In 1988, Michael Barnsley developped the theory of Iterated Function Systems (IFS in short), a way of describing and generating fractals well suited for programming. The functions used were all affine transforms, and can generate objects like ferns, Sierpinski gaskets… Apophysis (www. aphophysis.org) is a computer program, initiated by Scott Draves, which generalize IFS by using non linear functions instead of affine ones. Apophysis interface is quite complete and user friendly, allowing easy and fast experiments. This program is a good illustration of what experimental mathematics can offer : fast tests, large universe of possibilities.

Kardioid (kalp şekilli eğri) her yerde karşımıza çıkabilecek bir eğridir ve birçok şekilde oluşturulabilir: bir madeni parayı başka bir madeni paranın etrafında döndürünce ya da süt tenceresine düşen güneş ışığının yansımalarında… (Basit olmayan matematiksel bir nesneyle bir çocuğun herhalde ilk karşılaşmasıdır bu.) Pedoe yöntemi, bir kardioidi bir çember ailesinin zarfı olarak tanımlar. Bu çemberleri 3 boyutlu bir nesne elde edecek şekilde döndürür ve bu döndürme işlemini değişik yollarla komtrol ederseniz binlece cazip şekil elde edebilirsiniz. Tıpkı bu resimdeki gibi… Bu resimden aldığı ilhamla, İngiliz şapkacı Gabriela Ligenza dünyada üç boyutlu yazıcıdan çıkarılmış ilk şapkayı üretti. Şu makalede biraz daha açıklama ve örnek bulabilirsiniz:
http://archive. bridgesmathart.org/2014/bridges2014-349.pdf

It is know, that every archimedean solid has an hamiltonian circuit relying al of its vertices.
It is not so easy to find one of those circuits when your are more a computer scientist than a mathematician !
I just succeeded in finding a 104 vertices path on a truncated icosidodecahedron (instead of 120)
i then connected those 104 points with a curved line, and duplicated five time this path, with the same transformations that produces the compound of five tetrahedra from a single initial .
The last trick was to rotate the structure in order to hide the less symetric parts.
Mathematics can be beautiful, even if not perfect.

Doyle Spirals are a particular case of circle packing: each circle is surrounded with 6 tangent circles. When the parameters are well chosen (and this is the hard stuff), this circle packing can tile the plane. Applying to this tiling geometric transformations preserving the tangency property, one can create elegant patterns. This work was initiated while looking at the beautiful and challenging images created by Jos Leys.

Circle packing can be seen as the art of placing tangent circles on the plane, leaving as little unoccupied space as possible. The so called Apollonian Gasket recursively fills the space between tangent circles. But this method leaves large empty spaces inside the initial circles. The Steiner Chain method puts a chain of tangent circles between two circles. Here, I combined two methods and applies them repeatedly. Moreover, in order add more exploration possibilities, transforms that preserves circles and tangency are applied : circle inversions and mobius transforms