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Tangram

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A tangram is a type of dissection puzzle made of seven flat polygons. These individual pieces are known as tans.

Tangram set 00
Tangram set 00

The primary goal of the puzzle is to replicate a specific pattern. These patterns are often provided as simple outlines in puzzle books. To succeed, a player must use all seven pieces without any overlap. Some people use the tans for artistic purposes instead. They create minimalist designs that are valued for their aesthetic beauty.

squares
squares

To understand how a tangram works, one must look at its specific geometry. The seven pieces are designed to fit together to form a single large square. The set includes two large right triangles and one medium right triangle. There are also two small right triangles, one square, and one parallelogram. The parallelogram is a unique piece in the collection. It lacks reflection symmetry, meaning it only has rotational symmetry. Because of this, it is the only piece that may require flipping over to complete certain shapes.

convex tangram shapes
convex tangram shapes

Researchers have studied the many ways these pieces can be arranged. In 1942, Fu Traing Wang and Chuan-Chih Hsiung proved a specific mathematical limit. They showed there are only thirteen possible convex tangram configurations. A convex shape is one where any line drawn between two points stays inside the outline. While there are only thirteen convex shapes, the number of total possible patterns is massive. Over 6,500 different tangram problems were created from texts in the 19th century alone.

The exact origins of the tangram remain a subject of historical interest. It is widely believed to have been invented in China around the late 18th century. Some historians suggest the puzzle was inspired by the Song dynasty. During that era, a polymath named Huang Bosi used small banquet tables for entertainment. These six small tables, called feast tables or swallow tables, were arranged into patterns. It is possible a seventh table was added later to create the modern set.

Tangram caricature France 1818
Tangram caricature France 1818

As trading ships traveled, the puzzle moved from China to the West. In 1815, an American Captain named M. Donnaldson received books on the subject. He brought these books to Philadelphia in February 1816. This led to the first tangram books being published in America. The puzzle soon became a major craze in Europe. In Denmark, interest peaked around 1818 due to new publications. In Germany, an industrialist named Friedrich Adolf Richter marketed sets as "The Anchor Puzzle" around 1891. These sets were often made of stone or earthenware. The puzzle even saw a resurgence of popularity during World War I.

Tangrams are also used to demonstrate mathematical paradoxes, which are known as dissection fallacies. These occur when two shapes appear to be different sizes but use the same pieces. One famous example is the "two-monks paradox" attributed to Henry Dudeney. This paradox uses two similar shapes where one appears to be missing a foot. In reality, the area of the missing foot is compensated for by a slightly larger body. The change in area is hard to see because the math involves the square root of two.

tangram paradox explanation
tangram paradox explanation
Two monks tangram paradox
Two monks tangram paradox

Another interesting example is the "Magic Dice Cup" paradox found in Sam Loyd's work. This involves using the same seven shapes to create cups of different volumes. One cup might appear whole, while others contain different sized vacancies. These illusions show how the human eye can be tricked by geometric rearrangement.

The Magic Dice Cup tangram paradox
The Magic Dice Cup tangram paradox

Beyond simple play, the tangram connects to several broader fields. It is used in education to teach geometric reasoning and spatial skills. It also serves as a tool for art and minimalist design. The puzzle's history shows how a simple set of shapes can cross many cultures. It has moved from ancient banquet tables to modern classrooms and art galleries. This journey makes the tangram one of the most recognized dissection puzzles in the world.

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🖼️ Images & Media (7)
File:Tangram set 00.jpg
Tangram set 00.jpg
File:Tangram caricature France 1818.jpg
Tangram caricature France 1818.jpg
File:tangram_paradox_explanation.svg
tangram_paradox_explanation.svg
File:Two monks tangram paradox.svg
Two monks tangram paradox.svg
File:The Magic Dice Cup tangram paradox.svg
The Magic Dice Cup tangram paradox.svg
File:squares.GIF
squares.GIF
File:convex_tangram_shapes.svg
convex_tangram_shapes.svg
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