Little Octagons To A Big Octagon

A decoupage problem consists of cutting up one figure in order to reassemble its pieces in the shape of another figure. A closely related problem is one of assembling, starting with several figures of identical shape and then cutting them into pieces as economically as possible in order to reconstruct a similar figure of greater size.

Can you cut these three little octagons into a total of ten pieces and then reconstruct a single octagon with three times the area?

octagons11a8f

Solution

The dimensions of the little octagons must be multiplied by \sqrt{3} in order to obtain the big octagon with three times the surface. It is therefore necessary to produce the hypotenuse of a right triangle with sides \sqrt{2} and \sqrt{1}. The rest is inspired trial and error. (See Greg Frederickson, “More Geometric Dissections,” Journal of Recreational Mathematics [Summer 1974].)

octagons-dissectiona9b1.jpg

About Edmark M. Law

My name Edmark M. Law. I work as a freelance writer, mainly writing about science and mathematics. I am an ardent hobbyist. I like to read, solve puzzles, play chess, make origami and play basketball. In addition, I dabble in magic, particularly card magic and other sleight-of-hand type magic. I live in Hong Kong. I blog at learnfunfacts.com. You can find me on Twitter @EdmarkMLaw and Facebook. My email is learnfunfacts@gmail.com
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3 Responses to Little Octagons To A Big Octagon

  1. scifihammy says:

    Hmm I guess the short answer is – No I can’t! 😀
    Still, fun to see the solution. 🙂

    Liked by 2 people

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