▌MathLove Puzzle ▌The Pentagon Puzzle - The Golden Ratio
Most of the pentagon puzzle’s 17 pieces have the golden ratio. A large pentagon can be divided into 2 pieces, 4 little pentagons, and 5 equally sized little pentagons. What kind of principle makes this happen? Let’s find the answer by assembling the puzzle.
The Pentagon Puzzle is…
The Pentagon Puzzle contains 3 congruent regular pentagons and 14 isosceles triangles. As you reassemble these 17 pieces, you can transform the big pentagon into various sets of pentagons. You can make a set of two regular pentagons with
different sizes, a set of 4 with 3 different sizes, or a set of 5 congruent regular pentagons as in the figure below. Why do you not try these now?
1
2
3
Also, you can make some interesting shapes with the 17 pieces.
4
5
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We hope you enjoyed them. What do you think makes these transformations possible? the golden ratio. The golden It is ratio,
approximately 1.618, is considered to be a beautiful and comforting ratio and is In a regular
embedded in art and architecture since the ancient Greek period.
pentagon, the ratio of the length of a diagonal to that of a side is the golden ratio. In each isosceles triangle in the Pentagon Puzzle, the golden ratio is also found. The 14 isosceles triangles in the Pentagon Puzzle are either a 36°-72°-72° triangle or a 36°-36°-108° triangle (see the figure above). We call these
isosceles triangles “golden triangles” due to the ratio between the two different lengths of sides. As you draw diagonals in a regular
pentagon, you will see many golden triangles as in the figure. With the fact that the interior angle of a regular pentagon is 108°, golden triangles in a regular pentagon can help you understand how to transform a big regular pentagon into a set of pentagons or other interesting shapes.
Answer
Reference
What is the ‘ Golden Ratio’ ?
A line segment is said to be cut in the golden ratio when, as the whole line segment is to the greater segment, so is the greater to the lesser. the figure.
x x+1 To calculate the ratio, say AC = x and CB = 1. Then x +1:x = x: 1, x -x - 1 = 0
2
That is, AB: AC = AC : CB in
1
∴
x=
1± 5 2
Since x is a positive number, x ≒ 1.618
So an approximate value of the golden ratio is 1.618.
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