Henry Ernest Dudeney/Modern Puzzles/122 - Folding a Pentagon/Solution
Modern Puzzles by Henry Ernest Dudeney: $122$
- Folding a Pentagon
- If you are given a perfectly square piece of paper,
- how are you going to fold it so as to indicate by creases a regular pentagon,
- all ready to be cut out?
Solution
Take the square $\Box ABCD$.
Fold $\Box ABCD$ in half horizontally so $A$ lies on $B$ to get the midpoint $E$ of $AB$.
Fold $EC$.
Fold $EF$ to make $B$ lie on $EC$ at $G$.
Fold $CB$ onto $CE$ to make $H$ lie on $G$.
Fold $AB$ at $K$ to make $B$ lie on $H$.
Fold $\Box ABCD$ in half vertically along $PQ$ so as to make $K$ lie on $L$.
$BC$ is now said to be divided in medial section.
$KL$ is one of the sides of the regular pentagon.
Fold at $K$ so that $L$ lies on $AB$ at $M$.
Fold at $L$ so that $K$ lies on $CD$ at $N$.
Fold at $M$ so that $K$ lies on $PQ$ at $O$.
We can also fold at $N$ so that $L$ also lies on $PQ$ at $O$.
Folding $KM$, $LN$, $MO$ and $NO$ completes the construction.
Historical Note
Dudeney credits the $1893$ work Geometrical Exercises in Paper Folding by T. Sundara Row for the solution published in his Modern Puzzles of $1926$.
Sources
- 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $122$. -- Folding a Pentagon
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $362$. Folding a Pentagon