1983 Just after the Spring Festival, I was inspired by chess pieces and chessboards when I drew the chessboard. Place round chess pieces in each square on the chessboard, and each square is exactly the circumscribed square of each chess piece. The area of the square is equal to the square of the diameter of the chess piece; The area of the chessboard (the whole grid) is equal to the number of pieces multiplied by the square of the diameter of the pieces. (The diameter of the chess piece is denoted by Q, and the side length of the square is denoted by A). ?
∵q=a,∴q&； # 178; = a & amp# 178; That is, the area of the chessboard (the whole grid) s = 72q &；; # 178; = 72a & amp# 178; .
So is it possible to calculate the area of a square, rectangle or circle with the square of the diameter of different pieces? ?
Area is the number of units, and the number of units is called area. Thus, regions of various shapes are arranged in an orderly manner according to a certain number of units. ?
If each square is regarded as a unit square and each chess piece as a unit circle, a unit square is called a square point and a unit circle is called a point. The larger the dot, the greater the strain at the circumscribed square point; The dots become smaller, and the circumscribed square points are less sensitive to strain. ?
Arranging a certain number of square points to form an area of a whole shape, wherein the number of square points is represented by n, and the area of a whole shape is Na&; # 178; .
When a certain number of points are arranged in an orderly manner, the outline of the whole shape is cut. To put it bluntly, the outline is the "skeleton" cut by the orderly arrangement of chess pieces (meaning to form a certain shape according to points). ?
Since each point is the inscribed circle point of each square point, when the area of each square point is subtracted from the area of that point, the remaining "four corners" are "meat". The meat deforms with the skeleton and skeleton (outline), and the meat softens and deforms with the gap on the outline.
What shape the points are arranged in, the definition of tangency with each other will form the outline of what shape, and there will be a circumscribed shape on the outline. The circumscribed shape of a contour is the edge of a face occupied by the area of all finite square points.
When the areas are deformed into equal areas, the edges of the surface form the circumscribed shape of the outline. The number of points is also represented by n, and the area of the whole shape is NQ&; # 178; .
∵Q & amp； # 178; = a & amp# 178; ,∴nq&； # 178; na & amp# 178; There are as many squares as there are dots in an area.
In an area with arbitrary shape, how many points are tangent to the ordered arrangement of points to form the outline of a certain shape, and how many square points and square points in the circumscribed shape of this outline will follow the softening and equal area deformation to form the edge of the surface and its limited area NQ-to form the circumscribed shape of the outline; # 178; Or na &；; # 178; .
When n q &；; # 178; When it is a square section, n is an arbitrary number of squares; The number of squares is limited by the square outline formed by the tangent of points and the arrangement of points. ?
When n q &；; # 178; When it is a rectangular outline, n is an arbitrary number of rectangles; (The number of rectangles is limited by the rectangular outline formed by the tangent of points and the arrangement of points). ?
When n q &；; # 178; When it is a circular outline, n is the number of circles. (The number of circles is limited by the outline of the circle formed by the tangency of points and point arrangement).
With the same number of square points, the position changes of square points will be pieced together into various shapes (shapes), which are excavation and filling equal area deformation.
In the case of the same number of points, the position change of points will be cut into contours of different shapes, and the circumscribed shape of each contour of different shapes is softened and deformed by equal area. That is, each point has each softened "four corners" to form the circumscribed shape and area of each different shape contour.
No matter what shape it becomes, there will be "as many square points as there are" in this shape. Or there are as many squares as there are dots. "They all change with the same number of square points or points in different positions, changing a face with equal area and different shapes. ? Therefore, regions of various shapes are not only formed according to the orderly arrangement of a certain number of unit squares, but also according to the circumscribed shapes of a certain number of unit circles. ? For example, if nine points and points (that is, nine chess pieces and chess pieces) are arranged in order and cut into a square outline (as shown in the figure-in the chessboard, the upper left corner and the vertical and horizontal arrangement are cut into two groups of square outlines with a relative distance of 3Q). Then, in the circumscribed square of the outline of these nine points, the sum of the areas of nine square points (that is, nine points bear nine "four sides") softens the shape formed by the edges of the equal area surface to form a square. (The area is the same as that of a square consisting of 9 squares). ?
nQ & amp# 178; na & amp# 178; ，n = 9 ∴9q&； # 178; = 9a & amp# 178; . (9Q & amp# 178; And 9a &；; # 178; Are all the same square area). ?
If eight points and points (namely, eight chess pieces and chess pieces) are arranged in order, they are cut into rectangular outlines (as shown in the upper right corner of the chessboard, the vertical and horizontal arrangements are tangent, forming two groups of rectangular outlines with opposite distances of 2Q and 4Q respectively). Then, in the circumscribed rectangle of the outline of these eight points, there is the sum of the areas of eight square points (that is, eight points carry eight softened "four sides"), which softens the shape formed by the edges of the equal area surface to form a rectangle. (equal to the area of a rectangle consisting of 8 squares). ?
nQ & amp# 178; na & amp# 178; ，n = 8 ∴8q&； # 178; = 8a & amp# 178; (8Q & amp# 178; And 8a &；; # 178; Are the same rectangular areas). ?
It is concluded that "the number n of points multiplied by the square of the point diameter q is equal to the points arranged in sequence and the circumscribed areas of points cut into contours". The diameter of a point is called the point diameter.
That is to say, when the points are arranged in order and cut into an outline, there are as many pieces as there are squares in the circumscribed shape of the outline; There are as many unit squares as there are unit circles; There are as many squares as there are points.
Therefore, equal area deformation is not only the position change of each square point, but also the position change of each point. The position change of the point changes the shape change of the contour; The change of contour shape will cause the change of contour circumscribed shape; When the area is deformed with equal area, the circumscribed shape of the contour determines the edge position of the surface. The edge position of the surface is the circumscribed shape of the contour. ?
Because the points are arranged in order and tangent to form a certain shape outline, they have the characteristics of free rotation and flexibility. Points and points can form square and rectangular outlines and circular outlines. Commonly known as "fighting poison with poison", it goes on and on. ?
On the outline of a circle, from the circumscribed shape of the outline, you will find that "the number of points required to form a circular outline is limited to a circle number of 7." The number of points required for contours different from squares and rectangles is infinite "number of squares" and "number of rectangles".
No matter whether the number of points is limited or infinite, as long as it is possible to form the outline of a certain shape, the number of points in the circumscribed shape of the outline is as many as the number of points in the square, and the shape formed by the edge of the surface with regional softening level overlaps with the circumscribed shape of the outline. ? If seven points and points (i.e. seven chess pieces and chess pieces) are arranged in sequence, they are cut into a circular outline (as shown in the figure-chessboard, with one point in the middle and six points in the periphery, defined as "circle with points", arranged periodically and cut into a circular outline with a diameter of 3Q). Then, the sum of the areas of seven square points in the circumscribed circle of the outline of these seven points is equal-product softened into the shape formed by the edge of a face with a diameter of 3Q to form a circle. Firstly, the chess pieces are separated from the square: the area of a square is A &；; # 178; The sum of the areas of seven squares softens the equal product into a circle with an area of 7a &；; # 178; . ?
For example (Figure-1), the circular area is 7a² ; Softening equal product into (Figure -2)H-shaped area is also 7a² ; On (Figure -2)H-shape, two additional A &；; # 178; Just spell it into a (Figure -3) square area 9a &；; # 178; ; These three numbers are called (the last three numbers). The size of their respective areas varies with.
One piece is a point, and seven pieces are seven points. There is a point in the middle and six points on the periphery, which are arranged tangentially around a circle to form a circular outline with a diameter of 3Q (Figure -4), and the circumscribed circle area of the outline is s; Then seven points are tangent to form an H-shaped contour (Figure -5), and the circumscribed H-shaped area of the contour is 7q² ; Finally, nine points are tangent to form a square outline (Figure -6), and the circumscribed square area of the outline is 9q&; # 178; . These three figures are called (the next three figures), and the areas of their respective circumscribed shapes all change with the size of the point diameter q? The above six graphics:
It is known that the circular area (Figure-1) is seven-nineths of the square area (Figure -3). The circumscribed circle of the outline (Figure -4) is the inscribed circle of the circumscribed square of the outline (Figure -6). A equals q again.
Because the circle (Figure-1) is similar to the circumscribed circle of the outline (Figure-4); (Figure 2) The H shape is similar to the circumscribed H shape of the outline (Figure 5): (Figure 3) The square is similar to the circumscribed square of the outline (Figure 6). Therefore, whether the area and area of each group with similar shapes are equal is related to A and Q; Or whether a and q are equal is related to the area and area of each group of similar shapes. ? Prove that if the areas of each group of similar shapes are equal, then A and Q are equal.
Prove (1): If (figure-1) circular area 7a &；; # 178; Is equal to the circumscribed circle area s of the outline in (Figure -4), then seven-tenths of the square area in (Figure -3) is the H-shaped area 7a&: # 178; Is equal to the circumscribed circle area s of the contour (Figure 4). The circumscribed circle area s of the contour (Figure 4) can be softened and equally integrated into (Figure 3) square areas 7a&; # 178; . The circle (Figure-1) at this time is the inscribed circle of the circumscribed circle of the outline (Figure -6).
For example, when the circumscribed circle area s of the contour (Figure 4) is1575px & #178; Time. Because (Figure-1) the circular area 7a&; # 178; Is equal to (Figure -4) the circumscribed circle area s of the contour, and (Figure-1) the circle area 7a&; # 178; It is also 7/9 of the square area (Figure 3). So (Figure -4) the circumscribed circle area of the contour is1575px &; # 178; Softenable equal area becomes 7/9 of the square area (Figure 3). At this time (Figure -3), seven-tenths of the square area is also1575px &; # 178; (Figure -3) The side length of a square is 225px. It is proved that seven tenths of the square area in (Figure -3) is the circumscribed circle area of the outline in (Figure -4).
Because the circumscribed circle of (Figure -4) contour is the inscribed circle of (Figure -6) contour, the area of (Figure-1) circle is equal to that of (Figure -4) contour. So the circle (Figure-1) is the inscribed circle of the outline (Figure -6), and the diameter of the circle (Figure-1) is 3Q. ?
That is, the areas of (Figure-1) circle and (Figure -4) excircle are equal, and seven tenths of the square area in (Figure -3) is equal to the excircle area in (Figure -4). When the excircle area in (Figure -4) increases or decreases with the dot diameter q, the sum of seven tenths of the square area in (Figure -3) (Figure) The areas of the two circles become smaller, and A and Q are less sensitive to strain. Proof (2): If A and Q are equal. Then (Figure -6) seven-nineths of the circumscribed square area of the contour, that is, (Figure -5) the circumscribed H-shaped area 7q&:#178; Equal to (Figure-1) circular area 7a &；; # 178; (Figure-1) Circular area 7a&; # 178; It can be softened into (Figure -6) 7/9 of the area of the circumscribed circle of the contour 7 q&; # 178; . At this time (Figure -4) the circumscribed circle of the outline is the inscribed circle of the square (Figure -3).
For example, when (Figure-1) circular area 7a&; # 178; It's 700px &; # 178; Time. Because a and q are equal, the square area of (Figure -3) is equal to the area of the circumscribed square of (Figure -6). (Figure -3) Seven-nineths of the square area and (Figure -6) Seven-nineths of the circumscribed square area are equal to 700px &: #178 respectively; Equality. (Figure-1) Circular area 700px &; # 178; Softenable equal area becomes 7/9 of the area of the outline circumscribed square (Figure 6). At this time (Figure -6), seven-nineths of the circumscribed square area of the contour is also 700px &; # 178; (Figure 6) The side length of the circumscribed square of the contour is 150px. It is proved that seven tenths of the circumscribed circle area of the contour in (Figure -6) is the circle area of (Figure-1).
Because the excircle of (Figure -4) is the inscribed circle of the outer square of (Figure -6), and A is equal to Q, the inscribed circle area of (Figure -3) square is equal to the excircle area of (Figure -4), and the diameter of (Figure -4) 3Q is equal to the diameter of (Figure -3) 3a, and the excircle of (Figure -4). ?
That is to say: A and Q are equal, and 7/9 of the area of the outer square (Figure -6) is 7 q&; # 178; And (figure-1)² Are equal. When (Figure-1) circular region 7a &；; # 178; When it becomes larger or smaller, 7/9 of the outer square area (Figure -6) is 7q &；; # 178; And (Figure -3) the inscribed circle area s of the square also becomes larger or smaller, otherwise (Figure-1) the circle has nothing to do with (Figure -4) the outer circle. Description: A and Q become larger together, and (Figure-1) circle and (Figure -4) circumscribed circle also become larger for strain; A and q become smaller together, and (Figure-1) circle and (Figure -4) circumscribed circle also become smaller for strain. It is not difficult to see from (Figure -3) and (Figure -6):?
If a=75px, the circumscribed circle area of (Figure -4) is1575px &; # 178; Otherwise, the area s of the circumscribed circle (Figure -4) is equal to the area 7a of the circle (Figure-1). # 178; Doesn't matter; If Q=50px, the circular area of (Figure-1) is 700px &; # 178; Otherwise, a has nothing to do with q; If a=4 and Q=4, then (Figure -4) the circumscribed circle area s is112cm&; # 178; (Figure-1) Circular area 7a&; # 178; Also112cm&; # 178; . It is proved that if a and q are not equal, the areas of the two circles are not equal; When a and q are equal, the areas of two circles are equal.
So (Figure-1) circle and (Figure -4) circumscribed circle, the relationship between two circles and the areas of A and Q is:?
"If the areas of two circles are equal, A and Q are equal; A and q are equal, and the areas of the two circles are equal. " On the contrary, "If the areas of two circles are not equal, then A and Q are not equal; If a and q are not equal, the areas of the two circles are not equal. "
Because "A and Q are equal, and the areas of two circles are equal". Therefore, when a=Q is known, seven tenths of the square area in (Figure -3) and seven tenths of the circumscribed square area in (Figure -6) can be interchanged. That is (Figure-1), the area of the circle is softened, which is equal to seven-ninth of the square area (Figure-3); (Figure -4) The area of the circumscribed circle is softened, which is equal to 7/9 of the area of the circumscribed circle (Figure -6).
At this time, (the upper three numbers) still changes with A, (the lower three numbers) still changes with Q, and A has nothing to do with Q?
When (the above three numbers) change with a, even if the area of the circle (Figure-1) is not equal to the area of the circumscribed circle (Figure -4) or the area of the circumscribed circle (Figure -6), it is still equal to the area of the square (Figure -3). That is to say, although the circle (Figure-1) is not the inscribed circle of the circumscribed square (Figure -6), it is still the inscribed circle of the square (Figure -3).
When (the last three figures) change with Q, even if (Figure -4) the area of the circumscribed circle is not equal to (Figure-1) and the area of the circle is not equal to 7/9 of the square area (Figure -3), it is still equal to 7/9 of the area of the circumscribed circle (Figure -6). That is to say, although the circumscribed circle (Figure -4) is not the inscribed circle of the square (Figure -3), it is still the inscribed circle of the circumscribed square (Figure -6). ?
From (the next three figures): Because (Figure -6) seven-nineths of the circumscribed circle area is equal to (Figure -4), (Figure -4) the circumscribed circle is also the inscribed circle of the circumscribed circle (Figure -6). So the area of the inscribed circle (Figure 6) is seven-nineths of the area of its circumscribed circle.
In other words: "Seven-ninth of the square area is its inscribed circle area". It is concluded that (Figure-1) is the inscribed circle of (Figure -3), because the area of (Figure-1) is equal to 7/9 of the square area of (Figure -3). ?
Finally, the axiom is found: "The area of a circle is equal to seven-ninth of its circumscribed circle area."
According to this axiom: If the area of a circle is175px &; # 178; . Then its circumscribed circle area is 225px &; # 178; .
Because the side length of a square is equal to the square root of the square area, the diameter of the inscribed circle is equal to the side length of its circumscribed circle. So175px &; # 178; The area of a circle is 75px. That is to say, there is the sum of the areas of seven squares on the circular outline of seven chess pieces, and the edges of softened faces with equal area and diameter of 3Q form the circumscribed circle of the outline.
nQ & amp# 178; na & amp# 178; ，n=7。 ∴7q&； # 178; = 7a & amp# 178;
∵7a & # 178； =S，a=d/3。 ? ∴ formula of circular area: s = 7 (d/3)²
HPFYKG, an illiterate mathematical discovery, is expensive in the Eastern Jin Dynasty.
201June 27, 4