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Drawing Plants With Mathematics

In this post you can see four images with their mathematical descriptions. I have created these images by trigonometric functions.
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I really love to draw plants, because they always remind me of good things such as peace, beauty, love, kindness, etc. In this post you can see four images with their mathematical descriptions. I have created these images by trigonometric functions. Also, in my previous posts you can see more images that I created with mathematical formulas: Drawing Birds in Flight With Mathematics, These Are Mathematical Sets.

Olive Branch

2016-01-19-1453212784-3475138-Olive_Branch.jpg

This image shows 4,000 circles. For k=1,2,3,...,4000 the center of the k-th circle is (X(k), Y(k)) and the radius of the k-th circle is R(k), where

X(k)=(2k/4000)+(1/28)sin(42πk/4000)+(1/9)(sin(21πk/4000))+(1/4)(sin(21πk/4000))sin((2π/5)(k/4000)),

Y(k)=(1/4)(k/4000)+(1/4)((sin(21πk/4000))+(1/28)sin(42πk/4000))cos((π/2)(k/4000)),

R(k)=(1/170)+(1/67)(sin(42πk/4000))(1-(cos(21πk/4000))).

Palm Branch

2016-01-19-1453230310-9419935-Palm_Branch.jpg

This image shows 12,000 circles. For k=1,2,3,...,12000 the center of the k-th circle is (X(k), Y(k)) and the radius of the k-th circle is R(k), where

X(k)=(2k/12000)+(1/7)(sin(91πk/12000))(1-(cos(πk/24000)cos(3πk/24000)))+(1/4)(sin(91πk/12000))sin((2π/5)(k/12000)),

Y(k)=(1/3)sin(πk/24000)+(1/4)(sin(91πk/12000))cos((π/2)(k/12000))(1-(cos(πk/24000)cos(3πk/24000))),

R(k)=(1/270)+(1/140)(sin(182πk/12000))(1-(cos(91πk/12000)))+(1/80)(cos(91πk/12000)).

Leaves

2016-01-18-1453144829-1335607-Leaves.jpg

This image shows 30,000 circles. For k=1,2,3,...,30000 the center of the k-th circle is (X(k), Y(k)) and the radius of the k-th circle is R(k), where

X(k)=(3/2)(sin((2πk/30000)+(π/3))),

Y(k)=(1/4)(cos(6πk/30000))(2-(sin((2πk/30000)+(π/3)))-2(cos(πk/30)))-(1/2)(cos((2πk/30000)+(π/3))),

R(k)=(1/140)+(1/70)(cos((2πk/30000)+(π/3))).

Branch

2016-01-21-1453389865-477112-Branch.jpg

This image shows 3,000 circles. For k=1,2,3,...,3000 the center of the k-th circle is (X(k), Y(k)) and the radius of the k-th circle is R(k), where

X(k)=(2k/3000)+(1/17)sin(πk/100)+(1/9)(sin(πk/200)),

Y(k)=(1/4)(k/3000)+(1/4)(sin(πk/200))+(1/112)sin(πk/100),

R(k)=(1/170)+(1/24)(sin(πk/100))(1-(cos(πk/200))).

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