Who invented lines of symmetry

For example, the rotation U sends abc to cab. Because rotations and reflections in the plane are linear transformations, each of the six symmetries can be specified by writing down a 3-by-3 table of numbers. So the symmetries allow the permutation group to be represented as a group of matrices. Although mathematical symmetry may bring to mind a regular polygon or other geometric pattern, its roots pun unavoidable lie in algebra, in the solutions to polynomial equations.

Thus Stewart begins his account in ancient Babylon with the solution to quadratic equations. The Babylonians didn't have the algebraic notation to write down such a formula, but they had a recipe that was equivalent to it.

By the 18th century, mathematics and mathematical notation had matured to the point of finding explicit formulas for the roots of general polynomials of degree three cubic and degree four quartic.

The formulas look complicated, but they are made of simple building blocks: addition, subtraction, multiplication, division and radicals—such as square roots, cube roots, fourth roots and so on.

The existence of such formulas is summarized by the statement that all polynomials of degree four or less are solvable by radicals. For polynomials of degree five quintic and higher, no such formula has been found, because none exists: Although some fifth-degree polynomials are solvable by radicals, other fifth-degree polynomials are not.

This was proved by Niels Henrik Abel in , although nearly correct proofs were proposed as far back as These days, the result is not considered surprising: The requirements for an equation to be "solvable by radicals" are very restrictive. With such a small vocabulary of operations, one would expect that most interesting numbers cannot be written in that form. But Abel's proof was not the end of the story. A mystery remained: What distinguishes those polynomials of degree five that are solvable?

The official mathematical definition of a group does not explicitly mention symmetries; it concerns a set of elements that can be combined in pairs to form another element of the set, and certain axioms have to be satisfied.

The precise definition of a group is not important for the purpose of this review; also, Why Beauty Is Truth does not provide one. Suffice it to say that the mathematical concept of a group captures the essence of symmetry in abstract terms. The focus is on the operation that reveals the symmetry. The collection of symmetries of any object is a group, and every group is the collection of symmetries of some object. The symmetric objects of interest to mathematicians are not physical objects but mathematical entities.

In Galois's study of polynomials, the object was the set of roots of the polynomial. For polynomials of degree five, this is just a list of five numbers. And for Galois, the operation was rearranging the list of roots. When using an object as a pattern motif, it is convenient to assign it to one of the crystallographic point groups: in 2-D, there are 10 of these shown below ; in 3-D, there are In common notation, called Schoenflies notation after Arthur Moritz Schoenflies, a German mathematician:.

A lattice is a repeating pattern of points in space where an object can be repeated or more precisely, translated, glide reflected, or screw rotated. To make a pattern, a 2-D object which will have one of the 10 crystallographic point groups assigned to it is repeated along a 1-D or 2-D lattice. A 2-D object repeated along a 1-D lattice forms one of seven frieze groups. A 2-D object repeated along a 2-D lattice forms one of 17 wallpaper groups.

The various 3-D point groups repeated along the various 3-D lattices form varieties of space group. Also important is invariance under a fourth kind of transformation: scaling. Here, the idea of symmetry transformation is introduced. We look at some of the designs and find rotations and reflections and see that a rotation followed by a rotation is another rotation, a reflection followed by a reflection is a rotation, and the composition of a reflection and a rotation, in either order, is a reflection.

Further investigation reveals the fundamental properties of finite designs: 1 A pattern can have only rotations, but not only reflections and 2 If the identity is treated as a rotation and there are reflections, then there are just as many rotations as reflections. At this point, I introduce the Bhagavad-Gita verse and we spend quite a bit of time understanding the verse and how it can be interpreted in terms of symmetry transformations.

The first application of this way of thinking comes when we start working out the group table for the symmetry group of an equilateral triangle. After two symmetries are performed, one needs to determine what one symmetry is equivalent to the composition. We look at what stays the same. If one vertex of the triangle is left fixed, the composition is a reflection.

If no vertices are left fixed so that only the center is fixed , then the composition is a rotation. If we want to determine the type of a given transformation, look at what is fixed: if only a point is fixed, it is a rotation about that point; if a line is fixed, it is a reflection across that line; if everything is fixed, it is the identity.

As we move on to frieze ornaments and wallpaper patterns, to identify all possible transformations becomes more challenging. Locate a motif or small design that is repeated throughout the whole pattern; then see if there is a way to transform that motif or design to as many of its repetitions as possible. If any of these transformations are symmetries of the pattern as a whole, then we have located a symmetry transformation.

And the best way to describe the transformation is to say what stays the same: the center of rotation, the axis of reflection, the direction vector for a translation and the direction vector for a glide reflection the lines determined by these vectors are fixed.

When students begin design their own patterns, they start thinking in terms of aesthetics, what patterns they like and want to work on themselves. Symmetry is beautiful and fascinating. Can the understanding of symmetry that we have gained here help us in any way to understand this role? We have seen that a symmetrical pattern gives rise to symmetries or transformations of the pattern which leave it essentially unchanged. From the Bhagavad-Gita, we see that life has two aspects, active and inactive.

According to Maharishi [M], the silent level of life is pure consciousness, the source of thought, and it is subjectively experienced as bliss; whenever the active level of the mind begins to move in the direction of the silent level of the mind, there is increasing bliss. An artistic pattern or structure of nature expresses the diversity of relative existence, yet in the repetition of aspects of the design or structure, that is, in the symmetry, an underlying sameness or unifying value is indicated.

The mind is spontaneously led to experience activity and silence simultaneously. This is in the direction of the nature of the experience described in the verse of the Bhagavad-Gita that we have examined. Thus, our analysis can help shed light on the charming nature of symmetry. Mathematics is part of life; mathematicians doing mathematics are subject to the same natural laws that govern all of life.

A deep understanding of the whole of life should give us the kind of insight that will help us understand the parts of life, including some very specific aspect of mathematics. This paper presents how one expression of knowledge about the nature of life from the Bhagavad-Gita can be used to go deeply into the mathematical study of symmetry and, hopefully, acts as a suggestion that this bringing together of mathematics and life as a whole can be done in other ways.