These Beautiful Images Are Created By Drawing Ellipses

2015-11-10-1447155493-3061905-8000_Ellipses_1.jpg

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)

6,000 Ellipses (1)

Ring
2015-11-10-1447155683-6298082-Ring.jpg

9,000 Ellipses (1)
2015-11-10-1447155762-3972557-9000_Ellipses_1.jpg

8,000 Ellipses (2)
2015-11-10-1447155865-6713934-8000_Ellipses_2.jpg

9,000 Ellipses (2)
2015-11-10-1447155949-5959918-9000_Ellipses_2.jpg

12,000 Ellipses
2015-11-10-1447156035-6686789-12000_Ellipses_1.jpg

8,000 Ellipses (3)
2015-11-10-1447156107-9632152-8000_Ellipses_3.jpg

6,000 Ellipses (2)
2015-11-10-1447156164-2096560-6000_Ellipses_2.jpg

9,000 Ellipses (3)
2015-11-10-1447156238-7126963-9000_Ellipses_3.jpg

Butterfly
2015-11-10-1447156309-1961063-Butterfly.jpg
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)
2015-11-10-1447156361-3806880-6000_Ellipses_3.jpg
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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