An exploration in charting and "mapping" -- complex polynomials

[MrShorty]

The pure math side of this will probably be of almost no interest to anyone (some may even be convulsing on the floor in shock and dismay). For those who are up to it, I found this an interesting look into Excel charts.

The challenge when trying to visualize equations in complex space is that one needs 4 values to describe what is going on -- two values (a+bi) to describe the "domain" or "x" values of the function, and two values (c+di) to describe the "range" or "y" values of the function. It is almost the challenge of graphing equations in 4D space.

In searching for strategies, the first one I came across (that made easy sense) was this idea of a "map". We have two 2D scatter plots -- one will represent the "x" values we are putting into the function, and the other will represent the "y" values we are getting out of the function. If we can adequately vary the color/symbol for each point in each chart, we can use the symbols to show what happens to point1 in going from the domain to the range. I found that this was quite easy to do by having a single data series in each chart, and selecting the "vary by point" option in the Marker Fill section of the format series dialog.

As I tinkered with this, I noted that Excel seems to cycle through the nine built in symbols, varying the colors as it goes from point to point. I wanted a "gap" between each "contour" on both plots, so I found that a sequence of 8 points + 1 gap (for a 9 data point cycle) would make it so that the same symbol would correspond to the same point on each contour, which would make it easier to read the charts.

This naturally flowed out of my "root finding" spreadsheet, so the current arrangement is looking at those points where y=0+0i. These are represented by the "y" contours that seem to go across the origin of the range/y chart.

I find that the charts are easiest to read for up to 3rd order polynomials. I have programmed the spreadsheet to handle up to 5th order, but these charts become difficult to read (though some of the patterns created are interesting).

Things to do:

1) Study the charts and see if you can "see" where the roots (y=0+0i) to the equation occur. Check in C9:D13 to see if you are correct.
2) change the coefficients of the polynomial (D2:I3) to see how different equations look on the charts.
2a) Try some well known functions like y=x^2+1 (real coefficients, with imaginary roots).
2b) Copy and paste special-as values the random numbers in L2:Q3 to get different polynomials.
3) change the values in C16:D17 to see different "contours" on the two plots. (Correct choices here can really help visualize where the roots are).