ABSTRACT
In this project, the concept of art o dancing and the areas of mathematics which are applied to dancing were discussed. Then, how the knowledge of geometry can help choreographers, dancers, etc. maximize their potentials was also discussed.
CHAPTER ONE
1.0 GENERAL INTRODUCTION
Over the years our society has considered dance and mathematics to be near polar opposites. Dance is a fun activity weather performing or observing; it involves other people and can be source of great satisfaction. On the other hand, the general populace considers mathematics to be a dull and overly complicated source of constant frustration. The two seem to have nothing in common. And yet upon close investigation, many connections and similarities reveal themselves. Mathematics is present in dance. It is not the mathematics of simple number manipulation; we do not attempt to add or integrate through movement, instead we would like to employ abstract mathematics and various methods of analysis to understand dance at a deeper level.
Dance can be used to teach the fundamentals of mathematics and provide the students with basic intuition about the abstract concepts involved. Actually getting to experience math at work might be more exciting to students than “Two trains leave cities A and B going at 60mls…” Applying mathematics to more familiar ‘real life’ situations would certainly remove the stigma of the field being dry and inaccessible. Thus the awareness of how mathematics and dance interact and draw from each other can help us understand both areas on a whole new level and keep the inquiry exciting.
1.1 BACKGROUND OF THE STUDY
Dance is a performance art form consisting of purposefully selected sequences of human movement. This movement has aesthetic and symbolic value, and is acknowledged as dance by performers and observers within a particular culture. Dance can be categorized and described by its choreography, by its repertoire of movements, or by its historical period or place of origin. An important distinction is to be drawn between the contexts of theatrical and participatory dance, although these two categories are not always completely separate; both may have special functions, whether social, ceremonial, competitive, e.t.c. Others disciplines of human movement are sometimes said to have a dance-like quality, including martial arts, gymnastics, figure skating, synchronized swimming and many other forms of athletics.
Theatrical dance, also called performance or concert dance, is intended primarily as a spectacle, usually a performance upon a stage by virtuoso dancers. It often tells a story, perhaps using mime, (body motion), costume and scenery, or else it may simply interpret the musical accompaniment, which is often specially composed. Examples are western ballet and modern dance, Classical Indian dance and Chinese and Japanese song and dance dramas. Most classical forms are centred upon dance alone, but performance dance may also appear in opera and other forms of musical theatre. Participatory dance, on the other hand, whether it be a folk dance, a social dance, a group dance such as a line, circle, chain or square dance, or a partner dance such as Western ballroom dancing, is undertaken primarily for a common purpose, such as social interaction or exercise, of participants rather than onlookers. Such dance seldom has any narrative. Participatory dancers often a employ the same movements and steps but, for example, in the rave culture of electronic dance music, vast crowds may engage in free dance, uncoordinated with those around them. On the other hand, some cultures lay down strict rules as to the particular dances in which, for example, men, women and children may or must participate.
ALGEBRAIC EQUATIONS
An equation of the type fn = 0, where fn is a polynomial of degree n in one or more variables (n0) is called an algebraic equation. An algebraic equation is one variable is an equation of the form
A non-negative integers, are called the coefficients of the equation and x is an unknown to be found. It is assumed that the coefficients of the algebraic equation are not all equal to zero if if , then n is called the degree of the equation.
The values of the unknown x which satisfy an algebraic equation are known as the roots of the equation. The roots of a polynomial are related to its coefficients by viete’s formula. To solve an equation means to find all its roots contained in the range of values of the unknowns (s) under consideration. As far as applications are concerned, the most important case is that of coefficients and roots of an equation that are numbers of a certain kind e.g rational, real or complex. The case of the coefficients and roots being elements of an arbitrary field may also be considered. If a given number (or element of a field ) c is a root of the polynomial fn(x) then, in accordance with the Bezout theorem, fn(x) is divisible by x - c without remainder. The division may be performed according to the Horner scheme. A number (or element of a field) c is called a root of multiplicity k of a polynomial f(x) where k, if f(x) is divisible by (x-c)k, but is not divisible by (x-c)k+1, Roots of multiplicity one are called simple roots of the polynomial, other roots are called multiple roots. Each polynomial f(x) of degree n > 0 with coefficients, in a field p has at most n roots in this field p, each root being counted the number of times equal to its multiplicity (consequently, there are not more than n different roots).
In an algebraically closed field, any polynomial of degree n has exactly n roots (counted according to those multiplicity). In particular, this statements also applies to the field of complex numbers.
A polynomial, with coefficients from a field p is called irreducible over p if the polynomial Fn(x)= is irreducible over this field i.e cannot be represented as the product of other polynomials of degree lower than n over p. otherwise, both the polynomial and the corresponding equation are called reducible.
Polynomials of degree zero and zero itself are not considered to be reducible or irreducible. Whether a given polynomials is reducible or irreducible over a field p depends on the field in question. Thus, the polynomial x2 – 2 is irreducible over the field of rational numbers, since it has no rational roots, but is reducible over the field of real numbers;
x2 – 2=
Similarly, the polynomials x2 – 2 is irreducible over the field of real numbers, but is reducible over the field of real numbers, but is reducible over the field of complex numbers. Only polynomials of the first degree are irreducible over the field of complex numbers, and any polynomial can be decomposed into linear factors. Only polynomials of the first degree and polynomials of the second degree without real roots are irreducible over the field of real numbers (and all polynomials can be decomposed into products of linear and irreducible quadratic polynomials). Irreducible polynomials of all degree exist over the field of rational numbers, examples are the polynomials of the form xn – 2. The irreducibility of a polynomial over the field of rational numbers can often be established by Eisensteins criterion. If, for a polynomial of degree n > 0 with integral coefficients, there exists a prime number p such that the leading coefficient a0 is not divisible by p, all the remaining coefficients are divisible by p, and the constant term an is not divisible by p, then this polynomial is irreducible over the field of rational numbers. Let p be an arbitrary field, for each polynomial f(x) b of degree n > 1 i.e irreducible over p, there exist an extension of p containing at least one root of f(x). Moreover, there exists a splitting field of f(x) i.e a minimal extension of p in which this polynomial can be decomposed into linear factors. Every field has an algebraically closed extension.
1.2 STATEMENT OF THE PROBLEM
Over years, mathematics had been seen as a different subject whether at elementary levels or at advance levels. This is because people could not discover the applications of mathematics in our daily activities. In order to show the attractive effects of mathematics, this project examines the cordial relations between mathematics and art of dancing
1.3 AIM AND OBJECTIVE
The aim and objective of this project are as follows:
- To discuss an algebraic equations and its roles in the field of mathematics
- To discuss art of dancing, its importance, and different types of dance in accordance with different culture.
- To examine the relationship between algebraic equation and art of dancing.
- To look into some other areas of mathematics that are also related to dancing apart from algebraic equation.
- To show that dancers can maximize their potentials through perfect knowledge of mathematics.
1.4 SCOPE OF THE STUDY
Different aspects of mathematics such as trigonometry, Geometry, algebraic equations e.t.c are applicable to art of dancing. In this project, we examine majorly the relation of algebraic equations to art of dancing.
1.5 SIGNIFICANCE OF THE STUDY
This project shows that mathematics is not only dealing with differentiation, integration, and trigonometry e.t.c but also has many benefits in daily activities of human being. Hence, people will like to create more interest in it rather than running away from it
1.6 DEFINATION OF TERMS
The following are the definitions of some basic term relating to dance and algebraic equations;
1.6.1 CULTURE
This is the way of life, especially the general customs and beliefs, of a particular group of people at a particular time.
1.6.2 CHOREOGRAPHY
This is the art or practice of designing sequences of movements of physical bodies in which motion, firm, or both are specified. Choreography is used in a variety of fields including cinematography, show choir, theatre, synchronised swimming, animated art etc.
In the performing arts, choreography applies to human movement and from. In dance, choreography is also known as dance choreography or dance composition.
1.6.3 FOLK DANCES
These are dances developed by groups of people that reflect the traditional life of the people of a certain country or region.
1.6.4 STAGE
This is a designated space for the performance of productions. It serves as a space for actors or performers and a focal point for the members of the audience.
1.6.5 MUSICAL THEATRE
This is a form of theatrical performance that combines songs, spoken dialogue, acting, and dance.
1.6.6 ETHNOCHORIOLOGY
This is the study of dance through the application of a number of disciplines such as anthropology, musicology, ethnography etc.
1.6.7 ALGEBRAIC EQUATION
This an equation of the form M =N, where M and N are polynomials with coefficients in some field, often the field of the rational numbers.
e.g x3 – 5x + 2 = 0 is an algebraic equation with integer coefficients.
1.6.8 POLYNOMIAL
A polynomial in a variable x is an expression of the form
,
Where are constants
Examples of polynomials are:
3x3 – 5x + 4 is a polynomial of degree 2
-7X5 is a polynomial of degree 5.
1.6.9 FUNDAMENTAL THEOREM OF ALGEBRA
It states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with an imaginary part equal to zero. Equivalently, the theorem states that the field of complex numbers is algebraically closed.
1.6.9 VIETE THEOREM
This is a theorem which establishes relations between the roots and the coefficients of a polynomials.
Let f(x) be a polynomial of degree n with coefficients from some field and with leading coefficient 1. The polynomial f(x) splits over a field containing all the roots of F into linear factors.
F(x) =
Where are the roots of F(x), i=1,2,..,n
Vietes theorem asserts that the following relations hold;
Viete proved this relation for all n, but for positive roots only.
Examples
Given the polynomials F(x) =
Here a0 =-6, a1 =11, a2 = -6, a3 =1 and n = 3
The polynomials F(x) =
=(x-1)(x-2)(x-3)
Thus the zeros of f(x) are
, ,
By viete’s theorem
A0 =
An-2= a3-2 =a1=
(1)(2)+(1)(3)+(2)(3)=1
an-1= a3-1 =a2 = - ( )
=(1+2+3) =-6
1.6.10 ALGEBRAICALLY CLOSED FIELD
A field K is algebraically closed If any polynomials of non-zero degree over k has at least one root in k. in fact, it follows that for an algebraically closed field k each polynomial of degree n roots in k I.e each irreducible polynomials from the ring of polynomial k [x] is of degree one. A field k is algebraically closed if and only if it has no proper algebraic extension. For any field k, there exists a unique (up to isomorphism) algebraic extension of k that is algebraically closed; it is called algebraic closure of k and is usually denoted by . Any algebraically closed field containing k contains a subfield isomorphic to k.
1.6.11 EXTENSION OF A FIELD
A field extension K is a field containing a given field K as a subfield. The notation K/k means that k is an extension of the field k. in this case, K is sometimes called an over field of the field k.
Let k/k and L/k be two extensions of a field k. an isomorphism of fields K is called an isomorphism of extensions (or a k- isomorphism) of fields of is then identity on k. if an isomorphism of extensions exists, then the extension are said to be isomorphic if k = L, is called an automorphism of the extension k/k. the set of all automorphisma of an extension forms a group, Aut (k/k)
1.6.12 SYSTEM OF POLYNOMIAL EQUATIONS
This is a set of simultaneous equations f1=0, f2=0, f3=0 ... fn=0 where fi are polynomials in several variables x1, x2, x3, --- xn, I = 1,2…n, over Some field k. usually the field k is either the field of rational numbers or a finite field, although most of the theory applies to any field. A solution is a set of the values for the xi which make all of the equations true and which belong to some algebraically closed field extension k of k where k is the field of complex numbers and k is the field of rational numbers.
1.6.13 OVERDETERMINED AND UNDERDETERMINED SYSTEM
A system is over determined if the number of equations is higher than the number of variables. System is inconsistent if it has no solutions. By Hilbert’s Nullstellensatz, this means that 1 is a linear combination (with polynomials as coefficients) of the first members of the equations. Most but not all overdetermined systems are inconsistent. For example the system
X3 – 1 = 0
X2 – 1 = 0
Is overdetermined (having two equations but only one unknown), but it is consistent since it has the solution x=1.
A system is undetermined if the number of equations is lower than the number of the variables.
An undetermined system is either inconsistent or has infinitely many solutions in an algebraically closed extension k of k.
1.6.14 ALGEBRAIC GEOMETRY
This is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problem about those sets of zero.
The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory.
1.6.15 ELEMENTARY FUNCTION
This is a function of one variable which is the composition of a finite number of arithmetic operations (+, -, x, /), exponential, logarithms, constants, and solutions of algebraic equations (a generalization of nth roots).
The elementary functions include the trigonometric and hyperbolic functions and their inverse, as they are expressible with complex exponentials and logarithms. Elementary functions are analytic at all but a finite number of points.
Example of elementary functions include:
- Adding e.g x+1
- Multiplication e.g 7x
An example of a function that is not elementary is the error function
erf(x) =
1.6.16 TRANSCENDENTAL FUNCTION
This is an analytic function that does not satisfy a polynomial equation. Examples include the exponential function, the logarithm, and the trigonometric functions. For instance
Any function that is not transcendental is algebraic. The indefinite integral of many algebraic functions is transcendental. For instance, the logarithm function arose from the reciprocal function in an effort to find the area of a hyperbolic sector. Differential algebra examines how integration frequently creates functions that are algebraically independent of some class, such as when one takes polynomials with trigonometric functions as variables.
