THE BLOG
02/25/2016 02:59 pm ET Updated Dec 06, 2017

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)


2016-02-25-1456387958-8203180-Butterfly_1.jpg

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)


2016-02-25-1456395640-5824030-Butterfly_2.jpg

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)


2016-02-25-1456397956-2255511-Butterfly_3.jpg

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).