## These Beautiful Images Are Created By Drawing Ellipses

We can create an unlimited number of beautiful images by drawing ellipses. The following images are some examples that I have created by some computer programs. At the end of this post you can see the mathematical descriptions of “*Butterfly*” and “*6,000 Ellipses (3)*“.

**8,000 Ellipses (1)**

**Butterfly**

This image shows 8,000 ellipses. For each k=1,2,3,…,8000 the foci of the k-th ellipse are:

A(k)+iB(k)+C(k)e^(300πik/8000)

and

A(k)+iB(k)-C(k)e^(300πik/8000)

and the eccentricity of the k-th ellipse is D(k), where

A(k)=(sin(14πk/8000))^3,

B(k)=sin(14πk/8000)sin(10πk/8000),

C(k)=(1/200)+(1/20)+(1/20)cos(22πk/8000),

D(K)=(199/200)-(1/7)(sin(22πk/8000))^8.

**6,000 Ellipses (3)**

This image shows 6,000 ellipses. For each k=1,2,3,…,6000 the foci of the k-th ellipse are:

A(k)+iB(k)+C(k)e^(200πik/6000)

and

A(k)+iB(k)-C(k)e^(200πik/6000)

and the eccentricity of the k-th ellipse is D(k), where

A(k)=cos(16πk/6000)cos(6πk/6000),

B(k)=sin(16πk/6000)cos(6πk/6000),

C(k)=(1/200)+(1/14)+(1/14)(sin(10πk/6000))^3,

D(k)=(49/50)-(1/7)(sin(6πk/6000))^2.

See more images at: mathematics.culturalspot.org.

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