Hamid Naderi Yeganeh

Naderi Yeganeh has introduced two methods to draw real-life objects with mathematical formulas.[4] By the first method, he creates tens of thousands of computer-generated mathematical figures to find a few interesting shapes accidentally. Then he changes the equations a little bit in order to increase the resemblance of the accidentally found shapes to real life objects.[40][41][42] For example, by using this method, he found some shapes that resemble birds, fishes and sailing boats.[43][44][45][46] In the second method, he draws a real life object with a step-by-step process. In each step, he tries to find out which mathematical formulas will produce the drawing.[47][4] For example, by using this method, he drew birds in flight, butterflies, human faces and plants using trigonometric functions.[48][49][50][47][51][52] Naderi Yeganeh says: "In order to create such shapes, it is very useful to know the properties of the trigonometric functions".[53][48] In 2018, in an interview with the Sharif University of Technology Public Relations Naderi Yeganeh said: I use mathematical concepts in a work of art in a way that it could be thoroughly explained in a paragraph. That makes the understanding of an artwork's scientific underpinning easier.[54]

A version of A Bird in Flight made up of 500 line segments, which Naderi Yeganeh created by his first method.

Another version of A Bird in Flight made up of 20,001 circles, which Naderi Yeganeh created by his second method.[49]

An instance of drawing real things by using Yeganeh's methods is A Bird in Flight, which is the name of a number of bird-like geometric shapes introduced by Naderi Yeganeh. Yeganeh created those drawings by using the two methods mentioned above. An example of A Bird in Flight that was created by his first method is made of 500 segments defined in a Cartesian plane where for each ${\displaystyle k=1,2,3,\ldots ,500}$ the endpoints of the ${\displaystyle k}$-th line segment are:

${\displaystyle \left({\frac {3}{2}}\sin ^{7}\left({\frac {2\pi k}{500}}+{\frac {\pi }{3}}\right),\,{\frac {1}{4}}\cos ^{2}\left({\frac {6\pi k}{500}}\right)\right)}$

and

${\displaystyle \left({\frac {1}{5}}\sin \left({\frac {6\pi k}{500}}+{\frac {\pi }{5}}\right),\,{\frac {-2}{3}}\sin ^{2}\left({\frac {2\pi k}{500}}-{\frac {\pi }{3}}\right)\right)}$.

The 500 line segments defined above together form a shape in the Cartesian plane that resembles a flying bird. Looking at the line segments on the wings of the bird causes an optical illusion and may trick the viewer into thinking that the line segments are curved lines. Therefore, the shape can also be considered as an optical artwork.[49][8][13][55][52][56] Another version of A Bird in Flight that was designed by Naderi Yeganeh's second method is the union of all of the circles with center ${\displaystyle \left(A(k),B(k)\right)}$ and radius ${\displaystyle R(k)}$, where ${\displaystyle k=-10000,-9999,\ldots ,9999,10000}$, and

The set of the 20,001 circles defined above form a subset of the Cartesian plane that resembles a flying bird. Although this version's equations are a lot more complicated than the version made of 500 segments, it has a far better resemblance to a real flying bird.[49] It uses the periodic nature of trigonometric functions to form the feathers.[57] Other works similar to this version of A Bird in Flight that was released by Naderi Yeganeh in 2016 are in the form of a flying parrot, magpie and stork.[48][58]

Naderi Yeganeh has designed some fractals and tessellations inspired by the continents.[59][60] For example, in 2015, he described the fractal Africa with an Africa-like octagon and its lateral inversion.[43][14][61][26] And he has created tessellations with Northern America-like and South America-like polygons.[62]

Fractal Africa

Fractal Africa is a fractal made of an infinite number of Africa-like octagons which was introduced by Naderi Yeganeh.[43][63] The fractal's octagons are similar to each other and they have some resemblance to the map of Africa.[8][64] The numbers of octagons of different sizes in the fractal is related to the Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, 34, .... The height of the biggest octagon of the fractal is φ times larger than the height of second octagon; where φ is the well-known golden ratio.[14][13][65]