Perfect rectangle

Perfect rectangle made of 10 squares in concrete art (Image by the painter Irene Schramm-Biermann)

A perfect rectangle is a rectangle that can be divided into squares of different sizes. If a perfect rectangle is specifically a square, it is analogously called a perfect square.

A rectangle that is not perfect is also called an imperfect rectangle.[1]


Discoverers of Perfect Rectangles (Selection)

Many mathematicians have been involved in the discovery of perfect rectangles and perfect squares.

Below is a selection of important discoveries in this field.

  • 1925: Zbigniew Moroń decomposed a perfect smallest possible 33x32 rectangle into nine squares.
  • 1939: The German mathematician Roland Sprague published a large perfect square with 55 squares.
  • 1978: A. J. W. Duijvestijn dissected a perfect square into 21 squares with a total side length of 112, where 21 is the lowest possible number of subsquares of perfect squares.[2]

Perfect Rectangles with Special Properties

Among the numerous perfect rectangles and squares, the following selected examples are intended to highlight some special features.[3]

(The numbers in the squares indicate their respective side lengths.)

  • Smallest possible perfect rectangle (9 squares, Moroń)
    Smallest possible perfect rectangle (9 squares, Moroń)
  • Perfect rectangle with many squares (22 squares)
    Perfect rectangle with many squares (22 squares)
  • Almost symmetrical perfect rectangle (12 squares)
    Almost symmetrical perfect rectangle (12 squares)
  • Elongated perfect rectangle (17 squares)
    Elongated perfect rectangle (17 squares)
  • Perfect rectangle with a remarkably large side length of 7 for the smallest sub-square (10 squares)
    Perfect rectangle with a remarkably large side length of 7 for the smallest sub-square (10 squares)
  • Smallest possible simple perfect square (21 squares, Duijvestijn)
    Smallest possible simple perfect square (21 squares, Duijvestijn)

References

  1. ^ Perfect rectangle Wolfram MathWorld
  2. ^ Perfect Square Dissection Wolfram MathWorld
  3. ^ Perfect rectangles: an extensive collection of perfect rectangles

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