Turtle Geometry
Turtle geometry is applied to create complex symmetric patterns. The change of inclination angles of lines follows some prescribed rules. Lines are combined with special functions to generate colorful surroundings.
Links to related videos:
https://youtu.be/cFmSeORtSbA
https://youtu.be/PVXQ_kgKDHg
https://youtu.be/6rDUGWSmxTk
https://youtu.be/0DQOnLm3BS4
https://youtu.be/iBBuF546wqM
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- \varphi_{i+1} = \varphi_i + 170.5° \cdot i+180°
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- \varphi_{i+1} = \varphi_i + 165.5° \cdot i+180°
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- \varphi_{i+1} = \varphi_i + 71.5° \cdot i+180°
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- \varphi_{i+1} = \varphi_i + 5.5° \cdot i+180°
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- \varphi_{i+1} = \varphi_i + 60° \cdot i^2-180°
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- \varphi_{i+1} = \varphi_i + 30° \cdot i^2-180°
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- \varphi_{i+1} = \varphi_i + 150° \cdot i^2-180°
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- \varphi_{i+1} = \varphi_i + 20° \cdot i^2-180°
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- \varphi_{i+1} = \varphi_i + 40° \cdot i^2-180°
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Turtle Geometry engages students in exploring mathematical properties visually via a simple programming language. This example shows a polygonal line consisting of a series of points connected by lines with unit length. The coordinates of each pointare calculated iteratively following simple rules.
The result is a beautiful figure with sixfold symmetry. By modification of the parameters interesting figures of different shape and symmetry are obtained.
Formule
- \varphi_{i+1} = \varphi_i + 100° \cdot i^2-180°
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- \varphi_{i+1} = \varphi_i + 160° \cdot i^2-180°
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- \varphi_{i+1} = \varphi_i + 15° \cdot i^2-180°
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- \varphi_{i+1} = \varphi_i + 75° \cdot i^2-180°
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- \varphi_{i+1} = \varphi_i + 105° \cdot i^2-180°
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- \varphi_{i+1} = \varphi_i + 90°/7 \cdot i^2-180°
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- \varphi_{i+1} = \varphi_i + 84° \cdot i^2-180°
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- \varphi_{i+1} = \varphi_i + 132° \cdot i^2-180°
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- \varphi_{i+1} = \varphi_i + 168° \cdot i^2-180°
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- \varphi_{i+1} = \varphi_i + 78.75° \cdot i^2-180°
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- \varphi_{i+1} = \varphi_i + 10° \cdot i^2-180°
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- \varphi_{i+1} = \varphi_i + 50° \cdot i^2-180°
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