Mathematical Logic for Computer Science
Abstract
Art and Mathematics: Analogy
The beauty of art relies on a person, who visualizes it, feels it and creates it. It stimulates us to see, hear and feel things that aren’t part of the material world. There has been a striking relation between mathematics and art. There are several patterns of nature that seems artistic and follows mathematical principles. In fact, mathematics and art have long momentous connection. Many painters turned into mathematicians and many art works have underlying principles in it.
Visual Mathematics
Right from Da Vinci’s Golden ratio to modern Fractal art and computer animation, art has been a medium to express certain mathematical principles. The modern way of expressing art begins with the play of binary numbers defining certain algorithms to create unique masterpieces. Mathematics is inherent in all artworks. Even soap bubbles follow certain symmetry. Geometry has become a form of art leading to evolution of origami. Southern India is famous for their patterned art form called as kolam/ Rangoli created from dots which have line, mirror rotational symmetry. Even these decorative patterns follow mathematical calculations and some of them are based on Fibonacci numbers as well. A simple system to generate symmetric square kolams is based on the identity n^2 = (n-2) ^2 + 4(n-1). In this way, patterns are made and then converted into geometrical loops in a decorative way.
Both math and art breaks the boundaries of reality. It’s all about expressing the boundless realm we have within us and which in our real life can only come in finite ways. Algorithms, Codes and Patterns are all forms of art and we have to explore the possibility of linking several unnoticed beautiful creations of nature.
The sciences of collaborating the computer programming consist of mathematical and logical expressions and art forms are case studied with some of the innovative approaches. In the contemporary world of changing technologies within the creative arena the conclusion of the thesis will bring the opportunity to learn more about the smart approach to create art in any forms. The method of code generated artworks has created unusual designs using interesting platforms and has been modus operandi in painting technology. Generative art evolved from artwork and programming allows the artists to evolve them digitally and create novel masterpieces involving intrinsic mathematical codes.
Fractal Art
It is kind of algorithmic art that exist as electronic images derived from certain calculations. These mathematical calculations are represented as images and animations. The power of modern computers allows us to visualize complex objects. Fractal art is used in modeling mountains, fire, and other natural substances. It is basically a fragmented geometric shape arranged in random fashion.
Evolution of Fractal Art
A snowflake evolved from an equilateral triangle is a kind of fractal art . The mathematical phenomena of recursive self-similarity are exhibited in these art works. Ocean waves, DNA, heartbeat and crystals can be depicted using fractal art. From Generation of new music to computer graphic design for organic environments Fractal art has been a pioneer in projecting these. These art follow equations like Zn+1 = Zn^2 + C where n is the iteration number and each iteration number is given a different color.
Mathematics and geometry in art is clearly illustrated in these forms of art. To the mathematician they bring a glut of very difficult surmises that no one can unravel and to the artist they provide pillars around which imagination can play at will. In this way art and math are used to create graphic of intangible quality.
Fractal art allows us to use iterative procedure to generate unique designs and provides a new experience of the beauty of mathematical calculations and parameters.
Dynamic Painting
Dynamic painting is a movement in visual arts that has been generated algorithmically by a computer system. It can be considered a generative art. An artist begins creation of a Dynamic Painting by visualizing an idea for the painting, choosing colors, shapes and principles of their development over time. Using a many procedures that have deep origins in the styles of conventional paintings, an artist transforms the brush strokes into algorithms that can exactly express his original design of a digital painting. These algorithms govern the vitality of the developed art.
The deployment of computational algorithms for generating dynamic artworks might lead to a never ending picture that will not repeat itself from which animation, abstract designs and novel creations can be derived. The algorithm creates a slight mutation in the original system and evolves it into numerous paintings. A prudent algorithm uses a computer to generate a series of unique images that follow the style and pattern of original painting. The painting is always in the state of an everlasting transformation.
The concept of dynamic painting made us realize the value of self-shifting digital paintings and how algorithms are used to generate dynamism. This makes us think to deploy and work on novel algorithms or at computational level to generate unique artworks that has even the possibility of creating painterly animations. This tells how interesting platforms/ libraries are used to generate and different algorithms like ‘The Evolutionary algorithm’ are employed.
Platonic Solids in Art
Platonic solids are recurring subject in western art. These belong to the group of geometric figures called polyhedral. The regular solids that are possible are: Cube, Tetrahedron, Octahedron, Icosahedron and dodecahedron. These were the elements that were linked with four elements of nature. The cube to earth (most immobile), icosahedron to water (least mobile), tetrahedron to fire (most mobile) and octahedron to air (intermediate) and the fifth construction used for adorning the constellations. In this way, geometric solids were unified with nature. The golden ratio was contained in pentagons and several illustrations from nature follow the rule.
Modern day algorithms are cleverly used to produce smooth, rounded forms from coarse polygons. One can generate many forms with entirely diverse attributes by modifying certain algorithms and process parameters. Mostly the process is iterative and this produce entirely new forms unlike those seen in nature. This is a kind of generative art that employs algorithms to make out masterworks. The use of computational algorithms establishes a strong interconnection between mathematics and art till date and the future is unpredictable. The artwork can be fed into computer which can find out the underlying principle or fundamental notion inside it. Viral structures are built of regular icosahedron, grids based on atmospheric flow are based on icosahedron and the common dice is a platonic solid.
This kind of art that employs algorithms display a novel aesthetic and an amazing complexity whose degree of branching, porosity, topography and fractalization can be modified. These determine the surface development and variation in textures. Platonic solids date back to century of Plato but till date it has been an interesting area to explore for novel graphic design.
Conclusion
Generative art evolved from artwork and programming allows the artists to evolve them digitally and create novel masterpieces. The concept of dynamic painting, Fractal art and platonic solids boils down to concept of employing unified algorithm to create new style of artworks. These topics clearly establish a strong connection between mathematics and art. This enables us to proceed with an iterative algorithm to create generative artwork. In this way the artist chooses the attributes of his style to be incorporated and develop it into a distinctive work.
Bibliography
- Hyung W. Kang , Uday K.Chakraborty , Charles K.Chui , Wenjie He(2005);”Multi-Scale Stroke-based Rendering by Evolutionary Algorithm”.In Proc. International Workshop on Frontiers of Evolutionary Algorithms (JCIS),pp 546-549.
- David G. Stork (2009)”Computer Vision and Computer Graphics Analysis of paintings and drawings: An introduction to the literature” CAIP ’09 Proceedings of the 13th International Conference on Computer Analysis of Images and Patterns
- Dr. Gift Siromoney(1974)“Array grammars and Kolam” Computer Graphics and Image processing Vol 3, pp – 63-88
- Dr. T.Robinson, Nagata Shojiro(2006)”Digitalization of Kolam Patterns and Tactile Kolam Tools”- In Formal Models , Languages and Applications , Series in Machine Perception and Artificial Intelligence Vol 66, pp 353-362
- Glassner .A (1996)”Origami Platonic solids” Computer Graphics and Applications IEEE , Vol 16 Issue-4;pp-85-91
- Michael Hansmeyer (2010)”Subdivision Beyond Smoothness” Computational Aesthetics Conference Proceedings , London pp- 120-129
- Richard Taylor(2001) “Fractals : A Resonance between art and nature”, Symmetry : Art and Science , ISIS Journal Vol 1-2 pp-194
- Richard Taylor (1999)” Fractal Expressionism” Physics World Vol 12 , no 10 , pp-25
- Claude Paul Bruter (2002) Mathematics and art: Mathematical visualization in art and education
- Carla Farsi, Doug Craft (2005) Mathematics and the Arts, University of Colorado
- Ouellette, Jennifer (2001) Pollock’s Fractals, DISCOVER Vol.22 No11.
