 32 toruses (sqrt((sqrt((sqrt((sqrt(x^2+0^2) -24)^2 +0^2) -12)^2 + (sqrt(a^2+0^2) -12)^2) -6)^2 + (sqrt(y^2+0^2) -8)^2) -3)^2 + (sqrt(z^2+0^2) -4)^2 = 1 • (((((I))(I))())(I)) : [±R1C] intercepts are 2 places of 4x2x2x[R1 pair] of 32 toruses (sqrt((sqrt((sqrt((sqrt(x^2+0^2) -24)^2 +0^2) -12)^2 + (sqrt(y^2+0^2) -12)^2) -6)^2 + (sqrt(a^2+0^2) -8)^2) -3)^2 + (sqrt(z^2+0^2) -4)^2 = 1 • (((((I))(I))(I))()) : [±R1D] intercepts are 2 places of 4x2x2x[Rm pair] of 32 toruses (sqrt((sqrt((sqrt((sqrt(x^2+0^2) -24)^2 +0^2) -12)^2 + (sqrt(y^2+0^2) -12)^2) -6)^2 + (sqrt(z^2+0^2) -8)^2) -3)^2 + (sqrt(a^2+0^2) -4)^2 = 1 ----------------------------------------------- 3D Explore Functions , Single Translate+Rotate ----------------------------------------------- • (((((a))(A))(I))(I)) : Single Trans+Rotate; {±R1A±R2 of 4x2x2} to {±R1B of 8x2x2} (sqrt((sqrt((sqrt((sqrt(0^2+(x*cos(b)-a*sin(b))^2) -24)^2 +0^2) -12)^2 + (sqrt((x*sin(b)+a*cos(b))^2+0^2) -12)^2) -6)^2 + (sqrt(y^2+0^2) -8)^2) -3)^2 + (sqrt(z^2+0^2) -4)^2 = 1 --- -48 < a < 48 --- 0 < b <1.57 • (((((I))(a))(A))(I)) : {±R1C of 4x2x2x[R1-pair]} to {±R1B of 8x2x2} (sqrt((sqrt((sqrt((sqrt(x^2+0^2) -24)^2 +0^2) -12)^2 + (sqrt((y*cos(b)-a*sin(b))^2+0^2) -12)^2) -6)^2 + (sqrt((y*sin(b)+a*cos(b))^2+0^2) -8)^2) -3)^2 + (sqrt(z^2+0^2) -4)^2 = 1 --- -24 < a < 24 --- 0 < b <1.57 • (((((I))(I))(A))(a)) : {±R1D of 4x2x2x[Rm-pair]} to {±R1C of 4x2x2x[R1-pair]} (sqrt((sqrt((sqrt((sqrt(x^2+0^2) -24)^2 +0^2) -12)^2 + (sqrt(y^2+0^2) -12)^2) -6)^2 + (sqrt((z*sin(b)+a*cos(b))^2+0^2) -8)^2) -3)^2 + (sqrt((z*cos(b)-a*sin(b))^2+0^2) -4)^2 = 1 --- -13 < a < 13 --- 0 < b <1.57 • (((((a))(I))(I))(A)) : {±R1A±R2 of 4x2x2} to {±R1D of 4x2x2x[Rm-pair]} (sqrt((sqrt((sqrt((sqrt(0^2+(z*cos(b)-a*sin(b))^2) -24)^2 +0^2) -12)^2 + (sqrt(x^2+0^2) -12)^2) -6)^2 + (sqrt(y^2+0^2) -8)^2) -3)^2 + (sqrt((z*sin(b)+a*cos(b))^2+0^2) -4)^2 = 1 --- -48 < a < 48 --- 0 < b <1.57 • (((((A))(I))(a))(I)) : {±R1C of 4x2x2x[R1-pair]} to {±R1A±R2 of 4x2x2} (sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2+0^2) -24)^2 +0^2) -12)^2 + (sqrt(y^2+0^2) -12)^2) -6)^2 + (sqrt((x*cos(b)-a*sin(b))^2+0^2) -8)^2) -3)^2 + (sqrt(z^2+0^2) -4)^2 = 1 --- -48 < a < 48 --- 0 < b <1.57 --- Zrange = -15,15 • (((((I))(A))(I))(a)) : {±R1D of 4x2x2x[Rm-pair]} to {±R1B of 8x2x2} (sqrt((sqrt((sqrt((sqrt(x^2+0^2) -24)^2 +0^2) -12)^2 + (sqrt((y*sin(b)+a*cos(b))^2+0^2) -12)^2) -6)^2 + (sqrt(z^2+0^2) -8)^2) -3)^2 + (sqrt((y*cos(b)-a*sin(b))^2+0^2) -4)^2 = 1 --- -24 < a < 24 --- 0 < b <1.57 --- Zrange = -15,15 ---------------------- Dual Translate+Rotate ---------------------- • (((((a))(Ac))(C))(I)) : Dual T+R , explores 4x2 dual-void rectangle array of ((((())(II))())(I)) (sqrt((sqrt((sqrt((sqrt((x*cos(b)-a*sin(b))^2) -24)^2) -12)^2 + (sqrt((x*sin(b)+a*cos(b))^2+(y*cos(d)-c*sin(d))^2) -12)^2) -6)^2 + (sqrt((y*sin(d)+c*cos(d))^2) -8)^2) -3)^2 + (sqrt(z^2) -4)^2 = 1 --- [a,b,c,d] = [±24±12 ; 0 ; ±8 ; 1.57] : ((((())(II))())(I)) : [±R1A±R2]x[±R1C] are 4x2 places of ((((II)))(I)), 1x1x2x[R1-quartet] • (((((A)c)(C))(a))(I)) : Dual T+R (sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2) -24)^2 +(y*cos(d)-c*sin(d))^2) -12)^2 + (sqrt((y*sin(d)+c*cos(d))^2) -12)^2) -6)^2 + (sqrt((x*cos(b)-a*sin(b))^2) -8)^2) -3)^2 + (sqrt(z^2) -4)^2 = 1 • (((((A))(Ca))(-))(Ic)) : Dual T+R, Preset [±R1C]=8, explores multiple hypervoids (sqrt((sqrt((sqrt((sqrt((x*sin(b)+a*cos(b))^2) -24)^2) -12)^2 + (sqrt((y*sin(d)+c*cos(d))^2+(x*cos(b)-a*sin(b))^2) -12)^2) -6)^2 + (sqrt(8^2) -8)^2) -3)^2 + (sqrt(z^2+(y*cos(d)-c*sin(d))^2) -4)^2 = 1 • (((((-)A)(Ca))(-))(Ic)) : Dual T+R, Preset Void Locations [±R1A]=24;[±R1C]=8, explores dual-hypervoid intercept region (sqrt((sqrt((sqrt((sqrt(24^2) -24)^2 +(x*sin(b)+a*cos(b))^2) -12)^2 + (sqrt((y*sin(d)+c*cos(d))^2+(x*cos(b)-a*sin(b))^2) -12)^2) -6)^2 + (sqrt(8^2) -8)^2) -3)^2 + (sqrt(z^2+(y*cos(d)-c*sin(d))^2) -4)^2 = 1 ---------------------------------------------------------------- • Dual-Empty Hypervoid Intercepts <[For Navigational Purposes]> (((((R1a)R2)(R1b))(R1c))(R1d)) (((((±24)±12)(±12))(±8))(±4)) ------------------------------ (((((a)b)(c))(d))(e)) - [±a]x[±b]x[±c]x[±d]x[±e] : Hypervoid Locations of Ring Intercept Activity ------------------------------------------------------------------------------------------------- (((((I)I)(a))(b))(I)) - [±12]x[±8] are 2x2 places of (((((I)I)))(I)), 2x1x2x[R1-quartet] of 16 tori (((((a))(II))(b))(I)) - [±24±12]x[±8] are 4x2 places of ((((II)))(I)), 1x1x2x[R1-quartet] of 8 tori (((((a))(I))(b))(II)) - [±24±12]x[±8] are 4x2 places of ((((I)))(II)), 1x1x8 column of 8 tori (((((I))(a))(b))(II)) - [±12]x[±8] are 2x2 places of (((((I))))(II)), 1x1x16 column of 16 tori (((((II))(a))(b))(I)) - [±12]x[±8] are 2x2 places of (((((II))))(I)), 1x1x2x[R1-octet] of 16 tori (((((a)I)(b))(c))(II)) - [±24]x[±12]x[±8] are 2x2x2 places of ((((I)))(II)), 1x1x8 column of 8 tori (((((a)I)(b))(I))(I)) - [±24]x[±12] are 2x2 places of ((((I))(I))(I)), 4x2x2 array of 16 tori (((((a)I)(I))(b))(I)) - [±24]x[±8] are 2x2 places of ((((I)(I)))(I)), 2x2x2x[R1-pair] of 16 tori 10D (((((II)I)I)((II)I))I)(II)) - Tigritorus Diduotoric Torus , T4xC2xC3 , (((((R1a)R2a)R3)((R1b)R2b)R4)R5)(R1c)min) ---------------------------------------------------------------------------------------------------------------------- (sqrt((sqrt((sqrt((sqrt((sqrt(x^2 + y^2) -R1A)^2 + z^2) -R2A)^2 + w^2) -R3)^2 + (sqrt((sqrt(v^2 + u^2) -R1B)^2 + t^2) -R2B)^2) -R4)^2 + s^2) -R5)^2 + (sqrt(r^2 + q^2) -R1C)^2 -R6^2 = 0 • (((((I)))((I))))(I)) (sqrt((sqrt((sqrt((sqrt((sqrt(x^2 + 0^2) -R1A)^2 + 0^2) -R2A)^2 + 0^2) -R3)^2 + (sqrt((sqrt(y^2 + 0^2) -R1B)^2 + 0^2) -R2B)^2) -R4)^2 + 0^2) -R5)^2 + (sqrt(z^2 + 0^2) -R1C)^2 -R6^2 = 0 diameter adj (sqrt((sqrt((sqrt((sqrt((sqrt(x^2 + 0^2) -a)^2 + 0^2) -(a/2))^2 + 0^2) -(a/4))^2 + (sqrt((sqrt(y^2 + 0^2) -(a/2))^2 + 0^2) -(a/4))^2) -c)^2 + 0^2) -d)^2 + (sqrt(z^2 + 0^2) -b)^2 = 1 -Set 80 cubes -Set ranges: 0 < a < 30 0 < b < 3.5 0 < c < 4 0 < d < 1.75 XYbox = -60 / +60 Zrange = -10 / +10