Drawings of real-life objects Naderi Yeganeh has introduced two methods to draw real-life objects with mathematical formulas. For example, by using this method, he found some shapes that resemble birds, fishes and sailing boats. 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. Naderi Yeganeh says: "In order to create such shapes, it is very useful to know the properties of the trigonometric functions". More recently, he has introduced a method to describe an image pixel-by-pixel by using a network of mathematical functions.
A Bird in Flight 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 k=1, 2, 3, \ldots, 500 the endpoints of the k-th line segment are: : \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 : \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. 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 \left(A(k), B(k)\right) and radius R(k), where k=-10000, -9999, \ldots, 9999, 10000, and : A(k)=\frac{3k}{20000}+\sin\left(\frac{\pi }{2}\left(\frac{k}{10000}\right)^7\right)\cos^6\left(\frac{41\pi k}{10000}\right)+\frac{1}{4}\cos^{16}\left(\frac{41\pi k}{10000}\right)\cos^{12}\left(\frac{\pi k}{20000}\right)\sin\left(\frac{6\pi k}{10000}\right), : \begin{align} B(k)= & -\cos\left(\frac{\pi}{2}\left(\frac{k}{10000}\right)^7\right)\left(1+\frac{3}{2}\cos^6\left(\frac{\pi k}{20000}\right)\cos^6\left(\frac{3\pi k}{20000}\right)\right)\cos^6\left(\frac{41\pi k}{10000}\right) \\ & +\frac{1}{2}\cos^{10}\left(\frac{3\pi k}{100000}\right)\cos^{10}\left(\frac{9\pi k}{100000}\right)\cos^{10}\left(\frac{18\pi k}{100000}\right), \\ \end{align} : R(k)=\frac{1}{50}+\frac{1}{10}\sin^2\left(\frac{41\pi k}{10000}\right)\sin^2\left(\frac{9\pi k}{100000}\right)+\frac{1}{20}\cos^2\left(\frac{41\pi k}{10000}\right)\cos^{10}\left(\frac{\pi k}{20000}\right). 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. 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.
Fractals and tessellations Naderi Yeganeh has designed some fractals and tessellations inspired by the
continents. For example, in 2015, he described the fractal Africa with an
Africa-like octagon and its lateral inversion. And he has created tessellations with
Northern America-like and
South America-like polygons.
Fractal Africa Fractal Africa is a
fractal made of an infinite number of
Africa-like
octagons which was introduced by Naderi Yeganeh. The fractal's octagons are
similar to each other and they have some resemblance to the map of Africa. 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
golden ratio. File:Fractal Africa and Fibonacci Sequence Animation by Hamid Naderi Yeganeh.gif|alt= File:Fractal Africa and Fibonacci Sequence Animation 2 by Hamid Naderi Yeganeh.gif|alt= == Exhibitions and conferences ==