Enhancement ideas for existing visualizers

[gwhitney]

This discussion is for ideas on how to extend/enhance existing visualizers, that may not yet actually be planned/intended at the level of creating an issue, but are worth considering/discussing as possible future changes.

[gwhitney]

Add color schemes to Mod Fill visualizer

Currently ModFill colors a cell if a sequence entry has the corresponding residue, with the option to change what color is used under certain circumstances. You can watch the diagram fill in to get some sense of when in the sequence a given residue occurs, but there is no explicit indication in the final diagram (when the end of the sequence is reached). You can get some sense of it if the colors used have transparency, as the most recent colorations of a given cell have the most effect, but there's still nothing explicit.

I don't think it would be overcomplicating the visualizer to offer a couple more color schemes. The ones that come to mind (a subset or superset of these could be implemented) involve the user choosing a gradient (from some standard options like greyscale or typical heatmap gradients) and then using it to indicate one of

The position of the first term where that residue occurs
The position of the latest term visited where that residue occurs
The absolute number of times that residue occurs (not sure if this one makes sense)
The fraction of the visited entries in which that residue occurs

In (4) there is the question of whether the gradient should have the same dynamic range for different moduli -- for higher moduli, we might expect that all individual residues have low(er) frequencies, since there are so many residues, so we might want to (automatically, or under user control?) "turn the gain up" on the larger moduli.

This was originally #515, but moved here to update it and to avoid too many different discussions.

[gwhitney]

Note that with the version of ModFill currently in ui2, all of the specific proposals here can be arranged with judicious use of the color formulas, with the probable exception of the first one, which would seem to require knowing whether a residue has occurred before when calculating what color to paint it with -- not sure if there is currently any way to get that information (I don't think so).

[gwhitney]

Generalize Grid with formulas

Currently Grid has three possible "paths" along which the entries can be laid out in the grid, and then the grid cells are colored by "properties" of the contents of each cell. The properties are in the process of being generalized by formulas to be more or less arbitrary, but the "path" it seems could be generalized as well: the idea is that you could specify a formula with inputs x, y, maybe c for the number of columns and r for the number of rows in the layout, maybe s for the "spiral distance" from the center, for convenience, and which is allowed to use a as a function symbol to get sequence values, and then the formula returns the contents of the cell. With the convenience s variable, the spiral path would be a(s), the row layout would be a(y*c+x), and the row augment layout would be a(x) + y. But you could then also lay out vertically by a(x*r + y) or go back and forth along the rows ("boustrophedonically") by a(y*c + x + (c-2x)*(y % 2)) or go by diagonals via a(T(x+y-1)+y) (where here T(n) is an abbreviation for the nth triangular number). This last is important because for many OEIS sequences, that diagonal layout is the natural/intended layout for the numbers, so you could then display/highlight entries in their natural 2D arrangement. (e.g., show the modular behavior of the arithmetic triangle A007318).

[gwhitney]

This is implemented in FormulaGrid in ui2, and so this conversation should be hidden once alpha is released.

[gwhitney]

We could add a "legend" parameter for the Turtle visualizer that will display arbitrary values, say in the corner of the screen. This came up when animating the eta- (alternating) series expansion for the zeta function on the critical line as the imaginary part varied; it would be much easier to interpret each frame if the current imaginary part and the estimated distance of the limit point to the origin were displayed.

[gwhitney]

Possible enhancements to, or features of a future "advanced" version of, the Differences visualizer, that came up in the discussions concerning #555

• Control of font size
• Ability to scroll the display, at least left-to-right, but perhaps in both dimensions
• A row indicating the sequence indices
• A legend column (labeling indices, values, differences, 2nd differences, etc.)
• Ability to perform other operations on the pairs of entries (e.g., ratio, or p-adic distance, etc.)
• A formula to control the color an entry is displayed in
• Ability to display an entry as a shape, possibly with adjustable size

[gwhitney]

Many visualizers present text in one way or another: drawn in the graphics screen, or as popups or tooltips. It would be helpful if there were a uniform way of copying such text to the clipboard, for insertion in other contexts.

[gwhitney]

Would it be worth contemplating adding a uniform "fade" capability common to all P5 Visualizers?

[gwhitney]

(Factor)Histogram visualizer possible enhancements:

• Generalize to an arbitrary Histogram of any definable sequence(s)
• Indicate the total number of terms analyzed in some way, as the data is trickier to interpret without that and that information can be scrolled out of view in the sequence chooser.
• Animate the accumulation of data (i.e., in frame n, show the histogram of the first kn values of the sequence)

[gwhitney]

Possible enhancements to NumberGlyph visualizer.

The motivation for these two is that alternate layouts may allow better tracking of divisibility properties and the like.

• Provide an option to enforce a perfect square layout of glyphs (which would be centered in available space).
• Provide an option to specify exactly how many glyphs should be on each row (since some choices, like 12, may make it easier to see periodic patterns in the results).