The Puzzling Pythagorean Pebbles

Feature photo above by meichimite (CC BY 2.0).

Alexandria Jones and her family flew to Italy for spring break. Her father, the famous archaeologist, had to visit an excavation.

It was late when their plane landed in Crotone, a small coastal city near the instep of Italy’s boot. Dr. Jones had used the Internet to find a hotel that allowed pets, so Alex was able to snuggle down with her favorite pillow — her trusty dog, Ramus.

The Original School of Mathematics

The next day, Dr. Jones introduced his family to Sonya Theano, a former student of his and the director of this dig. “Come, let me show you around,” Dr. Theano said. “We’ve uncovered several buildings of a small compound, set apart from the city of Crotona, as it was called then. From the pottery and trade goods, we estimate these buildings were in use around 550-500 BCE.”

She pointed out the ruins of cell-like rooms in a U-shaped building. “We think this was the dormitory, surrounding a courtyard where the students could gather to discuss ideas.”

“Students?” Alex asked.

Alex’s father nodded. “Didn’t I tell you? Sonya believes she’s found the school Pythagoras started.”

Dr. Theano smiled. “I have a special interest in that old mathematician. We might even be related. After he settled in Crotona, Pythagoras married a local girl named Theano. She had been Pythagoras’s student for several years, and she wrote an important treatise on the Golden Ratio. She also wrote the old man’s biography.”

“Is that how you knew where to dig, by reading her book?” Leon asked.

The archaeologist shook her head. “No. All of Theano’s writings, like many ancient manuscripts, have been lost to history.”

“Not even one copy survived?” Alex asked. “How do you know she wrote it?”

Her father answered, “The same way we know about many ancient writers. Other people mentioned or quoted her book. Often, we get information 3rd- or 4th-hand: One writer quotes another writer telling what he heard about a third person who originally wrote the story down.”

To Catch a Thief

While they talked, Ramus had been sniffing around the base of a wall. Suddenly, he gave a low growl and strained at the leash.

“That’s strange,” Dr. Theano said. “I wonder if he smells our intruder.” She explained that several artifacts had disappeared from her dig over the past few weeks.

Alex glanced at her father and saw by his frown that he was thinking the same as she — Simon Skulk. “Maybe we could try a stake-out,” she said. “I bet Rammy could catch him.”

That night, Alex and Rammy crouched in the dark shadows cast by a pile of debris from the dig. The others were hiding in a loose circle around the site. Alex heard the crunch of a boot on rocks. She jumped up and saw a man with a shovel, scraping the ground near where the workers had been digging that afternoon.

“Get him, Rammy!”

“You!” the man said. “I thought you were still in Egypt.”

“Hello, Simon.” Dr. Jones walked up behind him. “I think you are finished here.”

The man kicked Rammy, making the dog yelp. Then he rushed at Alex and pushed her. The rubble slipped under her feet and she fell, slightly stunned. Her father ran to kneel by her side. By the time she sat up, rubbing her head, Simon Skulk was gone.

Building Up the Natural Numbers

Leon leaned over where the man had dropped his shovel. He held up a piece of cloth. “Looks like Rammy took a chunk out of his pants leg.”

The dog sniffed at the scrap, growling. Then he ran toward the road, looked back at them, and barked. He led them to a small building near the edge of town. The rooms were a jumbled mess.

“Looks like he packed in a rush,” Alex said. “He’s probably on a plane by now. At least he left these artifacts behind.”

Leon held up a painted bowl, filled with pebbles. “What’s this?”

Dr. Theano lifted out a pebble. “These are calculi, from which we get our word calculate. The Pythagoreans counted out pebbles to make their numbers. 1 was the beginning, the number of reason. 2 was the female, 3 the male.”

As she talked, she set each pebble on the table, making a triangle.

“4 was justice. 5, which is 2 + 3, was marriage.” The triangle kept growing. “…And 10 was the holiest number of all, the tetractys, the divine number of the universe.”

Pythagorean triangular number

To Be Continued…

Read all the posts from the March/April 1999 issue of my Mathematical Adventures of Alexandria Jones newsletter.

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