One day, a very long time ago, a young princess lived in a golden castle. She was twenty- one years old. She was so very beautiful, and everyone loved her because she too had golden proportions. Her name was Geometra. All of her family was golden. Even her dog, Fibonacci, and her magic golden winged horse, Goldielocks, were indeed golden. One day, her father, King Pentagram came to see Geometra. Geometra was outside, basking in the sun, and enjoying nature. She loved to observe all plants and animals, and most of her time was spent doing just that.
"Geometra, my beautiful daughter, I have something to tell you. You understand how important it is for the world to be golden, don't you Geometra?"
"Of course father, without a golden world, people would only see the pain and suffering, and not stop to look at the beauty."
"Exactly, Geometra. But I have a problem. Our followers are planning a revolt against us, because they believe only man made things can be golden."
"Oh father! How terrible! How stupid of them! Of course, nature is full of the golden mean! What are we going to do, father? You are too busy to do anything about it, mother is also too busy, looking after Pent Jr.!"
"This is why I have come--to ask of you a favor. You are old enough now, I believe, to have a quest. Your mission is to find three examples of the golden mean in nature. If you complete this task, you will forever be remembered in history, and people will look up to you as a heroine. If you do not complete this task, our reign will end, and the world will be filled with chaos. You must prove it to the people. I know you can do it. Prove yourself a woman."
"But father, how do I do it?"
"Use the golden force, Geometra, trust yourself on this quest, and be brave. I love you, my daughter. Now go!"
Geometra jumped on her horse, grabbed her dog, and off they went, riding into the pinewoods, not knowing what to expect next. Pretty soon, she came upon a clearing. She got off of Goldielocks, and went exploring. No sooner had she walked four yards, when she came upon the oddest little house you ever did see. She went to the door and knocked. A little old man came to the door and let her in.
"Hello fair maiden", he said. "My name is Mathman. Hermit, painter, and mathematician."
"Good day. I am princess Geometra. Pleased to meet you. What a coincidence! I am in need of some math help."
"Oh good!", Mathman said. "I just love helping people with math! Sit down, have a bite to eat, and tell me what you need to know."
She told him about the quest, and having to find the golden proportion in nature. They got to work.
"First of all, you must bring me your three favorite things in nature. After that, we shall proceed. I will send along with you my son, Goldman, to protect you and keep you free from danger." His son, she had noticed, was very handsome, indeed, and she thought to herself that that was a good name for him. She was a very firm believer in feminism though, so she said:
"Thank you very much, sir, but I can do very well on my own without any man to protect me, but I appreciate the offer. Now if you'll excuse me, I have my three favorite things in nature to get: a pineapple, pine cone, and an artichoke."
And off she rode on her horse with her little dog. She flew to Hawaii on Goldielocks, and bought a pineapple at the market. She then flew to a garden where she picked an artichoke. Later, she realized that it was a witch's garden, and sort of wished Mathman's son was there, because the witch wasn't too happy about having her artichoke stolen. After getting the witch and artichoke dispute settled, she flew back to the pinewoods and picked a pine cone. She got off her horse and walked to Mathman's house. He opened the door, and greeted her. They soon got to work.
He said, "Now that you have the three things, we have to establish what error you will allow in your calculations of the golden mean. I mean, nature isn't going to be pure gold, so how much can it deviate from 1.61803398...or .61803398?"
"Well", said Geometra. "How about it can't be larger than.1.626 or .626, and no smaller than 1.588 or .588?"
"Sounds good to me," Mathman said. "The next thing we have to do is count the rows naturally present in these plant items. If you take the pineapple, and count the rows that curve to the left, you count 13 of them. If you count the rows that curve to the right, there are 8 of them. You also have to count the rows that are almost vertical. There are 21 of them. What do these three numbers have in common?"
"They are all fibonacci numbers!!"
"Correct as usual, Princess Geometra. Now, divide the numbers found in the pineapple by the numbers next to them in the fibonacci sequence, that are also found in the pineapple. In other words, 13/21, 21/13, 8/13, and 13/8. But, not 8/21 or 21/8 because they aren't next to each other."
"Okidoke," said princess Geometra. She pulled out her pocket calculator and found that 13/21= .619, 8/13=.615, and 13/8 = 1.615.
"Yes!!! These are all close enough to the golden mean! I know what to do now!"
She took the artichoke, this time counting the rows curving left, and the rows curving right. This time there was no almost vertical row. She found that there were 8 rows curving left and 5 curving right! She was so excited. It too was almost golden!!! She did the pine cone next. She turned the pine cone upside down, so she was looking at the base of the pine cone. She counted the rows going out and left from the center of the pine cone---8!!!!!! Then, she had trouble counting the rows going out and right from the center of the pine cone, because she was shaking with excitement! Some how she managed--there were 13!!!!!! She had done it!
"Do you want to see if the pineapple's leaves are golden too?," asked Mathman.
"Alright!" said Geometra happily. She counted 8 rows of leaves going right. So far so good. But what was this? She counted 10 rows of leaves going left!"
"Oh no! How can this be?," she asked.
"It's okay Geometra. Not everything is golden. So, what conclusions have you come up with?," Mathman said.
"Well, the ratio of consecutive fibonacci numbers gets closer and closer to the golden mean! And things in nature can be golden!---but not everything is."
"Excellent work, my dear! Your father will be proud."
"Thanks Mathman, but I couldn't have done it without you."
Just then, Mathman's son came to her. He was so taken by her beauty that he asked her to marry him. She said yes, for he was sooo fine. They said thank you and good-bye to Mathman. They rode back to the golden castle, and sat on Goldielocks together, in the middle of a chaotic mob. The shouting stopped. Geometra held up her golden pineapple for all to see and said loudly and clearly "Gold is found in nature!". The crowd cheered. Geometra was looked upon as a heroine, and peace was restored in the kingdom once again. When the king and queen died, Geometra and her husband became king and queen, and they were the best rulers the land ever had. Geometra loved her husband, and he loved her back, and they lived happily ever after.
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