Tangram World

What is the Tangram
The Tangram is nowadays the most popular dissection puzzle formed from 7 polygons. The aim of the puzzle is to seamlessly arrange all the geometric pieces to form problem figures (rules of the game). More than 100 years ago, this game has been played passionately by many as entertainment, educational or mathematical tool, because it boosts shape recognition, problem solving, and pattern design skills. It is said that the Pythagorean theorem was discovered in the Orient with help of Tangram pieces...

The 7 polygons (see image further below) that form the Tangram are:
  • 5 right triangles: 2 small (hypotenuse of n/2 and sides of n/2√2); 1 medium (hypotenuse of n/√2 and sides of n/2); 2 large (hypotenuse of n and sides of n/√2). The large triangle is 4 times the size of the small triangle, but curiously its perimeter is only 2 times as big!
  • 1 square (side of n/2√2).
  • 1 parallelogram/rhomboid (sides of n/2 and n/2√2).
Of these 7 pieces, the parallelogram (or rhomboid) is the only piece that may need to be flipped when forming certain shapes; in fact, it has no reflection symmetry but only rotational symmetry, and so its mirror image can only be obtained by flipping it over.

Rules of the Tangram
Classic rules state that :
  • all 7 pieces must be used
  • all pieces must lie flat
  • all pieces must touch
  • no pieces may overlap
  • pieces may be rotated and / or flipped to form the desired shap
The object of the Tangram game is to put the seven geometric shapes together so as to form a given outline/silhouette.

The Tangram Legend
A legend says: “Thousands and thousands of years ago, Yu (玉龍), the Great Dragon, lived among human beings. He was venerated by them because he was 'yang', good, and was always ready to help them. One day, the God of Thunder, jealous of the offerings the men brought to Yu, in a moment of anger, crushed the sky with his hatchet. Consequently, the sky fell on the Earth in seven pieces black like coal. The light disappeared taking with it all existing things.
Yu at first felt sad for the world, and then felt nostalgic. Therefore, he picked up the seven black pieces of the sky and in memory of the former world, began to reassemble different kinds of shapes: animals, plants and human beings that had disappeared. But every time he finished a shape, a shadow left it and wandered the deserted world crying about its misfortune. The complaints arrived until the ears of the God of Thunder who was touched, and to remedy the harm he caused, he pulled from every shadow the body of a living being to repopulate the Earth. From that time on, the shadow faithfully follows every move we do and with the seven pieces of the sky, called Qi Qiao Ban (literally 'seven boards of cunning'), everything on Earth can still be shaped”.

Tangram Variants

Ostomachion is a puzzle that is tightly tied with Archimedes, although it may be older than Archimedes (the puzzle is also known as loculus Archimedius or Archimedes' box). In any case, ostomachion is apparently most known from the mathematical study by Archimedes. It is a puzzle made of 14 polygons, 13 of which are different and two are the same. Those 14 polygons (11 triangles, 1 pentagon and 2 quadrangles, originally made of bone) can be assembled in a square, as is shown in the image below.

A question regarding the number of ways in which one can assemble the square from the given elements immediately arises. The answer is  536 without rotations and reflections. It is obvious that any assembly of the square can be rotated three times by 90 degrees, so that one gets the same square each time, formally differently assembled. Therefore, the rotational multiplicity of the assembly is 4. Ostomachion is, besides for interesting mathematics it represents, perhaps better known (and can be understood in such a way) as a set of elements that one can use to form interesting shapes. Such interpretation of ostomachion is probably best suited to children.One form of play to which classical texts attest is the creation of different objects, animals, plants etc. by rearranging the pieces: an elephant, a tree, a barking dog, a ship, a sword, a tower etc.

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