### 3.5.4 Ticks and labels

The style parameters regulating the location of the viewport, appearance of the tick marks, and so on are fairly straightforward: You can specify which of the four sides of the viewport will have tick marks, or you can select “axis” style, where coordinate axes and ticks run through the middle of the viewport. Tick marks can extend into or out of the viewport, or both (beware that tick marks which project into the viewport often overlay a portion of your data, making it harder to interpret). You can draw a frame line around the edge of the viewport, or just let the tick marks frame your picture. You can specify which, if any, of the four sides of the viewport will have numeric labels for the ticks, the distance from the ticks to the labels, the font and size of the labels, and so forth.

An elaborate artificial intelligence (or stupidity) algorithm determines tick and label positions. Four parameters control this algorithm: nMajor, nMinor, logAdjMajor, and logAdjMinor. Tick marks have a hierarchy of lengths. The longest I call major ticks; these are the tick marks which get a numeric label (if you turn on labels). Like the scale on a ruler, there is a hierarchy of progressively shorter ticks. Each level in the hierarchy divides the intervals between the next higher level into two or five subdivisions – two if the interval above had a width whose least significant decimal digit of 1 or 2, five if that digit was 5.

Only two questions remain: What interval should I use for the major ticks, and how many levels should I proceed down the hierarchy? The nMajor and nMinor parameters answer these questions. They specify the upper limit of the density of the largest and smallest ticks in the hierarchy, respectively, as follows: The tick interval is the smallest interval such that the ratio of the full interval plotted to that interval is less than nMajor (or nMinor). Only major tick intervals whose widths have least significant decimal digit 1, 2, or 5 are considered (otherwise, the rest of the hierarchy algorithm fails).

Log scaling adds two twists: First, if there are any decades, I make decade ticks longer than any others, even if the first level of subdecade ticks are far enough apart to get numeric labels. If there will be ticks within each decade, I divide the decade into three subintervals – from 1 to 2, 2 to 5, and 5 to 10 – then use the linear scale algorithm for selecting tick intervals at the specified density within each of the three subintervals. Since the tick density changes by at most a factor of 2.5 within each subinterval, the linear algorithm works pretty well. You expect closer ticks on a log scale than you see on a linear scale, so I multiply the nMajor and nMinor densities by logAdjMajor and logAdjMinor before using them for the subintervals.

A final caveat: On the horizontal axis, long numeric labels can overlap each other if the tick density gets too high. This depends on the font size and the maximum number of digits you allow for those labels, both of which are style parameters. Designing a graphics style which avoids this evil is not easy.