An Aesthetic Exploration of Relative Motions

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3. Case Study of Aesthetic Forms

3-3 Recursive Multiple-level Relative Motions

One of the things computers are good at is performing dull repetitive tasks. In other words, a recursive procedure repeats itself. Recursive relative motion is a concept of peculiar motions. Figure 3-3 is shown as the difference between relative motions and recursive relative motions.

Figure 3-3 Motions Illustration

Rules:

P₀: P_0(angle) = P_0(angle) + P_{0(angular increment)}

P_0(x) = P_2(x)+ P_0(radius)* P_{0(cos(angle))}

P_0(y) = P_2(y)+ P_0(radius)* P_{0(sin(angle))}

P₁: P_1(angle) = P_1(angle) + P_{1(angular increment)}

P_1(x) = P_0(x)+ P_1(radius)* P_{1(cos(angle))}

P_1(y) = P_0(y)+ P_1(radius)* P_{1(sin(angle))}

P₂: P_2(angle) = P_2(angle) + P_{2(angular increment)}

P_2(x) = P_1(x)+ P_2(radius)* P_{2(cos(angle))}

P_2(y) = P_1(y)+ P_2(radius)* P_{2(sin(angle))}

Trajectory Recorded: P₀

Table: Case 3-3.1 Parameters Assigned (Figure 3-3.1)

Point	P₁	P₂	P₀
Coordinate System	P₀	P₁	P₂
Initial Angle	0	0	0
Angular Increment	-0.961003	0.2410107	-0.4810090
Angular Increment	-0.961003	0.2410107	-0.4810090
Radius(pixel)	1	1	1
Radius Increment	None	None	None

* Positive angle means clockwise rotation, negative counterclockwise.

[Enlarge the image]

Figure 3-3.1 (12,000,000 iterations)

Table: Case 3-3.2 Parameters Assigned (Figure 3-3.2)

Point	P₁	P₂	P₀
Coordinate System	P₀	P₁	P₂
Initial Angle	0	0	0
Angular Increment	-0.48003	0.24007	-0.35009
Angular Increment	-0.48003	0.24007	-0.35009
Radius(pixel)	1	1	1
Radius Increment	None	None	None

* Positive angle means clockwise rotation, negative counterclockwise.

[Enlarge the image]

Figure 3-3.2 (12,000,000 iterations)

Table: Case 3-3.3 Parameters Assigned (Figure 3-3.3)

Point	P₁	P₂	P₀
Coordinate System	P₀	P₁	P₂
Initial Angle	0	0	0
Angular Increment	-0.291252	0.388320	0.711920
Angular Increment	-0.291252	0.388320	0.711920
Radius(pixel)	1	1	1
Radius Increment	None	None	None

* Positive angle means clockwise rotation, negative counterclockwise.

[Enlarge the image]

Figure 3-3.3 (2,500,000 iterations)

Table: Case 3-3.4 Parameters Assigned (Figure 3-3.4)

Point	P₁	P₂	P₀
Coordinate System	P₀	P₁	P₂
Initial Angle	0	0	0
Angular Increment	0.2589	0.2912	-0.7766
Angular Increment	0.2589	0.2912	-0.7766
Radius(pixel)	1	1	1
Radius Increment	None	None	None

* Positive angle means clockwise rotation, negative counterclockwise.

[Enlarge the image]

Figure 3-3.4 (7,500,000 iterations)

Table: Case 3-3.5 Parameters Assigned (Figure 3-3.5)

Point	P₁	P₂	P₃	P₀
Coordinate System	P₀	P₁	P₂	P₃
Initial Angle	0	0	0	0
Angular Increment	0.30196	0.46667	-0.10980	-0.27493
Angular Increment	0.30196	0.46667	-0.10980	-0.27493
Radius(pixel)	1	1	1	1
Radius Increment	None	None	None	None

* Positive angle means clockwise rotation, negative counterclockwise.

[Enlarge the image]

Figure 3-3.5 (24,000,000 iterations)

Table: Case 3-3.6 Parameters Assigned (Figure 3-3.6)

Point	P₁	P₂	P₃	P₀
Coordinate System	P₀	P₁	P₂	P₃
Initial Angle	0	0	0	0
Angular Increment	0.25882	0.40000	-0.42353	-0.11801
Angular Increment	0.25882	0.40000	-0.42353	-0.11801
Radius(pixel)	1	1	1	1
Radius Increment	None	None	None	None

* Positive angle means clockwise rotation, negative counterclockwise.

[Enlarge the image]

Figure 3-3.6 (24,000,000 iterations)

Table: Case 3-3.7 Parameters Assigned (Figure 3-3.7)

Point	P₁	P₂	P₃	P₄	P₀
Coordinate System	P₀	P₁	P₂	P₃	P₄
Initial Angle	0	0	0	0	0
Angular Increment	-2.2	0.3157286	-0.96	-0.12	-0.18
Angular Increment	-2.2	0.3157286	-0.96	-0.12	-0.18
Radius(pixel)	1	1	1	1	1
Radius Increment	None	None	None	None	None

* Positive angle means clockwise rotation, negative counterclockwise.

[Enlarge the image]

Figure 3-3.7 (12,000,000 iterations)

Table: Case 3-3.8 Parameters Assigned (Figure 3-3.8)

Point	P₁	P₂	P₃	P₄	P₀
Coordinate System	P₀	P₁	P₂	P₃	P₄
Initial Angle	0	0	0	0	0
Angular Increment	0.08003	0.13	0.21	0.34	0.76
Angular Increment	0.08003	0.13	0.21	0.34	0.76
Radius(pixel)	1	1	1	1	1
Radius Increment	None	None	None	None	None

* Positive angle means clockwise rotation, negative counterclockwise.

[Enlarge the image]

Figure 3-3.8 (1,800,000 iterations)

Table: Case 3-3.9 Parameters Assigned (Figure 3-3.9)

Point	P₁	P₂	P₃	P₄	P₀
Coordinate System	P₀	P₁	P₂	P₃	P₄
Initial Angle	0	0	0	0	0
Angular Increment	0.12	0.19416	0.38832	0.72	-1.55328
Angular Increment	0.12	0.19416	0.38832	0.72	-1.55328
Radius(pixel)	1	1	1	1	1
Radius Increment	None	None	None	None	None

* Positive angle means clockwise rotation, negative counterclockwise.

[Enlarge the image]

Figure 3-3.9 (5,400,000 iterations)

Table: Case 3-3.10 Parameters Assigned (Figure 3-3.10)

Point	P₁	P₂	P₃	P₄	P₀
Coordinate System	P₀	P₁	P₂	P₃	P₄
Initial Angle	0	0	0	0	0
Angular Increment	0.144000	0.232992	-0.698976	-1.164690	1.152000
Angular Increment	0.144000	0.232992	-0.698976	-1.164690	1.152000
Radius(pixel)	1	1	1	1	1
Radius Increment	None	None	None	None	None

* Positive angle means clockwise rotation, negative counterclockwise.

[Enlarge the image]

Figure 3-3.10 (5,400,000 iterations)

Table: Case 3-3.11 Parameters Assigned (Figure 3-3.11)

Point	P₁	P₂	P₃	P₄	P₅	P₀
Coordinate System	P₀	P₁	P₂	P₃	P₄	P₅
Initial Angle	0	0	0	0	0	0
Angular Increment	0.220066	0.066	-0.242	0.374	-0.506	-0.572
Angular Increment	0.220066	0.066	-0.242	0.374	-0.506	-0.572
Radius(pixel)	1	1	1	1	1	1
Radius Increment	None	None	None	None	None	None

* Positive angle means clockwise rotation, negative counterclockwise.

[Enlarge the image]

Figure 3-3.11 (18,000,000 iterations)

Table: Case 3-3.12 Parameters Assigned (Figure 3-3.12)

Point	P₁	P₂	P₃	P₄	P₅	P₀
Coordinate System	P₀	P₁	P₂	P₃	P₄	P₅
Initial Angle	0	0	0	0	0	0
Angular Increment	0.35	0.2900007	0.42	0.6400003	-2.10001	-1.30002
Angular Increment	0.35	0.2900007	0.42	0.6400003	-2.10001	-1.30002
Radius(pixel)	1	1	1	1	1	1
Radius Increment	None	None	None	None	None	None

* Positive angle means clockwise rotation, negative counterclockwise.

[Enlarge the image]

Figure 3-3.12 (11,700,000 iterations)

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CONTENTS:

Case Study of Aesthetic Forms