Sean filed out of class with the rest of the herd. The crowd spreadoutlike distributaries of some great river as students headed to their respective destinations.
"Hey dude, wait up," the low deep voice didn''t belong to his tiny circle of acquaintances.
"Oh... oh," Sean glanced over his shoulder andpicked up his pace when he saw it was Brandon Cox towering a head above other students. Strategic withdrawlwas Sean''singrainedresponse tobeing pursued by one of Jason''s buddies. Oddly enough he was never inclined to flee whenever he ran into Jason himself. The sight of Jason with Tiffany and Carmen brought out the worst in Sean, a primeval rage that drove out any instinct of self-preservation.
"Why''re you running?" Brandon kept pace effortlessly on longer legs.
"Don''t want... to get punched," Sean slowed to catch his breath. Even dodging between students hadn''t slowed down thehalfback.
"Not gonna punch you," Brandon sounded aggreived andheld up abackpack, "Just want to talk... about my studies."
"What?" Sean stopped.
"Can we go somewhere quiet?" Brandon looked up and down the corridor uneasily.
"Um... whatever," Sean started walking, "Stayatleastten feet from me. And no texting your buddies."
Brandon followed after a moment. Sean led him on a tour of the ground floor and finally chosean empty classroom with two doors.
"Sit down," Sean pointed a few rows down. There was enough furtniture in between that Brandon couldn''t rush him. Sean could flee througheitherdoor if needed, "Talk."
"I can''t do it anymore, dude," Brandon emptied his backpack on a table and stared morosely at the heap of textbooks that tumbled out.
"Start making sense," Sean said testily. His limited patience was reservedfor Jason''s prettygirl friends.
"It... it''s just too much," Brandon gestured athis books, "Coach says if I don''t meet minimum GPA, I can''t stay on the team. He isn''tgoing to intercede with the principal anymore, like he did in my freshman and sophomore years. Says he''s used up too much goodwillon my behalf."
"Get a tutor," Sean suggested, "Look up online tutorials. I can recommend some good ones."
"There''s no way I can catch up on my coursework," Brandon seemed close to tears, "I''ve been neglecting my studies for too long. My folks hired a math tutor, but there''s too many rules to remember. Nothing makes sense."
"So you came to me?" Sean frowned.
"Kyle told me what you did for him," Brandon nodded, "Maybe you can help me figure this out?"
Why should I? Sean stared resentfully at Brandon''s broad muscled shoulders, his dark curly hair thatwomen loved, even hisexcessflabthatthey described as''snuggly''. Pretty girls came up to Brandon all the time to have long chats, to laugh with him and give him a hug. Why should Sean keephelping guys like Brandon and Kyle who had hit the genetic lottery? Sean suddenly felt ashamed at the turn of his thoughts. His own academic position would be precarious if not for the whim of an alien God trapped in dead cosmos. How often had Sean felt guilt at the unfair advantage that his power gave him over other students. Besides Brandon was part of Jason’s circle that Sean was drivinga wedge into, one person at a time. Tiffanywould have nudged Brandon towards Sean sooner or later.
"Tellme the parts that you do understand,” Sean nodded, “and we’ll go from there.”
Sean paced before the whiteboard as Brandon talked, his dismay growing as he realizedhow far behindthe other boy was. Sean continued to pace in silence after Brandon stopped talking.
"Since we are short on time,” Sean picked up adry-erasemarker, "I''m going try something drastic. Your problem is that youlook atmath as a bunch of rules to be memorized instead of seeing the big picture. Like missing the forest for the trees."Unauthorized use of content: if you find this story on Amazon, report the violation.
"What big picture?" Brandon looked skeptical.Sean drew a square on the whiteboard, markingit with sides one unit long. Then he connected two corners with a diagonal.
"How long is the diagonal of a unit square?" Sean tapped the board.
"Square root of 2, of course," Brandon frowned, "I''m not stupid, you know."
"I know you aren''t," Sean nodded, "Now why is it root 2?"
"The Pythagorean Theorem states that..." Brandon began.
"I''m not asking you to quote the theorem," Sean held up a hand, "I''m asking you why it is so."
"You expect me toprove the fucking Pythagorean Theorem?" Brandon demanded.
"I am asking you to see it," Seansketched three more unit squares touching the first square in a 2x2 grid, then drew diagonals for each such thatevery diagonal was a mirror of the diagonal of the adjacent squares.The four diagonals together formed another square standing on its tiplike a diamond, "See these diagonals? They are connected together like a diamond, right? If these diagonals are d-units long what''s the area of the diamond?"
SKETCH:https://www.cut-the-knot.org/do_you_know/SqRtOf2.shtml</a>
"d-squared," Brandon was looking bored.
"Right," Sean nodded briskly, tracing the biggest square formed by the 2x2 grid, "what''s the total area of all four unit squares?"
"Four," Brandon shrugged.
"Each diagonalsplitsits own square into two halves, right?" Sean tapped thediamond that the diagonals formed, "So this diamond is really made up of four triangles that are each half of theirunit squares. So thediamondmust be half the total area of all the unit squares. Half of four is two. Do you see now?"
"So... the diamond is two square units in area," Brandon nodded slowly, "which also equals d-squared. That''s why the diagonal length is root-2 ! Fuck... never knew it was that simple... that''s really cool, dude."
"It is that simple," Sean grinned as he erased thesketch and drew a slightly more complex mosaic involving right trangles, "and we canextend the same reasoning..."
Understandingdawned onBrandon''s face as he watched, "Shit... that''s whythe Pythagorean Theorem is true."
"Yes," Sean nodded earnestly, "Math isn''t about numbers or formulas. It''s the language of thought itself... a mosaic of reason that you cansee for yourself. A theorem is true because it can''t be any other way, anymore than a diagonal of a unit square can be anything other than root-2. Once I''m done with you, you''ll no longer see courseworkas boringcrap to slog through, but as mathematical structures you can hold and play with in your mind."
"You... you can do that?" Brandon blinked, "For all of them?"
Sean nodded, "If you are willing to put your mind to it."
"Bullshit," Brandon snorted, flipingthrougha book, "drawing squares and triangles is one thing, but you''re saying everything can be boiled down to pretty pictures? What about this... this calculus stuff? This chapter goes on and on about integral of something dx and dy/dx and crap like that. What does any of it mean?"
"What about it?" Sean laughed, "dx is just another way of saying a tiny piece of ''x''. Integral is justa fancy way of saying "add it all up". What do you get if add up all the tiny pieces of ''x''."
"x, probably," Brandon raised a brow.
"Correct," Sean smiled, "and if you split the area under a curve into tinyslivers and add it all up again?"
"The total area under the curve?" Brandonventured hesitantly.
"Correct again," Sean nodded, "If a tiny change in ''x'' causes another tiny change in a function, lets call the function ''y''. Then what is the ratio of their changes?"
"dy/dx, I think," Brandon said after a moment''s thought.
"Bingo," Sean raised his arms, "dy/dx is simply the ratio of tiny changes on the ouput and input of a function. It''s basically algebra dressed up to look fancier. The language of thought."
"It can''t be that simple" Brandon stared, "You are shitting me."
"Brandon," Sean sighed, "None of this is supposed to be hard. Isacc Newton invented calculus just to figure out some other problem he was trying to solve. Granted,most of us aren''t Newton, but we don''t need to be. The reason all this seems intimidating and boring is because the school curriculum hasn''t been updated since the dark ages. In the days before computers the only way to do calculus was by complicated analytical methods on toy problems. With computers none of that shit is necessary, but the school still expects us to go through the motions. It sucks, but once you start thinking in terms of higher dimensional structures, it''ll get easier, I promise. "
"Higher dimensional?" Brandon gaped.
"Some problems require N-dimensions to describe properly.," Sean nodded, "but there are tricks to visualize them as 3-D. And once you do, no problem in your textbook will be beyond your reach. Here, I''ll show anexample..."
As Sean sketched,Brandon acquired a far away look,"A thousand-fold thought."
"Huh?" Sean raised a brow.
"They say a picture is worth a thousand words," Brandon barked a laugh, "but, dude, your sketches are worth a thousand thoughts all connected to each another. It just clicks."
"Many-fold thought," Sean smiled, "I like it."
END OF CHAPTER