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:
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https://youtu.be/PVXQ_kgKDHg
https://youtu.be/6rDUGWSmxTk
https://youtu.be/0DQOnLm3BS4
https://youtu.be/iBBuF546wqM
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle_1705-720thick2.png?itok=twkZYAM_)
Formül
- \varphi_{i+1} = \varphi_i + 170.5° \cdot i+180°
Turtle Object No 1
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle_1655-720thick5.png?itok=qjqfN3Dt)
Formül
- \varphi_{i+1} = \varphi_i + 165.5° \cdot i+180°
Turtle Object No 2
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle_0715-720thick3.png?itok=TLIwkFvW)
Formül
- \varphi_{i+1} = \varphi_i + 71.5° \cdot i+180°
Turtle Object No 3
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle_0055-720thick7.png?itok=F8pbMdda)
Formül
- \varphi_{i+1} = \varphi_i + 5.5° \cdot i+180°
Turtle Object No 4
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle003x1_2.png?itok=ZE7drYyW)
Formül
- \varphi_{i+1} = \varphi_i + 60° \cdot i^2-180°
Turtle Object No 5
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle006x1_3.png?itok=9IhQ0lRT)
Formül
- \varphi_{i+1} = \varphi_i + 30° \cdot i^2-180°
Turtle Object No 6
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle006x5.png?itok=KK1e-F1A)
Formül
- \varphi_{i+1} = \varphi_i + 150° \cdot i^2-180°
Turtle Object No 7
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle009x1_2.png?itok=z1N_Oz6t)
Formül
- \varphi_{i+1} = \varphi_i + 20° \cdot i^2-180°
Turtle Object No 8
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle009x2.png?itok=TpGVf-TO)
Formül
- \varphi_{i+1} = \varphi_i + 40° \cdot i^2-180°
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Turtle Object No 9
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.
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle009x5_2.png?itok=TKqLcxI8)
Formül
- \varphi_{i+1} = \varphi_i + 100° \cdot i^2-180°
Turtle Object No 10
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle009x8.png?itok=svwQzSwT)
Formül
- \varphi_{i+1} = \varphi_i + 160° \cdot i^2-180°
Turtle Object No 11
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle012x01.png?itok=O5AJy6Yr)
Formül
- \varphi_{i+1} = \varphi_i + 15° \cdot i^2-180°
Turtle Object No 12
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle012x05.png?itok=m86EAc_s)
Formül
- \varphi_{i+1} = \varphi_i + 75° \cdot i^2-180°
Turtle Object No 13
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle012x07.png?itok=J1eWgYF3)
Formül
- \varphi_{i+1} = \varphi_i + 105° \cdot i^2-180°
Turtle Object No 14
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle014x01.png?itok=XIX00Axw)
Formül
- \varphi_{i+1} = \varphi_i + 90°/7 \cdot i^2-180°
Turtle Object No 15
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle015x07.png?itok=vp2L80--)
Formül
- \varphi_{i+1} = \varphi_i + 84° \cdot i^2-180°
Turtle Object No 16
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle015x11.png?itok=BXSdKxz2)
Formül
- \varphi_{i+1} = \varphi_i + 132° \cdot i^2-180°
Turtle Object No 17
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle015x14.png?itok=jmwmokE0)
Formül
- \varphi_{i+1} = \varphi_i + 168° \cdot i^2-180°
Turtle Object No 18
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle016x07.png?itok=-6GWOqUT)
Formül
- \varphi_{i+1} = \varphi_i + 78.75° \cdot i^2-180°
Turtle Object No 19
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle018x01_3.png?itok=Ul4Riw39)
Formül
- \varphi_{i+1} = \varphi_i + 10° \cdot i^2-180°
Turtle Object No 20
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle018x05_3.png?itok=IcrfIwQc)
Formül
- \varphi_{i+1} = \varphi_i + 50° \cdot i^2-180°
Turtle Object No 21
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle024x01.png?itok=ZfgfAW5a)
Turtle Object No 22
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle030x01_1.png?itok=tLKdlIN8)
Turtle Object No 23
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle128x01.png?itok=53RluESC)
Turtle Object No 24
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle180x013.png?itok=8IN_eIRM)
Turtle Object No 25
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle180x017.png?itok=-JYpSaXq)
Turtle Object No 26
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle180x019.png?itok=oF0L9nfI)
Turtle Object No 27
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle180x023.png?itok=RHul9ryy)
Turtle Object No 28
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle180x029.png?itok=Q4oJpMJw)
Turtle Object No 29
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle180x031.png?itok=MHrwAl9D)
Turtle Object No 30
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle180x041.png?itok=TsPlIoms)
Turtle Object No 31
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle180x043.png?itok=vdVeCiXf)
Turtle Object No 32
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle180x047.png?itok=BuP3b7Fv)
Turtle Object No 33
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle180x049.png?itok=uVsJICOP)
Turtle Object No 34
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle180x053.png?itok=1uDI_pVr)
Turtle Object No 35
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle180x059.png?itok=PZ9zbwLO)
Turtle Object No 36
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle180x061.png?itok=6jLsjalg)
Turtle Object No 37
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle180x089.png?itok=yDiewgL9)
Turtle Object No 38
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle180x091.png?itok=0NMqDwC-)
Turtle Object No 39
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle180x097.png?itok=SocS5j71)
Turtle Object No 40
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle180x101.png?itok=eqyeD6jd)
Turtle Object No 41
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle180x137.png?itok=oiEoyvdr)
Turtle Object No 42
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle180x173.png?itok=y7ZXGVIx)
Turtle Object No 43
![](https://www.imaginary.org/sites/default/files/styles/gallery-full/public/turtle180x179.png?itok=aTzMlsNc)