Mathematicians and artists continue to create stunning works in all media and to explore the visualization of mathematics--origami, computer-generated landscapes, tessellations, fractals, anamorphic art, and more
 

Mathematical Concepts Illustrated by Hamid Naderi Yeganeh

One of my goals is to create very beautiful images by using mathematical concepts such as trigonometric functions, exponential function, regular polygons, line segments, etc. I create images by running my program on a Linux operating system
Hamid Naderi Yeganeh

 
1,000 Line Segments (1)

This image shows 1,000 line segments. For each $i=1,2,3,\ldots\ldots,1000$ the endpoints of the $i$-th line segment are: $(-\sin(2\pi i/1000), -\cos(2\pi i/1000))$ and $((-1/2)\sin(8\pi i/1000), (-1/2)\cos(12\pi i/1000))$. I created this image by running my program on a Linux operating system.

1,000 Line Segments (2)

This image shows 1,000 line segments. For each $i=1,2,3,\ldots,1000$ the endpoints of the $i$-th line segment are: $(-\sin(4\pi i/1000), -\cos(2\pi i/1000))$ and $((-1/2)\sin(8\pi i/1000), (-1/2)\cos(4\pi i/1000))$. I created this image by running my program on a Linux operating system.

1,000 Line Segments (3)

This image shows 1,000 line segments. For each $i=1,2,3,\ldots,1000$ the endpoints of the line segment are: $(-\sin(8\pi i/1000), -\cos(2\pi i/1000))$ and $((-1/2)\sin(6\pi i/1000), (-1/2)\cos(2\pi i/1000))$. I created this image by running my program on a Linux operating system.

1,000 Line Segments (4)

This image shows 1,000 line segments. For each $i=1,2,3,\ldots,1000$ the endpoints of the $i$-th line segment are: $(-\sin(10\pi i/1000), -\cos(2\pi i/1000))$ and $((-1/2)\sin(12\pi i/1000), (-1/2)\cos(2\pi i/1000))$. I created this image by running my program on a Linux operating system.

Heart

This image contains a heart-like figure. It shows 601 line segments. For each $i=1, 2, 3, \ldots. , 601$ the endpoints of the $i$-th line segment are:
$(\sin(10\pi (i+699)/2000), \cos(8\pi (i+699)/2000))$
and
$(\sin(12\pi (i+699)/2000), \cos(10\pi (i+699)/2000))$.

I created this image by running my program.

A Bird in Flight (2014)

This image is like a bird in flight. It shows 2000 line segments. For each $i=1, 2, 3, \ldots , 2000$ the endpoints of the $i$-th line segment are:
$(3(\sin(2\pi i/2000)^3), -\cos(8\pi i/2000))$
and
$((3/2)(\sin(2\pi i/2000)^3), (-1/2)\cos(6\pi i/2000))$.

I created this image by running my program.

10,000 Circles

This image shows 10,000 circles. For each $i=1,2,3,\ldots,10{,}000$ the center of the $i$-th circle is:
$((\cos(38\pi i/10{,}000))^3, \sin(10\pi i/10{,}000))$ and the radius of the $i$-th circle is: $(1/3)(\sin(16\pi i/10{,}000))^2$.

A Bird in Flight (2015)

This image is like a bird in flight. It shows 500 line segments. For each $i=1,2,3,\ldots,500$ the endpoints of the $i$-th line segment are: $((3/2)(\sin((2\pi i/500)+(\pi /3)))^7, (1/4)(\cos(6\pi i/500))^2)$ and
$((1/5)\sin((6\pi i/500)+(\pi /5)), (-2/3)(\sin((2\pi i/500)-(\pi /3)))^2).$

Boat

This image is like a sailing boat. It shows 2,000 line segments. For each $k=1,2,3,\ldots,2000$ the endpoints of the $k$-th line segment are: $(\cos(6\pi k/2000)-i \cos(12\pi k/2000))e^{3\pi i/4}$ and $(\sin((4\pi k/2000)+(\pi /8))+i \sin((2\pi k/2000)+(\pi /3)))e^{3\pi i/4}.$

Fish

This image is like a fish. It shows 1,000 line segments. For $i=1,2,3,\ldots,1000$ the endpoints of the $i$-th line segment are: $(-2\cos(4\pi i/1000), (1/2)(\cos(6\pi i/1000))^3)$ and $(-(2/15)\sin(6\pi i/1000), (4/5)\sin(2\pi i/1000))$.

Olive Branch

This image shows 4000 circles. For $k=1,2,3,\ldots,4000$ the center of the $k$-th circle is $(X(k), Y(k))$ and the radius of the $k$-th circle is $R(k)$, where

$\begin{array}{llll}X(k)&=&&(2k/4000)+(1/28)\sin(42\pi k/4000)\\&&+ &(1/9)((\sin(21\pi k/4000))^8)\\&&+ &(1/4)((\sin(21\pi k/4000))^6)*\\ && &\sin((2\pi /5)(k/4000)^{12}),\end{array}$

$\begin{array}{llll}Y(k)&=&&(1/4)(k/4000)^2\\&&+&(1/4)(((\sin(21\pi k/4000))^5) \\&& + &(1/28)\sin(42\pi k/4000))*\\&&&(\cos((\pi /2)(k/4000)^{12})),\end{array}$

$\begin{array}{lll}R(k)& =& (1/170)+(1/67)((\sin(42\pi k/4000))^2)*\\& &(1-((\cos(21\pi k/4000))^4)).\end{array}$

Butterfly (1)

This image shows 40,000 circles. For $k=1,2,3,\ldots,40{,}000$ the center of the $k$-th circle is $(X(k), Y(k))$ and the radius of the $k$-th circle is $R(k)$, where

$\begin{array}{lll}X(k)&=&(6/5)((\cos(141\pi k/40{,}000))^9)(1-(1/2)(\sin(\pi k/40{,}000))^3)*\\&&(1-(1/4)((\cos(2\pi k/40{,}000))^{30})(1+(2/3)(\cos(30\pi k/40{,}000))^{20})-\\&&((\sin(2\pi k/40{,}000))^{10})((\sin(6\pi k/40{,}000))^{10})*\\&&((1/5)+(4/5)(\cos(24\pi k/40{,}000))^{20})),\end{array}$

$\begin{array}{lll}Y(k)&=&\cos(2\pi k/40{,}000)((\cos(141\pi k/40{,}000))^2)(1+(1/4)((\cos(\pi k/40{,}000))^{24})*\\&&((\cos(3\pi k/40{,}000))^{24})(\cos(19\pi k/40{,}000))^{24}),\end{array}$

$\begin{array}{lll}R(k)&=&(1/100)+(1/40)(((\cos(2820\pi k/40{,}000))^6)+\\&&(\sin(141\pi k/40{,}000))^2)(1-((\cos(\pi k/40{,}000))^{16})*\\&&((\cos(3\pi k/40{,}000))^{16})(\cos(12\pi k/40{,}000))^{16}).\end{array}$

Butterfly (3)

This image shows 40,000 circles. For $k=1,2,3,\ldots,40{,}000$ the center of the $k$-th circle is $(X(k), Y(k))$ and the radius of the $k$-th circle is $R(k)$, where

$\begin{array}{lll}X(k)&=&(3/2)((\cos(141\pi k/40{,}000))^9)*\\&&(1-(1/2)\sin(\pi k/40{,}000))*\\&&(1-(1/4)((\cos(2\pi k/40{,}000))^{30})*\\&&(1+(\cos(32\pi k/40{,}000))^{20}))*\\&&(1-(1/2)((\sin(2\pi k/40{,}000))^{30})*\\&&((\sin(6\pi k/40{,}000))^{10})*\\&&((1/2)+(1/2)(\sin(18\pi k/40{,}000))^{20})),\end{array}$

$\begin{array}{lll}Y(k)&=&\cos(2\pi k/40{,}000)*\\&&((\cos(141\pi k/40{,}000))^2)*\\&&(1+(1/4)((\cos(\pi k/40{,}000))^{24})*\\&&((\cos(3\pi k/40{,}000))^{24})*\\&&(\cos(21\pi k/40{,}000))^{24}),\end{array}$

$\begin{array}{lllcl}R(k)&=&(1/100)&+&(1/40)(((\cos(141\pi k/40{,}000))^{14})+(\sin(141\pi k/40{,}000))^6)*\\&&&&(1-((\cos(\pi k/40{,}000))^{16})((\cos(3\pi k/40{,}000))^{16})*\\&&&&(\cos(12\pi k/40{,}000))^{16}).\end{array}$

Heart (2)

This image shows 2500 ellipses. For each $k=1,2,3,\ldots,2500$ the foci of the $k$-th ellipse are:
$A(k)+iB(k)+C(k)e^{68\pi i k/2500}$
and
$A(k)+iB(k)-C(k)e^{68\pi i k/2500}$
and the eccentricity of the $k$-th ellipse is $D(k)$, where
$A(k)=(-3/2)((\sin(2\pi k/2500))^3)+(3/10)((\sin(2\pi k/2500))^7),$

$B(k)=\sin((2\pi k/1875)+(\pi /6))+(1/4)(\sin((2\pi k/1875)+(\pi /6)))^3,$

$C(k)=(2/15)-(1/8)\cos(\pi k/625),$

$D(k)=(49/50)-(1/7)(\sin(4\pi k/2500))^4.$

8,000 Ellipses

This image shows 8,000 ellipses. For each $k=1,2,3,\ldots,8000$ the foci of the $k$-th ellipse are:
$A(k)+iB(k)+C(k)e^{300\pi ik/8000}$
and
$A(k)+iB(k)-C(k)e^{300\pi ik/8000}$
and the eccentricity of the $k$-th ellipse is $D(k)$, where
$\begin{array}{llll}A(k)&=&&(3/4)\sin(2\pi k/8000)\cos(6\pi k/8000)\\&&+&(1/4)\sin(28\pi k/8000),\end{array}$

$\begin{array}{llll}B(k)&=&&(3/4)\cos(2\pi k/8000)\cos(8\pi k/8000)\\&&+&(1/4)\cos(28\pi k/8000),\end{array}$

$\begin{array}{lll}C(k)&=&(1/18)+(1/20)\cos(24\pi k/8000),\end{array}$

$ \begin{array}{lll} D(k)&=&(49/50)-(1/7)(\sin(10\pi k/8000))^4.\end{array}$

Ring

This image shows 5600 ellipses. For each $k=1,2,3,\ldots,5600$ the foci of the $k$-th ellipse are:
$A(k)+iB(k)+C(k)e^{44\pi ik/5600}$
and
$A(k)+iB(k)-C(k)e^{44\pi ik/5600}$
and the eccentricity of the $k$-th ellipse is $D(k)$, where
$\begin{array}{lll}A(k)&=&(\cos(28\pi k/5600))^3,\end{array}$


$\begin{array}{llll}B(k)&=&&\sin(28\pi k/5600)\\&&+&(1/4)(\cos((14\pi k/5600)-(7\pi /4)))^{18},\end{array}$


$\begin{array}{lll}C(k)&=&(1/70)+(1/6)+(1/6)\sin(28\pi k/5600),\end{array}$


$\begin{array}{lll}D(k)&=&(399/400)-(1/6)(\sin(28\pi k/5600))^8.\end{array}$

A Bird in Flight (2016)

This image shows all circles of the form:
$(x-A(k))^2+(y-B(k))^2=(R(k))^2$, for $k=-10000, -9999, \ldots , 9999, 10000$, where

$\begin{array}{lllcl}A(k)&=&(3k/20{,}000)&+&\sin((\pi /2)(k/10{,}000)^7)((\cos(41\pi k/10{,}000))^6)\\&&&+&(1/4)((\cos(41\pi k/10{,}000))^{16})((\cos(\pi k/20{,}000))^{12})\sin(6\pi k/10{,}000),\end{array}$

$\begin{array}{lll}B(k)&=&-\cos((\pi /2)(k/10{,}000)^7)*\\&&(1+(3/2)(\cos(\pi k/20{,}000)\cos(3\pi k/20{,}000))^6)*\\&&((\cos(41\pi k/10{,}000))^6)+(1/2)(\cos(3\pi k/100{,}000)\cos(9\pi k/100{,}000)\cos(18\pi k/100{,}000))^{10},\end{array}$

$\begin{array}{lllcl}R(k)&=&(1/50)&+&(1/10)((\sin(41\pi k/10{,}000)\sin(9\pi k/100{,}000))^2)\\&&&+&(1/20)((\cos(41\pi k/10{,}000))^2)((\cos(\pi k/20{,}000))^{10}).\end{array}$