The Shadows Lurking in the Equations
For all the history of computational mathematical visualization, graphing equations has been done in binary mode - where graphs show only where an equation is EXACTLY equal. But when you only see in black-and-white, some things are invisible. For all this time, lurking beneath the error == 0 surface, mathematical shadows have been lurking in the equations.
FuzzyGraph, on the other hand, visualizes equations in Non-Binary mode - showing not only where an equation are exactly equal, but also where the equation nearly equal and where the equation is far from equal (where the error is high). Sometimes, these high error areas form clear visual shadow-like features.
Let's look at some examples...
Example 1: Slash Dot Equation
Here is the "Slash Dot" Equation ( \( \frac{y}{x^2+y^2} = \frac{x+1}{x^2+y^2} \)) as both a conventional and fuzzy graph...
Note the giant black hole that is present in the Fuzzy/Non-Binary graph, but invisible in conventional/Binary graphing. This "black hole" feature represents a region of high error in the equation.
Example 2: Quasar Equation
Let's look at another example: \(y = \frac{x}{x^2 + y^2} \) Conventional graph of \(y = \frac{x}{x^2 + y^2} \) Fuzzy graph of \(y = \frac{x}{x^2 + y^2} \)
Notice that the black hole eye-looking features are COMPLETELY INVISIBLE in the conventional/binary mode of graphing.
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