The first half of last week’s comics are mostly ones from Comics Kingdom and Creators.com. That’s unusual. GoComics usually far outranks the other sites. Partly for sheer numbers; they have an incredible number of strips, many of them web-only, that Comics Kingdom and Creators.com don’t match. I think the strips on GoComics are more likely to drift into mathematical topics too. But to demonstrate that would take *so* much effort. Possibly any effort at all. Hm.

Bill Holbrook’s **On the Fastrack** for the 8th of July is premised on topographic maps. These are some of the tools we’ve made to understand three-dimensional objects with a two-dimensional representation. When topographic maps come to the mathematics department we tend to call them “contour maps” or “contour plots”. These are collections of shapes. They might be straight lines. They might be curved. They often form a closed loop. Each of these curves is called a “contour curve” or a “contour line” (even if it’s not straight). Or it’s called an “equipotential curve”, if someone’s being all fancy, or pointing out the link between potential functions and these curves.

Their purpose is in thinking of three-dimensional surfaces. We can represent a three-dimensional surface by putting up some reasonable coordinate system. For the sake of simplicity let’s suppose the “reasonable coordinate system” is the Cartesian one. So every point in space has coordinates named ‘x’, ‘y’, and ‘z’. Pick a value for ‘x’ and ‘y’. There’s at most one ‘z’ that’ll be on the surface. But there might be many sets of values of ‘x’ and ‘y’ together which have that height ‘z’. So what are all the values of ‘x’ and ‘y’ which match the same height ‘z’? Draw the curve, or curves, which match that particular value of ‘z’.

Topographical maps are a beloved example of this, to mathematicians, because we imagine everyone understands them. A particular spot on the ground at some given latitude and longitude is some particular height above sea level. OK. Imagine the slice of a hill representing all the spots that are exactly 10 feet above sea level, or whatever. That’s a curve. Possibly several curves, but we just say “a curve” for simplicity.

A topographical map will often include more than one curve. Often at regular intervals, say with one set of curves representing 10 feet elevation, another 20 feet, another 30 feet, and so on. Sometimes these curves will be very near one another, where a hill is particularly steep. Sometimes these curves will be far apart, where the ground is nearly level. With experience one can learn to read the lines and their spacing. One can see where extreme values are, and how far away they might be.

Topographical maps date back to 1789. These sorts of maps go back farther. In 1701 Edmond Halley, of comet fame, published maps showing magnetic compass variation. He had hopes that the difference between magnetic north and true north would offer a hint at how to find longitude. (The principle is good. But the lines of constant variation are too close to lines of latitude for the method to be practical. And variation changes over time, too.) And that shows how the topographical map idea can be useful to visualize things that aren’t heights. Weather maps include “isobars”, contour lines showing where the atmospheric pressure is a set vale. More advanced ones will include “isotherms”, each line showing a particular temperature. The isobar and isotherm lines can describe the weather and how it can be expected to change soon.

This idea, rendering three-dimensional information on a two-dimensional surface, is a powerful one. We can use it to try to visualize four-dimensional objects, by looking at the contour surfaces they would make in three dimensions. We can also do this for five and even more dimensions, by using the same stuff but putting a note that “D = 16” or the like in the corner of our image. And, yes, if Cartesian coordinates aren’t sensible for the problem you can use coordinates that are.

If you need a generic name for these contour lines that doesn’t suggest lines or topography or weather or such, try ‘isogonal curves’. Nobody will know what you mean, but you’ll be right.

Ted Key’s **Hazel** for the 9th is a joke about the difficulties in splitting the bill. It is archetypical of the sort of arithmetic people know they need to do in the real world. Despite that at least people in presented humor don’t get any better at it. I suppose real-world people don’t either, given some restaurants now list 15 and 20 percent tips on the bill. Well, at least everybody has a calculator on their phone so they can divide evenly. And I concede that, yeah, there isn’t really specifically a joke here. It’s just Hazel being competent, like the last time she showed up here.

Mark Anderson’s **Andertoons** for the 11th is the Mark Anderson’s **Andertoons** for the week. And it’s a bit of geometry wordplay, too. Also about how you can carry a joke over well enough even without understanding it, or the audience understanding it, if it’s delivered right.

Rick DeTorie’s **One Big Happy** for the 11th is another strip about arithmetic done in the real world. I’m also amused by Joe’s attempts to distract from how no kid that age has ever not known precisely how much money they have, and how much of it is fairly won.

Bill Griffith’s **Zippy the Pinhead** for the 11th is another example of using understanding algebra as a show of intelligence. And it follows that up with undrestanding quantum physics as a show of even greater intelligence. One can ask what’s meant by “understanding” quantum physics. Someday someone might even answer. But it seems likely that the ability to do calculations based on a model has to be part of fully understanding it.

I have even more Reading the Comics posts, gathered in reverse chronological order at this link. Other essays with **On The Fastrack** tagged are at this link. Other Reading the Comics posts that mention **Hazel** are at this link. Some of the many, many essays mentioning **Andertoons** are at this link. Posts with mention of **One Big Happy**, both then-current and then-rerun, are at this link. And other mentions of **Zippy the Pinhead** are at this link.