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The Shadows Lurking in the Equations

Slash Dot Equation comparison 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...

Conventional graph of \( \frac{y}{x^2+y^2} = \frac{x+1}{x^2+y^2} \)
Conventional graph of \( \frac{y}{x^2+y^2} = \frac{x+1}{x^2+y^2} \) 
Fuzzy graph of \( \frac{y}{x^2+y^2} = \frac{x+1}{x^2+y^2} \) Fuzzy
graph of \( \frac{y}{x^2+y^2} = \frac{x+1}{x^2+y^2} \)

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: \(y = \frac{x}{x^2 + y^2} \)
[quasar_equ] Conventional graph of \(y = \frac{x}{x^2 + y^2} \) 
[quasar_equ] 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.

Example 3: Simple Star and Black Hole

To get a better idea of what these black hole things are, let's look
at a simpler example. First let's look at the opposite of a black
hole - a simple star/particle example: \( x^2 + y^2 = 0 \). For this
equation, there is only 1 solution: (0, 0). So if you graph this in a
conventional graphing app, it will only show a single dot at (0, 0).
But in FuzzyGraph, it looks like a fuzzy particle or something.

[particle_e] Conventional graph of \( x^2 + y^2 = 0 \) [particle_e]
Fuzzy graph of \( x^2 + y^2 = 0 \)

But now, let's invert this to get the "Black Hole Equation": \( \frac
{1}{x^2+y^2} = 0 \)...

[black_hole] Conventional graph of \( \frac{1}{x^2 + y^2} = 0 \) 
[black_hole] Fuzzy graph of \( \frac{1}{x^2 + y^2} = 0 \)

In this case, there is absolutely nothing to show on a conventional
graph, as there are actual solutions to this equations. However,
there is still a mathematical topography which can be visualized (as
can be seen in the fuzzy graph).

Example 4: Shadow Line

Not all of the Shadows are like black holes.

In this example, let's start by combining 2 lines together: \(y=x\)
and \(y=-x\).

We can visually add 2 equations together by refactoring them so they
are both equal to 0, and then multiplying the two refactored
equations together. \(y=x\) can be changed to \(y-x=0\), and \(y=-x\)
can be refactored to \(y+x=0\).

We can then combine 2 into a single equation these like this: \(
(y-x) \times (y+x) = 0 \)

[x_equation] Graph of \( (y-x) \times (y+x) = 0 \) [x_equation] Fuzzy
graph of \( (y-x) \times (y+x) = 0 \)

And now, let's invert one of the equations using division: \( \frac
{x-y}{x+y} = 0 \)

[lines_divi] Graph of \( (y-x) \times (y+x) = 0 \) [lines_divi] Fuzzy
graph of \( (y-x) \times (y+x) = 0 \)

So as you can see, the line that was inverted (under the division
line) is now a Shadow Line. And this seems like a more "correct" way
to visualize this than as the conventional graph shows it (which is
indistinguishable from the simpler equation, \(y-x=0\)).

Example 5: Phi Equation

This equation works almost exactly as the previous. And like before,
let's start with multiplication to combine 2 equations (in this case,
a circle and a vertical line equation): \( x \times (x^2+y^2-1) = 0
\).

[line_and_c] Graph of \( x \times (x^2+y^2-1) = 0 \) [line_and_c]
Fuzzy graph of \( x \times (x^2+y^2-1) = 0 \)

But now, let's invert the circle by using division, which makes the
equation: \( \frac{x}{x^2+y^2-1} = 0 \).

[phi_equati] Graph of \( \frac{x}{x^2+y^2-1} = 0 \) [phi_equati]
Fuzzy graph of \( \frac{x}{x^2+y^2-1} = 0 \)

Note that the Shadow Circle is invisible in the conventional graph.
In fact, the conventional graph looks identical to a conventional
graph of the \(x=0\) equation (as if the denominator was not there).

Example 6: Underwater Islands

In all of these previous examples, the "shadows" have represented
areas of high error. But in this last example, we'll see some hidden
details that represent areas of low error - areas that are nearly
solutions to the equation.

Consider the equation, \( y=4 sin(x)+ sin(2.7y) \), as both a
conventional graph and a fuzzy graph:

[underwater] Conventional graph of \( y=4 sin(x)+ sin(2.7y) \) 
[underwater] Fuzzy graph of \( y=4 sin(x)+ sin(2.7y) \)

Note the floating dots in the fuzzy graph version that are not there
in the conventional/binary graph. These are like underwater islands -
underwater mountains that are just below the surface of the water (or
in this case, the \( error == 0 \) surface). These hidden islands
represent area that are near-solutions to the equation (which are
only visible in FuzzyGraph).

Their presense hints that we can tweak the equation slightly to cause
them to burst above the surface of the water (which should also make
them visible in conventional graphs).

So let's change the equation from:
\( y=4 sin(x)+ sin(2.7y) \) to:
\( y=4 sin(x)+ sin(2.8y) \)...

[abovewater] Conventional graph of \( y=4 sin(x)+ sin(2.8y) \) 
[abovewater] Fuzzy graph of \( y=4 sin(x)+ sin(2.8y) \)

And as you can see, those previously-hidden islands are now visible
in the conventional graph.

So Fuzzy/non-binary graphing can help us see features of the
mathematical topography that are completely invisible with
conventional/binary.



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Date published: 2025-11-05

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