TANGRAMS
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Photograph from The Tangram Book by Jerry Slocum |
A tangram is a puzzle formed by cutting a square into seven simple geometric pieces—five isosceles right triangles, one rhomboid, and one square. These pieces can be reassembled to make other geometric shapes, as well as playful, stylized figures of humans and animals. Tangrams were invented in China hundreds of years ago. The oldest known set, carved from ivory in 1802, is shown at right. Tangram books published in China in 1858 include 789 figures, each one made using all seven tangram pieces. Today tangrams are commonly used in elementary school classrooms to teach geometry, fractions, and spatial reasoning.
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A grid of nine tangram squares appears above. Eight geometric figures are shown to the right of the grid. Each figure comprises all seven tangram pieces. Three of these eight figures are outlined in the grid. Can you find the outlines of the other five figures? Hints: Figures may be rotated and may appear more than once. Some may overlap.
HINGED DISSECTIONS
Tangrams are just one of a large class known as dissection puzzles. Purdue computer science professor Greg Frederickson, author of Dissections: Plane and Fancy (Cambridge University Press, 1997), is a connoisseur of these geometric marvels. His recent book Hinged Dissections: Swinging and Twisting (Cambridge University Press, 2002) starts with a wonderful four-piece dissection of a triangle into a square that was presented by Henry Ernest Dudeney in his 1907 book The Canterbury Puzzles. As you can see below, pieces of a triangle swing around hinges at corners as the triangle turns into a square.
By swinging individual pieces, you can transform the hinged construction into a variety of shapes.
1. In each figure below, identify a single piece you could swing to turn the figure back into a triangle or form a square. (Hint: One figure requires swinging more than one piece.)
2. Which figure requires swinging two pieces of the hinged construction to form a triangle or a square?
3. In which figure would the swinging piece collide with another piece before it reached its destination?
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