Artie stared at the diagram that Ting had drawn on the section of slate, still mulling over its implications of an interconnected world. He wasn't quite sure about how he felt about the drawing his teacher had sketched for him. More than anything he was surprised at how quickly Ting had come up with an answer.
There was no doubt that Ting had felt the tension building over the past several weeks, right after he had begun introducing Artie to some of the more formal concepts found within Geometry. And who could blame Artie for his malaise when his younger sister, Alicia was still outside playing while he was expected to focus on his schoolwork.
Despite his interest in architecture, Artie had taken an instant dislike to Geometry. For a discipline dedicated to describing the principles of patterns found in Nature, the definitions of point, line, and surface made no sense. As far as he was concerned, there was no way to see a dimensionless point, or for that matter, a line with no width. When Ting had introduced the idea that a surface had no depth, Artie thought he was going to explode with anger, but he had way too much respect for Ting to take out his frustrations on his mentor.
Ting was widely respected as one of the finest clockmakers in the entire region around the village. In truth, if Ting had chosen to settle in a city, he would have been considered one of the finest clockmakers in the world. He had an uncanny sense of time and could instantly understand almost any mechanical device by tracing through the inner workings of its components.
Up until the past few weeks, Artie's lessons with Ting felt more like an apprenticeship than what most teenagers would have considered "schoolwork". Yet all that had begun to change, when the time came for Artie to learn Geometry. The formality of the lessons didn't bother Artie, nor did the need for him to memorize material. What bothered Artie the most is that he felt like Euclid's Elements were founded upon a set of lies. Even worse, his teacher appeared to be completely oblivious to the problems with Euclid's view of the world.
Ting's promise that they would get to some "practical applications" of Euclid's ideas, seemed further off than ever. This lesson started off the same as the past few sessions, with Ting asking Artie to recite Euclid's most basic definitions:
"A point is that which has no parts."
"A line is length without width."
"The extremities of a line are points."
"A surface possesses length and width without depth."
"The extremities of a surface are lines."
"A solid has length, width, and depth."
"The extremities of a solid are surfaces."
Although Artie complied with his teacher's request, his face became redder and redder with each exposition.
"What's wrong, Artie?", Ting asked, "You should be proud of yourself, you've worked hard at memorizing Euclid's definitions."
All the emotions that Artie had been holding back suddenly poured out in one long stream:
"Well, I'm not proud of myself, not at all", exclaimed Artie, "and I don't think you should be proud of me either!"
"Why not?" asked Ting, with just a twinkle of a smile in his eyes.
"Who's ever heard of a point without parts?", explored Artie, "Or a line without width?"
"Or a surface without depth?" queried Ting, a broad smile splitting the surface of his face.
"Yes!" exclaimed Artie.
"You sound like someone I know", said Ting, as he began doodling a design on the piece of slate that sat in front of them.
"Who?" asked Artie, with a genuinely curious tone.
"Me", replied Ting.
"You?", asked Artie.
"Yes", said Ting, "at one point I was just as frustrated as you are now."
"And what happened?"
"My mentor drew me a picture like this.", said Ting as he passed the slate over to Artie.
Artie stared down at the slate and saw a simple depiction of a cube, labeled in the following manner:
"I don't get it," murmured Artie, as he scanned over the slate.
"What is your definition for a solid?", asked Ting, as he pointed to the cube.
"A solid has length, width, and depth", answered Artie.
"Alright, so there's no problem there", replied Ting.
"What about a surface having no depth?" queried Artie, with genuine curiosity.
"The facets of the cube have no depth unto themselves, only length and width" stated Ting, "otherwise you start to descend into the cube's volume."
Artie's eyes widened with surprise, for what Ting told him was true — even though the surfaces acted as the limits of the cube they possessed no depth unto themselves. Artie immediately saw that the edges of the cube acted as lines, as any additional width would immediately bubble out into the area of the surface.
Artie was even able to find an analog to Euclid's conception of a point. Each corner of the cube was technically dimensionless and served as the limit of a line (a cube's edge) without possessing any length or dimension unto itself.
"Huh...", said Artie, still staring at the slate.
"I said the same thing myself", replied Ting, "but that was many many years ago."
Artie kept staring at the slate, suddenly realizing that Euclid's Elements was much more than a disconnected collection of concepts. For the first time in almost a month, he finally saw Geometry as a way to describe the continuously unfolding patterns he had long seen in the Natural World.
Ting meandered to another corner of his workshop to examining the inner workings of one of the clocks that sat at the local Tavern, while Artie continued to ponder the picture.
"So this means that all of Euclid's concepts are connected..?!", exclaimed Artie after an indeterminate amount of time.
"Yes, it does", replied Ting.
"Except for volumes..."
"No, volumes are connected as well."
Ting looked up from the mantel clock, looking Artie directly in the eyes, "Volumes are connected through time."
"Yes, time", replied Ting, "Volumes can be viewed as limits of time."
For the first time in his life, Artie caught a glimpse of the interconnected world his grandmother, the town healer spoke so frequently about. His mind oscillated between seeing the world as a disconnected series of parts and interconnected aspects of time.
"But what about time...", Artie whispered to himself.
"We'll tackle the question of what time might limit another day.", shouted Ting, whose hearing was as sharp as an owl's.
With that, Artie went back to staring at the slate and Ting went back to his clock, each one thoroughly enthralled with the focus of their study.