Mathematics, Computer Mathematical Systems, Creativity, Art

Mathematics, Computer Mathematical Systems, Creativity, Art

Sergiy Rakov (G.S. Skovoroda National Pedagogical University, Ukraine), Viktor Gorokh (G.S. Skovoroda National Pedagogical University, Ukraine) and Kirill Osenkov (G.S. Skovoroda National Pedagogical University, Ukraine)
DOI: 10.4018/978-1-60566-352-4.ch015
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The chapter discusses the possibilities modern IT opens for Mathematics and its applications to real life, in particular to Art – by an example of automated construction of caricatures. The research approach in math education which is based on computer modeling as a methodology of a competency approach is discussed as well as the future educational environment with free access to Internet resources and possibilities to operate software in distributive mode. Many examples are given – from simple illustrations to new math facts invented with original dynamic geometry package DG and proved with the help of computer algebra system Derive.
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Creativity as a common root of producing (discovering) new knowledge in Math and Science and producing new artifacts in Art (Philosophical aspects)

The tradition to oppose Math & Sciences to Art based on dichotomy “non-creative (algorithmic, reproductive) – creative (non–algorithmic, productive)” is wrong and may be concerned only in the framework of usage of the Math & Science and Art artifacts but not in the process of their creation; the processes of constructing new artifacts (both new Math & Science knowledge and new Art products) have the same base – CREATIVITY, and both these processes are similar in both cases (concentration, meditation, modeling, associative and critical thinking etc.). In wider context CREATIVITY is the main goal of a human being, CREATIVITY is the only thing that God can justify him/her (e.g. it is his/her anthropodecium (Berdyaev N., 1931)) – each personally or a mankind in a whole; only CREATIVITY transforms a human life into human being and approach human to God (Berdyaev N., 1931). It does not matter in which area you are working (are you a scientist, an artist, an engineer, a worker, a teacher etc.) – the only thing that matters is: are you CREATIVE or not. CREATIVITY is a God for mankind, the only God and the entire God, the Trinity of God: the origin, the existence and the soul of being. For mankind the human activities, which have sense only if they have the CREATIVE matter, are conditionally divided onto some distinct branches: Technologies, Science, Philosophy, Art, Ethic (in particular religion), each of which plays more or less definite role in definite historical periods of any individuum or community (tribe, nation, country etc.) but during the historical periods one of these branches in their leading role replace one other. Nowadays it seems that technologies, in particular IT, play the main role in civilization progress, stimulating other spheres: Science, Art, Ethic and that is why modern society is called an Information Society (I-Society). But technology itself can’t be the goal of civilization progress and it is obvious that this type of society will be changed by the knowledge society (K-Society), in which the main system leading role would be played by the Science, inflating and stimulating other branches of human activities. Poverty of mankind, growing the number and the scale of global ecological problems should demonstrate bounding of such models of society as I-Society and K–society and maybe Art and Ethic will be the base of a new type of society – Soul society (S–society) and the concept of Sustainable society (economic development, social development and ecological stability) is a step in this direction. IT helps to create crucially new possibilities in all other spheres of human being: human activities, human work, human leisure etc. In particular IT creates a base for new possibilities in Art, Science, Math. etc. (for example computer graphics is beneficial in a lot of IT applications). In the picture below the “magic curves” are shown and all of them are the graph of the one (!) parametric equation , depending from the only parameter – real number. What are these graphical compositions? Mathematics? IT? Art? The reader is proposed to answer these questions for himself/herself and try to estimate the approximate values of p corresponding to each of the Pictures 1– 61. Maybe these pictures are the synthesis of all these – somewhat entity of Math, IT, Art etc. and Math, IT, Art etc. are all only the sides of this entity which may exist separately only in our heads – if our heads are not too creative, but in creative heads they all compose to this entity which is the entity itself.

Figure 1.

Figure 2.

Dynamic drawing “Magic Curves” in the DG environment

Key Terms in this Chapter

Research Approach in Education: Research approach in education is a methodology of competence paradigm of education based at the principles of discovering new knowledge through learning explorations which reflect the ways of research work. Research approach in education is the reflection of educational systems on the challenges of new type of societies – knowledge society (K-society) and sustainable society (S-society) and widely use the possibilities of information and communication technologies (ICT) in computer modeling and computer based laboratories (CBL) with appropriate software for automatic and automate experiments.

Computer Algebra System: A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. The core functionality of a CAS is manipulation of mathematical expressions in symbolic form. The symbolic manipulations supported typically include: simplification to the smallest possible expression, substitution of symbolic, partial and total differentiation, symbolic constrained and unconstrained global optimization,solution of linear and some non-linear equations over various domains, solution of some differential and difference equations, taking some limits, some indefinite and definite integration, including multidimensional integrals, arbitrary-precision numeric operations etc.

Social Constructivism: Constructivism is an epistemology, a learning or meaning-making theory, which offers an explanation of the nature of knowledge and how human beings learn. It maintains that individuals create or construct their own new understandings or knowledge through the interaction of what they already know and believe and the ideas, events, and activities with which they come in contact (Cannella & Reiff, 1994; Richardson, 1997). Knowledge is acquired through involvement with content instead of imitation or repetition (Kroll & LaBoskey, 1996). Learning activities in constructivist settings are characterized by active engagement, inquiry, problem solving, and collaboration with others. Rather than a dispenser of knowledge, the teacher is a guide, facilitator, and co-explorer who encourages learners to question, challenge, and formulate their own ideas, opinions, and conclusions. “Correct” answers and single interpretations are de-emphasized.

Dynamic Geometry Systems: Dynamic Geometry Systems (DGS) or Interactive geometry software (IGS, also called “dynamic geometry environments”, DGEs) are computer programs which allow one to create and then manipulate geometric constructions, primarily in plane geometry. One starts construction by putting a few points and using them to define new objects such as lines, circles or other points. After some construction is done, one can move the points one started with and see how the construction changes.

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Editorial Advisory Board
Table of Contents
Gabriele Meiselwitz
James Braman, Giovanni Vincenti, Goran Trajkovski
James Braman, Giovanni Vincenti, Goran Trajkovski
Chapter 1
Adérito Fernandes Marcos, Pedro Branco, João Álvaro Carvalho
Art objects might be described as symbolic objects that aim at stimulating emotions. They reach us through our senses (visual, auditory, tactile, or... Sample PDF
The Computer Medium in Digital Art's Creative Process
Chapter 2
Salah Uddin Ahmed, Letizia Jaccheri, Guttorm Sindre, Anna Trifonova
The interaction between art and technology, especially computing technology, is an increasing trend in the recent years. The context of this... Sample PDF
Conceptual Framework for the Intersection of Software and Art
Chapter 3
Joseph William Pruitt
The purpose of this chapter is to define the roles of engineering and design within the product development cycle looking at both the scientific and... Sample PDF
The Design of Engineering
Chapter 4
Jim Bizzocchi, Belgacem Ben Youssef
The chapter describes the synergistic integration of distinct research and creation agendas, each firmly grounded in its own set of practices and... Sample PDF
Ambient Video, Slow-Motion, and Convergent Domains of Practice
Chapter 5
Ethan Ham
Randomness is a slippery term that conveys different meanings in different disciplines. In mathematics, an individual number is random when there is... Sample PDF
Randomness, Chance, & Art
Chapter 6
Holography: Re-Defined  (pages 103-112)
Martin Richardson, Paul Scattergood
When writing this chapter it became apparent that we were not only exponents of digital holography, but also the critics. This is a problem when it... Sample PDF
Holography: Re-Defined
Chapter 7
Lorenzo Picinali
What is the real potential of computer science when applied to music? It is possible to synthesize a “real” guitar using physical modelling... Sample PDF
3D Sound Simulation over Headphones
Chapter 8
Raphael DiLuzio
This is a guide for working with a visual art form using a digital time-based medium. This chapter will provide an overview of the necessary... Sample PDF
Broken Cinema: The Eye and Hand in a Time-Based Art
Chapter 9
Ambivalent Interplay  (pages 146-161)
Heejoo Kim
The human vision, the most ubiquitous receptor of the human senses, has been the prevailing sensory organ for a noticeable manifestation of visual... Sample PDF
Ambivalent Interplay
Chapter 10
Yueh Hsiu Giffen Cheng
The development of net art originates from the rising of net media generally. During the past two decades, Net art has overthrown the standards of... Sample PDF
The Aesthetics of Net dot Art
Chapter 11
Nicola Quinn, Annette Aboulafia
People have used tools for artistic expression for millennia. Relatively recent is the use of digital technology to afford the creation of art.... Sample PDF
A Graphics Tablet as a Fine Art Tool
Chapter 12
Greg J. Smith
This text seeks to contextualize the history of and discourse surrounding information visualization. It positions visualization in relation to... Sample PDF
Information Visualization and Interface Culture
Chapter 13
Benjamin David Robert Bogart
“Memory Association Machine” (also known as “Self-Other Organizing Structure #1”) is the first prototype in a series of site-specific responsive... Sample PDF
Memory Association Machine
Chapter 14
Stefano De Luca, Eugenia Benelli, Francesco Altarocca, Dario Dussoni
Designing good and sound architectural projects is a hard job. Generally these kinds of projects involve many stakeholders, everyone with his/her... Sample PDF
EVO-PARK: Designing Better Architecture Projects Using Participated and Interactive Genetic Algorithms
Chapter 15
Sergiy Rakov, Viktor Gorokh, Kirill Osenkov
The chapter discusses the possibilities modern IT opens for Mathematics and its applications to real life, in particular to Art – by an example of... Sample PDF
Mathematics, Computer Mathematical Systems, Creativity, Art
Chapter 16
Jim Barta, Ron Eglash
Students who may typically view mathematics as a sterile and disjointed subject are learning new skills and concepts using a suite of virtual design... Sample PDF
Teaching Artful Expressions of Mathematical Beauty: Virtually Creating Native American Beadwork and Rug Weaving
Chapter 17
Mia Kalish
One visualization in Diné philosophy is four small dots arranged in a circular sequence at 90°, 0°, 270°, and 180°. Each position is associated with... Sample PDF
Visual Analytics and Conceptual Blending Theory
Chapter 18
Lindsay Grace
Enculturation is the act of passing cultural ideologies from one person to the other. It is what breeds innovation instead of new creation. It is... Sample PDF
The Challenge of Enculturation on Art
Chapter 19
Lindsay Grace
Software is philosophical. Software is designed by people who have been influenced by a specific understanding of the way objects, people and... Sample PDF
The Philosophies of Software
Chapter 20
Technological Social-ism  (pages 343-374)
Judson Wright
Culture is a byproduct of our brains. Moreover, we’ll look at ways culture also employs ritual (from shamanistic practices to grocery shopping) to... Sample PDF
Technological Social-ism
Chapter 21
Stephen A. Schrum
As creative people inhabit virtual worlds, they bring their ideas for art and performance with them into these brave new worlds. While at first... Sample PDF
Theatre in Second Life® Holds the VR Mirror up to Nature
Chapter 22
Machinima in Second Life  (pages 396-416)
Stephany Filimon
This chapter provides a brief history of machinima, films created by computer users within virtual worlds, and focuses on machinima produced within... Sample PDF
Machinima in Second Life
Chapter 23
Andrew Jinman
Massively multiplayer online role-playing games (MMORPGs) are becoming an increasingly popular recreational activity for social engagement.... Sample PDF
Player Motivation and Understanding Game Dynamics
About the Contributors