## Drawing Plants With Mathematics

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:

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))^{8}+(1/4)(sin(21πk/4000))^{6}sin((2π/5)(k/4000)^{12}),

Y(k)=(1/4)(k/4000)^{2}+(1/4)((sin(21πk/4000))^{5}+(1/28)sin(42πk/4000))cos((π/2)(k/4000)^{12}),

R(k)=(1/170)+(1/67)(sin(42πk/4000))^{2}(1-(cos(21πk/4000))^{4}).

#### Palm Branch

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))^{8}(1-(cos(πk/24000)cos(3πk/24000))^{10})+(1/4)(sin(91πk/12000))^{6}sin((2π/5)(k/12000)^{12}),

Y(k)=(1/3)sin(πk/24000)+(1/4)(sin(91πk/12000))^{5}cos((π/2)(k/12000)^{12})(1-(cos(πk/24000)cos(3πk/24000))^{10}),

R(k)=(1/270)+(1/140)(sin(182πk/12000))^{2}(1-(cos(91πk/12000))^{4})+(1/80)(cos(91πk/12000))^{6}.

#### Leaves

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)))^{7},

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

R(k)=(1/140)+(1/70)(cos((2πk/30000)+(π/3)))^{10}.

#### Branch

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))^{8},

Y(k)=(1/4)(k/3000)^{2}+(1/4)(sin(πk/200))^{5}+(1/112)sin(πk/100),

R(k)=(1/170)+(1/24)(sin(πk/100))^{2}(1-(cos(πk/200))^{4}).

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