To start the last week of March Mathness, because it’s a Monday, I’m going to go easy on y’all with some light, easy topics. (Maybe I can lull you into paying attention for the major topic of the month: matrix rotation.)

It has occurred to me that, if I’m talking about math in March, I absolutely must mention one of my all-time favorite mathematical objects, the **Mandelbrot**. I’ll try to get to that today, but the main topic is a simple something that I ran into while working on my 3D model of the big island of Hawaii.

The question was: *How many miles are there per degree of latitude?*

Just to be clear, we’re talking about the *east-west* distance *along* a line of latitude. The north-south distance, along lines of longitude, does not vary. (By “distance” I mean the distance from one degree to another.)

What makes the question a bit interesting is that it is slightly more complicated than asking: *How many miles are there per degree of longitude?*

The reason they’re different questions is that the *circles of latitude* get smaller as we move from the equator to the poles.

How many miles (or kilometers) per degree along a line of latitude *depends on your latitude*. At the equator, the circle of latitude is “full size,” but at the poles, it shrinks to zero size.

By “full size” I mean that the circle of latitude is the same size as any circle of longitude.

More importantly, “full size” is the *maximum* size any such circle can be. The center of these circles coincides with the center of the Earth.

Therefore, at the equator, the east-west distance per degree is (a) at its maximum and (b) the same as north-south distance per degree.

As we move north, the east-west distance gets shorter and shorter until, at the poles, it becomes zero.

How can we calculate the distance at any given latitude?

**§**

The first thing we can do is calculate the distance per degree along the equator. (That will also be the distance per degree for any north-south meridian.)

This is very simple if we remember the formula for circumference (2πr):

Where * R* is the radius of the Earth (

**3958.8**miles or

**6371.0**kilometers). Plugging in R, we get:

Which, by the way, is 60 nautical miles (**nmi**). A nautical mile is defined as *one minute* of (oddly, under the circumstances) latitude. That is: **1/60** of a degree.

In kilometers, one degree is:

So now we know the maximum, the distance per degree along the equator, which is the same as the distance per degree along any meridian: **69.094** miles or **111.19** kilometers.

**§**

The next step in solving this starts with characterizing the size of a circle of latitude at a given latitude.

Essentially, we want its *radius*. Then we can calculate its circumference and divide that by 360.

The insight we need comes from recognizing what controls the circle’s size.

The latitude, obviously, but we need to *see* the latitude as *the angle* of a line drawn from the center of the Earth to a point on its surface.

At the equator, the angle is **0°**, and at the poles, it is **90°** (technically ±90° for north (+) and south (-)).

Otherwise, the angle is somewhere between **0°** and **±90°**.

Now consider a line drawn from a given latitude *horizontally* to the Earth’s axis (blue line, aka **y**-axis, in the diagram to the right).

The length of this line is the same as the distance as along the **x**-axis (green line) if we draw a vertical line straight down from our latitude point.

The point where the vertical line meets the x-axis is the *cosine* (of the angle) times the radius.

(The point where the horizontal lines meets the **y**-axis is the *sine* times the radius. That number isn’t useful here, and we’re ignoring it. I only mention it in case you wondered where it was.)

The bottom line is that **this distance is the radius of the circle of latitude.** We now have a direct connection between that radius and the angle (latitude).

This gives us the formula for a degree along a line of latitude:

It’s as simple as that.

If we remember that ** cos(0°)** is

**1**, and

**is**

*cos*(90°)**0**, we notice two things:

Firstly, at **0°** (i.e. at the equator), this equation is the same as the one above.

Secondly, at **90°** (at the poles), this equation gives us zero, which is what we’d expect at the poles.

**§**

Lastly, we can plug some figures in to see how it all works out:

degs |
cos(deg) |
mi/deg | km/deg |
---|---|---|---|

0° |
(1.00000) | 69.094 |
111.195 |

5° | (0.99619) | 68.831 | 110.772 |

10° | (0.98481) | 68.044 | 109.506 |

15° | (0.96593) | 66.740 | 107.406 |

20° | (0.93969) | 64.927 | 104.489 |

25° | (0.90631) | 62.621 | 100.777 |

30° | (0.86603) | 59.837 | 96.298 |

35° | (0.81915) | 56.599 | 91.086 |

40° | (0.76604) | 52.929 | 85.180 |

45° | (0.70711) | 48.857 | 78.627 |

50° | (0.64279) | 44.413 | 71.475 |

55° | (0.57358) | 39.631 | 63.779 |

60° | (0.50000) | 34.547 | 55.597 |

65° | (0.42262) | 29.200 | 46.993 |

70° | (0.34202) | 23.632 | 38.031 |

75° | (0.25882) | 17.883 | 28.779 |

80° | (0.17365) | 11.998 | 19.309 |

85° | (0.08716) | 6.022 | 9.691 |

90° |
(0.00000) | 0.000 |
0.000 |

**§**

Something I find a bit amusing about all this is that, although the answer is simple enough to figure out, it requires the dreaded **trigonometry**, which so many thought they’d never use in life.

The truth is, at least in my case, I’ve used “trig” quite a bit over the years!

It crops up all the time in image generation, especially 3D image generation. (Even in 2D work you often want to generate circles or circular things.)

There’s a reason why any decent calculator has * sin* and

*keys! Trig is super useful and pops up in all sorts of places.*

**cos**Now, part of what happens is that, having a tool in the toolkit means being able to whip it out in situations where it’s useful. *Not having it*, one may never realize how it could be used.

So it’s hard to tell if I’ve found trig so useful just because I have it available or because it really does crop up that often. It’s also likely that my interests bring me both to trig, itself, *and* the things it’s useful for.

It’s worth *not being afraid of*; I want to return to that later.

**§**

For whatever it may be worth, my intention is to wrap up March Mathness this coming Sunday (the last day of March) with a discussion of math phobia.

For now I’ll just mention what an odd thing it seems to me that people seem so *pleased* with themselves when they say they don’t do math. It’s almost as if it makes one more human to be bad at math.

I keep imagining people being the same way about reading and writing.

(It also amuses me in the context of AI fans who’ve long exclaimed how much the brain is *“like a computer.”* Ha! As if.)

The thing is, it has mostly to do with the dreadful way math is taught. It’s so bad, that I can’t really *blame* anyone who fears, hates, can’t do, or just plain has no use for, math.

I do feel bad for people missing out on all the fun, beauty, and utility.

The other consideration is that there is a big difference between math that is actually useful and math that is viewed as necessary or even foundational. Sometimes that view is out of date.

For instance, does anyone really need to know long division?

Maybe how companies set prices, or understanding lotto odds, or how populations grow, would be more valuable in understanding our modern world.

That said, there is something I find valuable about the rigor mathematical training gives one. There are other practices that do the same (computer programming can be one of them if approached correctly).

**§**

At this point I think I’ll leave talking about the **Mandelbrot** for another day.

It seems too much of a topic shift, and I have written about it before, so I don’t really have a good sense of what more to say about it. (I figured I’d think of something if I got started.)

I have been enjoying a YouTube channel, **Maths Town**, that specializes in some really (*really, really*) nice Mandelbrot zooms. I’m utterly enthralled by them.

Here’s one of my (latest) favorites (look at these on the *best* monitor you have at the *best* internet speed you have and the *best* resolution you can):

Here’s another (very different *look*, but same Mandelbrot underneath):

What makes them both so strikingly beautiful and so striking *different* is the palette and technique used to render the image. It’s the same old Mandelbrot in all cases.

Here’s one more that’s a little more *Mandelbrot-y* in how it’s rendered:

You’ll find many more, along with variations on the Mandelbrot itself (inversions, larger exponents, distortions) on the **Maths Town** channel.

I’ll try to re-visit the Mandelbrot topic later this week.

**§**

*Have a good latitude, my friends!*

March 25th, 2019 at 6:59 pm

(By the way, this all assumes the Earth is a perfect sphere, which isn’t the case. But close enough for this. There is math for more precision, if it is needed. See the Wiki latitude and longitude pages for details.)

March 26th, 2019 at 1:00 pm

For the record, the Kīlauea volcano has a latitude of just south of

19.5°North, so the answer I was originally looking for is:65.131miles, (104.817km).To show how the distance along latitude lines changes, just a half-degree south, at

19.0°N, it’s65.330mi (105.137km).Half a degree to the north, at

20.0°N, it’s64.927mi (104.489km).There isn’t a huge variation, because Hawaii is so close to the equator. The change becomes more extreme as we move towards the poles.

March 26th, 2019 at 3:30 pm

I used to wonder why astronomy papers always put distances in parsecs instead of light years. Eventually I discovered that scientific papers prefer to put things in the least contingent manner possible. Light years are an estimate. But parsecs are a direct mathematical calculation based on measurements, at least for relatively close stars.

But that direct calculation requires trigonometry. Another class I took where actual applications were rarely mentioned.

(For the same reason, the distance to cosmically distant objects is usually expressed using the z value, the amount of observed redshift, rather than parsecs or light years, both of which would be estimates based on assumptions that might change in the future.)

March 26th, 2019 at 8:27 pm

Yeah, parsecs really put a damper on my theory that it’s all just a backdrop somewhere just past Pluto (the planet). 😀

The Pioneer Anomaly was also a bit of a challenge. (

Obviouslythey were replaced with radio transmitters programmed to shift frequency according to very precise models of the spacecraft. So precise they surprised us, but not our alien zoo-keepers.)As I’ve mentioned, my interests and background made science topics a welcome ground for me. Here’s a story that says a lot: Must have been 5th or 6th grade, on the school bus, I was showing a girl how a weight on a string revealed the bus’ acceleration. She told me to be sure to become a “famous scientist” one day so she could say she knew me. The story shows I was a major geek who chased girls. 😀

(In high school I discovered art and decided I liked it better. Still chased the girls, though. And I obviously never lost interest in science and math. It made computer science in college a fairly easy minor.)

March 27th, 2019 at 7:24 am

Ha! The zoo-keeper’s plan is working if even you are starting to doubt 🙂

At my school, showing too much smarts got you beat up, and laughed at by the girls. Of course, if I had it to do all over again, I wouldn’t hide my smarts and just go with the minority of girls who would have been interested. With hindsight, they were much better catches anyway.

March 27th, 2019 at 9:56 am

Oh, much better! 😀

Looking back, I was surprisingly lucky with my high schooling. I discovered theatre in my sophomore year, so I was associated with the theatre groups, the dance troupe, the school choir, and the teenage fashion show. That really helped mitigate being seen as a geek or nerd. Always was (and am) a geek, but all that arts stuff expanded my horizons.

There is also that this was Los Angeles in the late 1960s, early 1970s. The Watts Riots were in 1965, and racial tensions were much more a distraction than worries about girls laughing at your glasses or whatever. Some days you’d come out of class, and you could feel the tension in the air… you knew something had gone down.

I will say this: Times have changed. The worst thing that

everhappened was a (non-fatal) knifing, and that was only the one time it got that ugly. Lotta scary gatherings, words, yelling, maybe a little throwing, possibly a broken window (and we found that scary enough, thank you).These days… Unbelievable what “scary times in high school” can mean these days.

March 27th, 2019 at 1:13 pm

I do have to admit that my own high school time (in the early 80s) was mild compared to what kids today deal with. I recall a kid pulling a knife on another kid, but not using it, and that was in middle school. (Middle school was much more a battle for survival for me than high school. Just about every fight I ever got into was during middle school.)

In high school, aside from the occasional teen pregnancy, the most common salacious event was someone getting expelled for drugs.

March 27th, 2019 at 2:12 pm

I had to think about that a moment, I didn’t go to a “middle school” — they called it “junior high school” back in my day. (I think the “middle school” label is more modern.) It was the grades 7 and 8.

Yeah, it was harder for me, too, than high school. I mostly tried to stay invisible.

(That was an odd time in my life. I fell in with the only neighborhood boys there were, but they were vandals and thieves, and thus began a somewhat lawless period of my life that ended when I got into some serious trouble — as in possible prison sentence — in college. Realized nothing was worth my freedom. Some wild times though.)

March 27th, 2019 at 2:43 pm

For me, middle school was grades 6,7, and 8. Not fun years. I suspect the main reason high school was better was because the main troublemakers dropped out. (At least the ones who gave me trouble did.)

I knew a mother who actually considered home schooling her daughter just through those grades due to their unique nastiness. (I don’t think she was able to make the economics work though.)

March 27th, 2019 at 3:09 pm

That 5-3-4 pattern seems pretty common now. My ex-wife’s kids did that pattern. I don’t see the 6-2-4 pattern much anymore (not that I’m paying attention, just what I’ve noticed in news articles).

Seems like I’ve heard of a 6-3-3 pattern, which I suppose tries to divide 9th and 10th grade for the same reason 5-3-4 breaks 5th and 6th: a perceived gap in child development that benefits from a new environment.

So,… 5-4-3… anyone? anyone? 🙂