How many sides is there in a square

What is a square

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A square is a rectangle with four right angles and four equal sides.

.......The same figure stands on top.

They are called a diamond, even if you know it is a square.
One wants to express the position of the square with this word.


Sizes of the square Top

......The square has 4 equal sides.
2 sides meet in a corner and are perpendicular to each other.
The length of one side is a.
The square has 2 diagonals of length d = a * sqr (2).
The area is A = a², the circumference is 4a.
The circumference has the radius R = a * sqr (2) / 2, the inscribed circle the radius r = a / 2.

A formula to the square Top

......
It is possible to just go through a square in one coordinate system a Describe equation.

| x | + | y ​​| = 2 or abs (x) + abs (y) = 2


Square numbers Top
The square numbers are obtained by multiplying the natural numbers by themselves.

......The square numbers can be represented by squares from the same figures.

Every square number is equal to the sum of two triangular numbers.
Formula: n² = [1 + 2 + 3 + ... + n] + [1 + 2 + 3 + ... + (n-1)]

Each square number n² is equal to the sum of the first n odd numbers.
Formula: n² = 1 + 3 + 5 + ... + (2 * n-1)

A square can be broken down into two smaller squares and two equal rectangles.
The first binomial formula (a + b) ² = a² + 2 * a * b + b² applies.

......
A square can be broken down into strips in a diagonal direction.
The following applies: 1 + 2 + 3 + ... + n + ... 3 + 2 + 1 = n².

You can find more on my page square numbers.

Halve a square Top
There are many ways to halve the area of ​​a square. Here are five.

Regarding 1 and 2: Usually a square is bisected by a diagonal or center line. This is how a square piece of paper can be folded.
To 3: Every straight line through the center of the square divides it into two congruent trapezoids.
To 4: Instead of the straight line, you can also use other lines, e.g. two semicircles.
To 5: The figure in the middle consists of four congruent triangles, the whole square of eight. So the halving is assured. If the starting square has the length 2, the green square has the length sqr (2). The meaning of this figure lies in the fact that the outer square has a rational side, the inner one an irrational side.


Squares in the square Top
The square through partial points

.......If you mark a distance (red) from each corner always in the same direction, you get four points. If you connect these four points, you get a square.
Claim: The rectangle is a square
proof: The four triangles are congruent according to SWS. W denotes the right angle. Hence the hypotenuses are the same and that is precisely the sides of the square. A simple angle consideration leads to right angles of the quadrangle.

A square inside
......If you connect the four partial points each with a corner point, a square is created inside.

If the side of the square is given with a and the division ratio with r: (a-r), then the side length b of the inner square is

b = (a²-ar) / sqrt (a² + r²).


Derivation
The auxiliary variables x, y and z are introduced.
According to the Pythagorean theorem, r² = x² + y².
According to the 2nd theorem of rays, y: x = a: r applies. This implies y = (ax) / r.
These are two equations with the variables x and y.
You have the solution x = r² / sqrt (a² + r²) and y = (ar) / sqrt (a² + r²).
For the line connecting a partial point with the corner point, z² = a² + r² applies. Then z = sqrt (a² + r²).

The side length of the inner square is
b = zxy = sqrt (a² + r²) -r² / sqrt (a² + r²) - (ar) / sqrt (a² + r²) = (a² + r²-r²-ar) / sqrt (a² + r²) = (a² -ar) / sqrt (a² + r²), wzbw.

A series of squares Top

......The red square has the side length a and thus it has the area a².
The sum of the areas of the inner squares results
A = a² / 2 + a² / 4 + a² / 8 + a² / 16 + ... = a² (1/2 + 1/4 + 1/8 + 1/16 + ...).
In the brackets there is a geometric series with the limit value 1.
Result: The sum of the areas of the inner squares tends towards the area of ​​the large square.

The binomial formulas and the Pythagorean theorem also deal with squares.
I refer to my website formulas in the picture.

Tiling with squares Top
It is easy to divide a square into equal squares.
But it's hard to find different squares that do this.

......You need at least 21 squares that form a square (Duijvestijn 1962, 1978).
solution
112² = 50²+42²+37²+35²+33²+29²+27²+25²+24²+19²+18²+17²+16²+15²+11²+9²+8²+7²+6²+4²+2²
Many mathematicians (and computers) have studied this problem (Book 2). More can be found on Erich Friedmann's homepage (URL below).

A geometric series Top

......The geometric series 1/4 + 1/16 + 1/64 + ... has the limit value 1/3.
Proof: s = a0[1 / (1-q)] = (1/4) [1 / (1- (1/4)] = 1/3
Left is a notable illustration.
Discovered at http://en.wikipedia.org/wiki/The_Quadrature_of_the_Parabola

Largest and smallest rectangles Top
The square is a border rectangle. This is expressed in the solutions to the following extreme value problems.
Exercise 1) Which of the rectangles with the same circumference has the largest area?
Exercise 2) Which of the rectangles with the same area has the smallest circumference?
Exercise 3) Which rectangle in the equilateral triangle has the largest area?
Exercise 4) Which rectangle in the square has the largest area?
Exercise 5) Which rectangle in the circle has the largest area?

In all cases, a square is the rectangle you are looking for.
The largest square in the cube can be found on my side of the cube.


Representation of digits by squares Top
Matrix of squares for characters
The pixels for displaying the characters of a font form a matrix of small squares that only become visible when enlarged sufficiently.

......
For example, one uses the number for the picture 2 a rectangle with 9x14 pixels in the font "Times New Roman" by Winword.
......

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If you double a square, you get the figure on the left, with which the ten digits can be represented well (right).
This display is known from the pocket calculator.
......

Ring of Solomon
......
According to an old legend, Lucas tells that a square with diagonals was engraved on a ring of Solomon (left).
This symbol contains the ten digits (right) [Book 1].
......


Calculator keyboard
In this context one can also mention the keyboard of a pocket calculator. The numbers 1 to 9 are on square keys. They form a 3x3 square. Strangely enough, the numbers run from bottom left to top right. The keypad of the button phone or the cell phone run from top left to bottom right like our writing.
I can remember trying to create a uniform keyboard, especially since the keypad phones came first. That failed. The first affordable computers came from the Far East, and the arrangement of the digits was based on local ideas.
......Some nostalgia:
I still have a working desk calculator from 1971 from Quelle ("Privileg"), which at that time cost almost 300 DM. It is connected to 220V. The digits are displayed with glow lamps. The calculator does not yet take into account point before line calculation, but already has leading zeros suppression.

It saved me a lot of arithmetic at the time ;-).


Optical illusion Top
The squares on the left differ in width and height, and the squares on the right differ in size.

You guessed it: the squares are the same size.

Four squares lie in front of a group of circles, squares or straight lines. The sides are distorted; you can guess: All four figures are squares!

Both grids have gray spots at the crossing points, on the left in the white, on the right in the black area.
One suspects it: the gray spots are just pipe dreams.

An arrangement of four three-quarter circles
......No square is drawn, but you can still see it.

Games with squares Top
If you look at the popular games from the point of view of whether they contain any form of squares, you will find what you are looking for. Almost all games use squares, be it as the shape of the game board, the playing field or the game piece.

I only name the games in alphabetical order that I describe elsewhere on this website:

Domino puzzle, Froebel star, game of fifteen, heaven and hell, magic squares, mill, Rubik's Magic, solitaire, tic-tac-toe, tangram


The topic of placing a square can be found elsewhere.

Squares on my homepage Top
Polyominos
You can put squares together to form increasingly complex figures. It is required that two squares are always connected on at least one side. The figures have been counted (up to 28 ominos).


One could imagine the way to the Pentominoes (plural also Pentominoes) as follows:

You can find more on my websites Domino Puzzle, Pentomino and Hexomino.


If bodies are also allowed, monoids are created.
......Three squares are given as an example. In addition to the two triominos, eight spatial monoids can be formed from them.
The most famous monoid is the cube, which consists of six squares.

You can find more on my Rubik's Magic page. With this puzzle you can find numerous bodies made up of eight squares.

Parquet flooring of the level

 

You can find more on my page Homogeneous tiling.

body
The cube is made up of 6 squares.

The image pairs enable a three-dimensional view.


The largest square in the cube
The red square is the largest square that can fit in a cube.
The corner points of the square divide the edges in a ratio of 1: 3. (Book 3)


7 of the 13 Archimedean solids are bordered by squares, among other things.

Truncated octahedron

Large diamond cuboctahedron

Large rhombicosidodecahedron

 

Cuboctahedron

Beveled cube

Small rhombicosidodecahedron



Small rhombic cuboctahedron

Squares on the internet

German

Wikipedia
Square (geometry), squaring the square, squaring the circle


English

Eric W. Weisstein (MathWorld)
Square, Square Packing, Perfect Square Dissection

Kevin L. Gong
counting polyominoes

Erich Friedman
Erich's packing center

Wikipedia
Square, squaring the circle



credentials Top
(1) Walter Lietzmann: Funny and Strange Things about Numbers and Shapes, Göttingen 1969
(2) Friedrich L. Bauer: Invitation to Mathematics, Deutsches Museum, Munich 1999
(3) Martin Gardner: Mathematical Carnival, Ullstein Berlin 1977 (page 61)

Feedback: Email address on my main page

This page is also available in German.

URL of my homepage:
http://www.mathematische-basteleien.de/

© 2002 Jürgen Köller

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