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))(1-(1/2)(sin(πk/40000)))(1-(1/4)(cos(2πk/40000))(1+(2/3)(cos(30πk/40000)))-(sin(2πk/40000))(sin(6πk/40000))((1/5)+(4/5)(cos(24πk/40000)))),
Y(k)=cos(2πk/40000)(cos(141πk/40000))(1+(1/4)(cos(πk/40000))(cos(3πk/40000))(cos(19πk/40000))),
R(k)=(1/100)+(1/40)((cos(2820πk/40000))+(sin(141πk/40000)))(1-(cos(πk/40000))(cos(3πk/40000))(cos(12πk/40000))).
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))(1-(1/2)(sin(πk/40000)))(1-(1/4)(cos(2πk/40000))(1+(2/3)(cos(30πk/40000))))(1-(1/3)(sin(2πk/40000))(sin(6πk/40000))((1/2)+(1/2)(sin(18πk/40000)))),
Y(k)=cos(2πk/40000)(cos(141πk/40000))(1+(1/4)(cos(πk/40000))(cos(3πk/40000))(cos(19πk/40000))),
R(k)=((9/8)-(sin(2πk/40000)))((1/100)+(1/40)((cos(141πk/40000))+(sin(141πk/40000)))(1-(cos(πk/40000))(cos(3πk/40000))(cos(12πk/40000)))).
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))(1-(1/2)sin(πk/40000))(1-(1/4)(cos(2πk/40000))(1+(cos(32πk/40000))))(1-(1/2)(sin(2πk/40000))(sin(6πk/40000))((1/2)+(1/2)(sin(18πk/40000)))),
Y(k)=cos(2πk/40000)(cos(141πk/40000))(1+(1/4)(cos(πk/40000))(cos(3πk/40000))(cos(21πk/40000))),
R(k)=(1/100)+(1/40)((cos(141πk/40000))+(sin(141πk/40000)))(1-(cos(πk/40000))(cos(3πk/40000))(cos(12πk/40000))).
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