Butterflies always remind me of the beautiful nature in the world. In this post you can see three butterfly drawings with their mathematical descriptions. I have created them by using sine and cosine. Also, in my previous posts you can see more images that I created with mathematical formulas: *Drawing Plants With Mathematics*, *Drawing Birds in Flight With Mathematics*, *These Are Mathematical Sets*.

#### Butterfly (1)

This image shows 40,000 circles. For k=1,2,3,...,40000 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)=(6/5)(cos(141πk/40000))^{9}(1-(1/2)(sin(πk/40000))^{3})(1-(1/4)(cos(2πk/40000))^{30}(1+(2/3)(cos(30πk/40000))^{20})-(sin(2πk/40000))^{10}(sin(6πk/40000))^{10}((1/5)+(4/5)(cos(24πk/40000))^{20})),

Y(k)=cos(2πk/40000)(cos(141πk/40000))^{2}(1+(1/4)(cos(πk/40000))^{24}(cos(3πk/40000))^{24}(cos(19πk/40000))^{24}),

R(k)=(1/100)+(1/40)((cos(2820πk/40000))^{6}+(sin(141πk/40000))^{2})(1-(cos(πk/40000))^{16}(cos(3πk/40000))^{16}(cos(12πk/40000))^{16}).

#### Butterfly (2)

This image shows 40,000 circles. For k=1,2,3,...,40000 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)=(6/5)(cos(141πk/40000))^{9}(1-(1/2)(sin(πk/40000))^{3})(1-(1/4)(cos(2πk/40000))^{30}(1+(2/3)(cos(30πk/40000))^{20}))(1-(1/3)(sin(2πk/40000))^{30}(sin(6πk/40000))^{10}((1/2)+(1/2)(sin(18πk/40000))^{10})),

Y(k)=cos(2πk/40000)(cos(141πk/40000))^{2}(1+(1/4)(cos(πk/40000))^{24}(cos(3πk/40000))^{24}(cos(19πk/40000))^{24}),

R(k)=((9/8)-(sin(2πk/40000))^{10})((1/100)+(1/40)((cos(141πk/40000))^{14}+(sin(141πk/40000))^{6})(1-(cos(πk/40000))^{16}(cos(3πk/40000))^{16}(cos(12πk/40000))^{16})).

#### Butterfly (3)

This image shows 40,000 circles. For k=1,2,3,...,40000 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)(cos(141πk/40000))^{9}(1-(1/2)sin(πk/40000))(1-(1/4)(cos(2πk/40000))^{30}(1+(cos(32πk/40000))^{20}))(1-(1/2)(sin(2πk/40000))^{30}(sin(6πk/40000))^{10}((1/2)+(1/2)(sin(18πk/40000))^{20})),

Y(k)=cos(2πk/40000)(cos(141πk/40000))^{2}(1+(1/4)(cos(πk/40000))^{24}(cos(3πk/40000))^{24}(cos(21πk/40000))^{24}),

R(k)=(1/100)+(1/40)((cos(141πk/40000))^{14}+(sin(141πk/40000))^{6})(1-(cos(πk/40000))^{16}(cos(3πk/40000))^{16}(cos(12πk/40000))^{16}).