# Using Mathematical Formulas to Draw Butterflies

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).