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Remember the famous story (that I totally made up) of Knot Dude and Papa Knot? You know, the one where the clever young son (that's Knot Dude) of a big-shot ancient Egyptian pyramid builder (that's Papa Knot) used his mathematical curiosity about knots to discover the Pythagorean Theorem and in so doing to solve his dad's problem of figuring out how to build pyramids with perfectly square bases?

Well, as it turns out, there are more mathematical lessons to be gleaned from this story. To begin with, today we're going to see how Papa Knot used his son's discovery to simplify the process of building perfectly shaped pyramids of any size—an ability which he used to revolutionize the pyramid building industry.

What exactly did he do? He made a graph. How did that help? And how did he do it? Those are exactly the questions we'll be answering today and over the next few weeks.

Why Are Graphs Useful?
Imagine we're once again headed back in time to visit the wonderful world of ancient Egypt. Our trip begins just as Papa Knot receives an order to build a modest yet perfectly shaped 15-by-15-foot pyramid to house the remains of a wealthy family's beloved feline friend. Using either the oh-so-handy rope with specially tied knots that Knot Dude made for him or his son's newly discovered relationship that the lengths of the two sides (dubbed a and b) and hypotenuse (dubbed c) of a right triangle are related by a^2 + b^2 = c^2 (which a few millenia later would be known as the Pythagorean Theorem), Papa Knot was able to figure out the length required for the diagonal across the base of the pyramid.

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