Henry Ernest Dudeney/Modern Puzzles/123 - Making an Octagon/Solution
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Modern Puzzles by Henry Ernest Dudeney: $123$
- Making an Octagon
- Can you cut the regular octagon from a square piece of paper without using compasses or ruler,
- or anything but scissors?
- You can fold the paper to make creases.
Solution
Let the square be $\Box ABCD$.
Fold $AD$ to $BC$ and $AB$ to $DC$ to obtain the midpoints $E$, $F$, $G$ and $H$ of $AD$, $BC$, $AB$ and $DC$ respectively.
Fold $EG$, $FG$, $EH$ and $FH$.
Fold at $H$ to place $DH$ against $EH$, then at $E$ to place $ED$ against $EH$.
Where those folds cross at $I$ is one of the vertices of the regular octagon we are seeking.
The other three vertices are obtained similarly.
Sources
- 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $123$. -- Making an Octagon
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $363$. Folding an Octagon