The symmetry beauty in Mathematics(数学中的对称美)
The symmetry beauty in Mathematics（数学中的对称
美）
The symmetry beauty in Mathematics(数学中的对称美)
The symmetry beauty in Mathematics
Abstract: symmetry usually refers to the graphic or object on a
certain point, line or plane, the size, shape and arrangement that has a corresponding relation, in mathematics, the concept of symmetry is
slightly extended to some related concepts have often regarded as symmetric or opposing, such symmetry has become an important component part of mathematics, symmetry is a vast subject in two aspects of art
and nature are of great significance, it is the fundamental mathematics, beauty and symmetry are closely linked.
Key words: symmetrical graphs, mathematics, symmetry, beauty
Symmetry in nature has many things, such as maple leaves, snowflakes, etc., symmetry itself is a kind of harmony, a kind of beauty. The application of mathematics is also very broad, such as: everyone is very familiar with the axial symmetry graphics, etc., in fact, according to the principle of symmetry, in primary school mathematics in the various knowledge areas, can be found in the application of this rule. How to
let the students grasp the basic principle of symmetry to solve some practical problems, find the intrinsic unity between things, go within this simple mathematical thinking, principle and contain profound philosophy essence, which requires us to understand the hidden deep in
the back problem, according to some cases the author found in teaching that explains how to find symmetry in mathematics.
First, get inspiration from a palindrome, solving arithmetic
sequence
Palindrome number has many, such as 0:2002 years is a palindrome number, the next palindrome number to wait until 2112, integer multiplication, the most interesting palindrome number is: 1 x 1=1, 11 x 11=121111 * 111=12321. According to this law can be calculated by: 111111111 * 111111111=12345678987654321, the students for this special palindrome results, all feel very surprised, it has strong interest, lamented the symmetry number. As a kind of symmetrical beauty, become an eternal theorem in the universe, like Yin there is Yang, black is white, that's more like some iffy, modern physics theory inference as a matter with antimatter, as we see in life, all things feel is the same material in the universe, there we see not see antimatter energy and matter are equal, so the universe was balanced, you like the universe, there are also "anti you", if one day you "a handshake, then you and you" suddenly disappear, like 5+ (-5) =0, somewhat absurd, but this assumption in answer some problems, but it is clever and easy to understand.
As in the primary school of good degree of students on the sum of arithmetic series, mostly using the formula: (a + end) * / 2 items of teaching, for students to master and is difficult to understand. As a "bad woman weaving" the ancient arithmetic: a woman at her weaving, woven Budoubi every day to reduce some day, reduction is equal to the
number of the first day, she made five feet, the last day of the texture of a ruler, a total of thirty days for a total of fabric, fabric how many feet of cloth? The difficulty of this question is that, apart from the first and last day, the cloth woven in the middle is not an integer, and it is not easy to weave less cloth every day than the previous day. The symmetry of the idea is this answer: suppose that there is another girl like weaving and the woman, but she and the woman Weaver's situation is just the opposite: the girl every day of the Budoubi fabric to increase the number and increase is equal to the number of the first day, she weaves a ruler, the last day the fabric is five feet, also made thirty days, the total length of the weaving girl and women are equal, the number of women cloth cloth every day to reduce the number and increase every girl weaving is equal, so the two day
total of cloth is six feet, thirty days a total of 6 fabrics * 30=180 feet, each 90 feet of fabric.
This problem lies in the subtleties of a set of arithmetic sequence Abstract sum into the image a palindrome number summation method is vivid, wonderful and different approaches but equally satisfactory results in physics that matter and antimatter.
In fact, as the sum of an arithmetic series can be used in this solution, using the symmetrical thinking to understand arithmetic sequence than the simple formula to emphasize and vivid multi.
Two. The application of the symmetry principle from the axial symmetry figure
According to the half of the axial symmetry figure and the symmetrical axis, the other half figure of the axial symmetry figure can be accurately drawn, which is a common problem after the teaching of the axial symmetry figure. In mathematics, axial symmetry also provides some insights into people's
research into mathematics, for example, it is often used in game theory. Such as: there are 21 pieces on the table, arranged in a row,
you can get one at a time, you can also take two pieces, and even can take three pieces. Where to get pawn OK, not in order to take, but with two or three piece must be adjacent to that no space or other pieces,
two people take turns to ask: "who got the last one who wins, if you
take to ensure the win?"
This problem seems complicated, according to the permutation and combination method in order to get many 11 analysis clearly too laborious, in fact by using symmetry principle is very simple, as long
as the first person to take away the middle grain, eleventh grain of chess, so the left and right sides of the remaining ten grains, so that the other party to get the pieces, you take on the right side of the piece, and the number and position and he is symmetrical, if the other party with the right to pawn, you will get the pieces according to him, anyway as long as keep the left and right sides. The rest of the number and location of the same, as long as he took some, you also have to take, so the last grain bound to fall into your hands therefore, to take the
win, if the pieces are 20 grains (even number), you will take the middle two, let both sides of the remaining 9 chess pieces, so you win.
There are similar topics such as: number one yuan coin two people take turns to put it in a large disk, the coin cannot overlap between requirements, who can put who loses, is the first place to win or win back? Apparently, according to the principle of symmetry, the first put people as long as the first to occupy the center, after which the other put you as he put on the opposite side of symmetry, as long as he has the space
pendulum, then known as the local relative also must have a space pendulum, until the other side can put so far, each other to lose. In fact, basic characteristics of thinking method of these two questions are from the axis of symmetry, the teacher in the teaching content of the end of axis symmetry can penetrate this knowledge properly, students willing to learn, and enhance the use of axial symmetry knowledge and deep understanding, find symmetrical beauty, feel the charm of mathematics.
Representation of duality by symmetric graphs. The symmetrical
beauty of mathematics is naturally expressed in the duality of mathematical elements and the duality of mathematical propositions. But duality has no geometric intuition. Projective geometry is an excellent way to make the "dual" visual symmetry in mathematics and to use it for mathematical problems.
Example 1 in Euclidean plane geometry, the proposition "over two points" can make a straight line". We change the "point" into "straight line", "straight line" to "point", and then change the relative words appropriately, then the proposition becomes "two straight line" and "one point"". This proposition is clearly wrong, because there is no intersection point when the two lines are parallel. This shows that the relation between point and line is not symmetrical in this proposition. Assume that two parallel lines intersect at infinity, when the point and the line form a symmetric relationship. It is here that Dishag established the projective geometry theory
preliminarily. We show the duality relation of example 1 intuitively, using the following graph: the uplink represents
two points, and a straight line can be determined; the meaning of
the arrow is "to determine"". The two arrows on the left and right indicate the position of the switching point and the straight line. The downlink means "two lines to one point", and the meaning of the arrow is "intersection", which is based on the assumption that the graph (1) becomes the perfect rectangle, and the dual relation between the point and the line is transformed into
Geometrical symmetry. The symmetry beauty in mathematics provides a unique method for mathematical research, that is, symmetry. In a simple way, symmetry is the method of thinking that uses symmetry.
The "assumption" used in example 1 is the use of symmetry. Mathematicians use this method to reveal and discover a lot of
mathematical mysteries and get very useful theories and conclusions. It is also the symmetry method that enlightens us to translate duality into symmetry". In addition to projective geometry, such applications are also present in modern algebra.
Example 2[1] sets K as a domain, and A is a k - space. A is called a K algebra, if the elements in A have multiplication and unit elements, and the multiplication satisfies the union law. We use: A, A, mapping with A multiplication, with A, K said: epsilon mapping unit of A, is a combination of law and unit respectively can be expressed as the following commutative diagram: all the arrows and reverse, replace, substitute, and exchange, and map: the delta is called the comultiplication, is called more than unit exchange graph (3) called the comultiplication coassociative law. Here, multiplication and multiplication, the combining law and the residue combining law are all dual elements. Algebra, a branch of modern mathematics, plays an important role in quantum physics.
Three, to solve the problem of equation, to infiltrate the thought
of symmetry, and help students change from arithmetic thinking to algebraic thinking
It is known that arithmetic thinking is converse thinking, while equation thinking is forward thinking. It is difficult to solve some arithmetic problems by using the thinking of equation. But the solution to the arithmetic of the pupils is deep-rooted, but the solution to the equation is always rejected. Such as positive and negative proportional
problems the sixth grade 2, many students use the arithmetic solution done out, but the proportion of solutions are always confused and proportion, because they affect the arithmetic solution by the negative transfer of knowledge, try to find the answer to the problem instead of looking for the amount of change, for lack of deep understanding of the equation, no to realize the equation itself is the use of symmetry principle, and the key is to find whether the proportion of the amount of change, the equation of the left and right as the axis of symmetry on both sides of the same size but it's not exactly the same. Found the same amount on the left and right and found the equation.
Principle in the solution of the equation can also use the same symmetry makes the problem more simple, such as: the solution of the equation: 5x+6=3x+11 this equation on both sides about X if the junior high school knowledge good answers can be transferred, are easy to answer with the principle of symmetry
in equation: if the primary school on both sides at the same time take 3 x equation on both sides also set up? Obviously still equal, so the question is simplified as follows: 2 x+6=11, this way of thinking is understood by every student, and at the same time deepens the understanding of equations. Duality is the natural representation of mathematical symmetry and the extension of symmetry. We then illustrate duality in plane geometry". In plane geometry, we say, "one point in a straight line."." This can also be changed to another saying: "a
straight line through a little."." Here, "point" and "straight line" two
nouns exchanged position. Such two propositions are geometrically called "self dual" propositions, and "points" and "lines" are called dual elements. In geometry, there are many propositions about positional relations, such as "two in one line" and "two line" at one point". This is the two different position relations, as long as the "point" into "straight line", "straight line" to "point", and then change the appropriate terms of relations, you can get the latter relationship from the front of a relationship. In these two propositions, geometry is
called the proposition of "mutual duality". Therefore, when we select
two or more than two mathematical elements in a mathematical proposition, exchange or rotate their positions, and then change the relative words, we obtain a new proposition, which is called the dual proposition of the original proposition.
"Symmetry" in mathematics is universal: axial symmetry and central symmetry, symmetric polynomial, from parity can be seen as the symmetric inverse operation can also see each other as symmetric relation from the operation angle, there are many places are reflected in its charm, just like Aristotle said:
Although no obvious references to the goodness and beauty of mathematics,
But goodness and beauty are not completely separate from mathematics. Because the main forms of beauty are order, symmetry, and certainty, these are the principles that mathematics studies. We do for the new curriculum under the guidance of teachers should not only teach students
the knowledge, more important is to cultivate the students' ability of discovering and creating beauty, let the students find the beauty of mathematics in mathematics, the charm of mathematics was deeply moved, to further improve the mathematical literacy, to explore the world the true, the good and the beautiful, like a physicist said: if a theory that it is beautiful, it must be the truth.
Reference:
[1] Chang Gengzhe, Li Jiongsheng. High school mathematics competition course, [M]. Jiangsu Education Publishing
Society, 1989.
[2] Zhang Herui, Hao Bingxin. Advanced algebra. [M. Higher education press, 1983.