A-Hunting They Will Go

Christmas tree farm

Alexandria Jones and her family piled into the car for a drive in the country. This year, they were determined to find an absolutely perfect Christmas tree at Uncle William Jones’s tree farm.

“I want the tallest tree in Uncle Will’s field,” Alex said.

“Hold it,” said her mother. “I refuse to cut a hole in the roof.”

“But, Mom!” Leon whined. “The Peterkin Papers…”

“Too bad. Our ceiling will stay a comfortable 8 feet high.”

The Jones Brothers’ Great Pruning Challenge

Dr. Jones glanced at his children in the rear-view mirror as he drove. “When Will and I were your age, Alex,” he said, “it was our job to prune the young trees. We used to split the field and compete for who could grow the best-shaped trees.”

“What makes a good Christmas tree?” Alex asked.

“Your grandpa used an old rule of thumb that the tree should be 1.5 times as tall as it was wide at the base. But I always preferred the Golden Section.”

“Christmas trees should be green,” Leon said.

Dr. Jones’s voice took on a familiar, lecturing tone. “The ancient Greeks strove for artistic perfection in all things. The believed the most pleasing relationship between two lengths was the Golden Section, and they used that proportion in many of their buildings, statues, and other creations.”

Balance Leads to Beauty

“I remember reading about that,” Alex said. “The Greeks believed the perfect way to split a line was to balance the ratio: The long section compared to the short section had to give the same ratio as the whole line compared to the long section.”

The \; Golden \; Section \; ratio

|———————A———————|————B————|

A \; is \; to \; B \; as \; \left(A + B \right) \; is \; to \; A, \; or . . .

\frac{A}{B}   =  \frac{A + B}{A}  = \: ?

“Very good, Alex,” said Dr. Jones. “And it turned out that the ratio was the same, no matter what the length of the original line.”

“But a Christmas tree isn’t a line,” Leon said.

“No, it is more like a cone,” his father agreed. “I used the Golden Section ratio to compare the height of the tree to its width. And I might add, it made for a nice-looking, well-balanced tree.”

Can You Solve These Christmas Tree Puzzles?

Christmas ornaments

  • What is the exact value of the Golden Section ratio?
  • If a 7-foot tree will fit in the Jones family’s living room, allowing for the tree stand and for a star on top, how wide will the tree be?
  • Approximately how much surface area will Alex and Leon have to fill with lights and ornaments?

[Note: Your answers to the last two questions will depend on whether you think Uncle William prunes his trees to the Golden Section ratio or to Grandpa Jones’s old rule of thumb. If you were Uncle Will, what would you do?]

Edited to add: Answers are now posted here.

To Be Continued…

Read all the posts from the November/December 1998 issue of my Mathematical Adventures of Alexandria Jones newsletter.


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