Mercator and polar projections

This post is a more quantitative version of the previous post. Before I said that straight lines on a Mercator projection map correspond to loxodrome spirals on a sphere. This post will make that claim more explicit.

So suppose we plot a straight path from Quito to Jerusalem on a Mercator projection.

Quito to Jerusalem on a Mercator projection

The red dot in the lower left corner represents Quito and the next red dot represents Jerusalem.

Mercator projection leaves longitude λ unchanged, but latitude φ is transformed via

φ ↦ log( sec φ + tan φ )

for reasons explained here. We can apply the inverse of the Mercator projection to put the path above on a globe, and when we do, it looks like the following.

Loxodrome path from Quito to Jerusalem on a polar plot

The path planned on a Mercator projection map when projected onto the globe becomes a logarithmic spiral in polar projection. The radial direction in the plot above shows the angle down from the North Pole rather than the angle up from the equator.

So if our flight of constant bearing keeps going rather than stopping at Jerusalem, it will spiral quickly toward the North Pole. It appears to stop at pole unless you look carefully. In theory the spiral keeps going and never actually reaches the pole. This is easy to see on the Mercator map because the North Pole is infinitely far away on the vertical axis.

Straight on a map or straight on a globe?

Straight lines on a globe are not straight on a map, and straight lines on a map are not straight on a globe.

A straight line on a globe is an arc of a great circle, the shortest path between two points. When projected onto a map, a straight path looks curved. Here’s an image I made for a post back in August.

Quito to Nairobi to Jerusalem and back to Quito

The red lines form a spherical triangle with vertices at Quito, Nairobi, and Jerusalem. The leg from Quito to Nairobi is straight because it follows the equator. And the leg from Nairobi to Jerusalem is straight because it follows a meridian. But the leg from Quito to Jerusalem looks wrong.

If you were flying from Quito to Jerusalem and saw this flight plan, you might ask “Why aren’t we flying straight there, cutting across Africa rather than making a big arc around it? Are we trying to avoid flying over the Sahara?”

But the path from Quito to Jerusalem is straight, on a globe. It’s just not straight on the map. The map is not the territory.

Now let’s look at things from the opposite direction. What do straight lines on a map look like on a globe? By map I mean a Mercator projection. You could take a map and draw a straight line from Quito to Jerusalem, and it would cross every meridian at the same angle. A pilot could fly from Quito to Jerusalem along such a path without ever changing bearing. But the plane would have to turn continuously to stay on such a bearing, because this is not a straight line.

A straight line on a Mercator projection is a spiral on a globe, known as a loxodrome or a rhumb line. If a plane flew on a constant bearing from Quito but few over Jerusalem and kept going, it would spiral toward the North Pole. It would keep circling the earth, crossing the meridian through Jerusalem over and over, each time at a higher latitude. On a polar projection map, the plane’s course would be approximately a logarithmic spiral. The next post goes into this in more detail.

loxodrome spiral

I made the image above using the Mathematica code found here.

Although straight lines the globe are surprising on a map, straight lines on a map are even more surprising on a globe.

Related posts

Quirks in Mathematica’s administrative division data for Mexico

If you ask Mathematica for a list of Mexican states via

    CountryData["Mexico", "RegionNames"]

you will get a list of strings:

    "Aguascalientes", "Baja California", ..., "Zacatecas"}

However, when you try to turn this into a list of objects representing these states via

    states = Entity["AdministrativeDivision", {#, "Mexico"}] & /@ 
                 CountryData["Mexico", "RegionNames"]

something strange happens. Some items in the list are turned into useful objects, and some are uninterpreted symbols.

For example, Aguascalientes is recognized as an administrative division, but Baja California is not. It recognizes Oaxaca but not Nuevo Leon. The pattern is that states with a space are not recognized. There is an inconsistency in Mathematica: output names do not always match input names. To create the object representing Baja California, you need to pass in the string BajaCalifornia with no space.

    Entity["AdministrativeDivision", {"BajaCalifornia", "Mexico"}]

OK, so let’s remove spaces before we try to create a list of geographic objects.

    names = StringReplace[#, " " -> ""] & /@ 
               CountryData["Mexico", "RegionNames"]

This mostly works, but it trips up on Mexico City. The output name for the region is Ciudad de México, but Mathematica does not recognize CiudaddeMéxico as an administrative division. Mathematica does recognize MexicoCity as the name of a city but not as the name of an administrative division.

Changing CiudaddeMéxico to MexicoCity in the list of names did not fix the problem. But when I directly edited the list of state objects by replacing the uninterpreted value with the output running

    Entity["AdministrativeDivision", {"MexicoCity", "Mexico"}]

by itself everything worked. Then I was able to find a Traveling Salesman tour as in earlier posts (Africa, Americas, Eurasia and Oceania, Canada).

Traveling Salesman tour of Mexico

The tour is

  1. Baja California
  2. Baja California Sur
  3. Sinaloa
  4. Durango
  5. Zacatecas
  6. Aguascalientes
  7. Nayarit
  8. Jalisco
  9. Colima
  10. Michoacán
  11. México
  12. Mexico City
  13. Morelos
  14. Guerrero
  15. Oaxaca
  16. Chiapas
  17. Tabasco
  18. Campeche
  19. Quintana Roo
  20. Yucatán
  21. Veracruz
  22. Puebla
  23. Tlaxcala
  24. Hidalgo
  25. Querétaro
  26. Guanajuato
  27. San Luis Potosí
  28. Tamaulipas
  29. Nuevo León
  30. Coahuila
  31. Chihuahua
  32. Sonora

The tour is 8,343 kilometers.

A traveling salesman tour of Canada

Here is a Traveling Salesman tour of Canada’s provinces and territories created by Mathematica. This is the shortest path connecting the geographic centers of the regions.

Here is a much larger (4.5 MB) PDF file of the same map with higher resolution.

Starting in the northwest, the tour is

  1. Yukon
  2. Northwest Territories
  3. Nunavut
  4. Quebec
  5. Newfoundland and Labrador
  6. Prince Edward Island
  7. Nova Scotia
  8. New Brunswick
  9. Ontario
  10. Manitoba
  11. Saskatchewan
  12. Alberta
  13. British Columbia

The tour is 11,070 km.

For more tours like this, see my earlier posts on tours of

Update: Here is an adjacency network for Canadian provinces and territories.

This is an SVG image so you can scale it to make it easier to read if you’d like.

More adjacency graph posts:

Two-letter vs Three-letter Country Abbreviations

The ISO 3166-1 standard defines three codes for each country: a 2-letter abbreviation, a 3-letter abbreviation, and a 3-digit code.

The 2-letter abbreviations may be familiar because it is very often (but not always [1]) also the country code top-level domain (ccTLD). For example, AU is the ISO abbreviation for Australia, and .au is the ccTLD.

I was curious about the relation between the two-letter and three-letter abbreviations. Sometimes the former is a prefix of the latter, such as US and USA. Sometimes the latter adds a letter in the middle, such as going from CN to CHN. How common are these two patterns?

I wrote a script using the iso3166 Python module to find out.

Turns out that the prefix pattern is most common, and occurs 63% of the time.

The infix pattern is the next most common, occurring 29% of the time.

Suffixes are rare. There are only for instances where the 2-letter name is the last two letters of the 3-letter name: ATF, MYT, SPM, and SGS.

The remaining possibilities are miscellaneous relations, such as IL and ISR for Israel.

Here’s a table of countries and ISO codes in plain text (org-mode) format.

[1] There are four ccTLDs that are not ISO 3166-1 alpha-2 names: uk, su, ac, and eu.

Finding similar world flags with Mathematica

A week ago I posted some pairs of similar flags on Twitter, and later I found that Mathematica’s CountryData database contains flag descriptions. So I thought I’d use the flag descriptions to see which flags Mathematica things are similar.

For example, the FlagDescription attribute for Chad in Mathematica is

Three equal vertical bands of blue (hoist side), yellow, and red; similar to the flag of Romania; also similar to the flags of Andorra and Moldova, both of which have a national coat of arms centered in the yellow band; design was based on the flag of France.

I had Mathematica output a list of countries and flag descriptions, then searched the output for the word “similar.” I then made the following groupings based on the output [1].

Chad / Romania


Bolivia / Ghana

Bolivia   Ghana

Colombia / Ecuador

Equador   Columbia

India / Niger


Ireland / Côte d’Ivoire

Ireland   Ivory Coast

El Salvador / Nicaragua / Honduras

El Salvador    

Egypt / Iraq / Syria / Yemen



Luxembourg / The Netherlands


Andorra / Moldova

Indonesia / Monaco

Indonesia   Monaco


Each flag has an emoji, so here are the groupings above using emoji icons

  • 🇹🇩 🇷🇴
  • 🇧🇴 🇬🇭
  • 🇨🇴 🇪🇨
  • 🇮🇳 🇳🇪
  • 🇮🇪 🇨🇮
  • 🇸🇻 🇳🇮 🇭🇳
  • 🇪🇬 🇮🇶 🇸🇾 🇾🇪
  • 🇱🇺 🇳🇱
  • 🇦🇩 🇲🇩
  • 🇮🇩 🇲🇨

Related posts

[1] The groupings are based on Mathematica’s output, but I did some editing. Strictly following Mathematica’s descriptions would have been complicated. For example, Mathematica’s description might say A is similar to B, but not say B is similar to A. Or it might cluster four flags together that could better be split into two pairs.

Graphing Japanese Prefectures

The two previous posts looked at adjacency networks. The first used examples of US states and Texas counties. The second post made suggestions for using these networks in a classroom. This post is a continuation of the previous post using examples from Japan.

Japan is divided into 8 regions and 47 prefectures. Here is a network diagram of the prefectures in the Kanto region showing which regions border each other. (In this post, “border” will be regions share a large enough border that I was able to see the border region on the map I was using. Some regions may share a very small border that I left out.)

This is a good example of why it is convenient in GraphViz to use variable names that are different from labels. I created my graphs using English versions of prefecture names, and checked my work using the English names. Then after debugging my work I changed the label names (but not the connectivity data) to use Japanese names.

To show what this looks like, my GraphViz started out like this

    graph G {
    AI [label="Aichi"]
    AK [label="Akita"]
    AO [label="Aomori"]
    AO -- AK
    AO -- IW
    AK -- IW

and ended up like this

    graph G {
    AI [label="愛知県"]
    AK [label="秋田県"]
    AO [label="青森県"]
    AO -- AK
    AO -- IW
    AK -- IW

Here’s a graph only showing which prefectures border each other within a region.

This image is an SVG, so you can rescale it without losing any resolution. Here’s the same image as a PDF.

Because this network is effectively several small networks, it would be easy to look at a map and figure out which nodes correspond to which prefectures. (It would be even easier if you could read the labels!)

Note that there are two islands—literal islands, as well as figurative islands in the image above—Hokkaido, which is its own region, and Okinawa, which a prefecture in the Kyushu region.

Here’s the graph with all bordering relations, including across regions.

The image above is also an SVG. And here’s the same image as a PDF.

Classroom exercise with networks

In the previous post I looked at graphs created from representing geographic regions with nodes and connecting nodes with edges if the corresponding regions share a border.

It’s an interesting exercise to recover the geographic regions from the network. For example, take a look at the graph for the continental United States.

It’s easy to identify Alaska in the graph. The node on the left represents Maine because Maine is the only state to border exactly one other state. From there you can bootstrap your way to identifying the rest of the states.

Math class

This could make a fun classroom exercise in a math class. Students will naturally come up with the idea of the degree of a node, the number of edges that meet that node, because that’s a handy way to solve the puzzle: the only possibilities for a node of degree n are states that border n other states.

This also illustrates that networks preserve topology, not geometry. That is, the connectivity information is retained, but the shape is dramatically different.

Geography class

Someone asked me on Twitter to make a corresponding graph for Brazil. Mathematica, or at least my version of Mathematica, doesn’t have data on Brazilian states, so I made an adjacency graph using GraphViz.

adjacency graph of Brazilian states

Labeling the blank nodes is much easier for Brazil than for the US because Brazil has about half as many states, and the topology of the graph gives you more to work with. Three nodes connect to only one other node, for example.

Here the exercise doesn’t involve as much logic, but the geography is less familiar, unless of course you’re more familiar with Brazil than the US. Labeling the graph will require staring at a map of Brazil and you might accidentally learn a little about Brazil.


The labeled version of the graph above is available here. And here are the GraphViz source files that make the labeled and unlabeled versions.

The layout of a GraphViz file is very simple. The file looks like this:

    graph G {


        AC [label="Acre"]
        AL [label="Alagoas"]
        AC -- AM
        AC -- RO

There are three parts: a layout, node labels, and connections.

GraphViz has several layout engines, and the sfdp one matched what I was looking for in this case. Other layout options lead to overlapping edges that were confusing.

The node names AC, AL, etc. do not appear in the output. They’re just variable names for your convenience. The text inside the label is what appears in the final output. I’ll give an example in the next post in which it’s very convenient for the variables to be different from the labels. The order of the labels doesn’t matter, only which variables are associated with which labels.

Finally, the lines with variables separated by dashes are the connection data. Here we’re telling GraphViz to connect node AC to nodes AM and RO. The order of these lines doesn’t matter.

Related posts

Adjacency networks

Suppose you want to color a map with no two bordering regions having the same color. If this is a map on a plane, you can do this using only four colors, but maybe you’d like to use more.

You can reduce the problem to coloring the nodes in a graph. Each node corresponds to a region, and there is an edge between two nodes if and only if their corresponding regions share a border.

Here is a sort of topologists’s or graph theorist’s view of the continental United States.

This was created using the following sample code from the Mathematica documentation.

    RelationGraph[MemberQ[#2["BorderingStates"], #1] &, 
            EntityClass["AdministrativeDivision", "ContinentalUSStates"]]]

You can recognize Maine in the graph because it’s the only state that only borders one other state. Alaska is also easy to locate. Exercise for the reader: mentally add Hawaii to the graph.

The analogous graph for Texas counties took much longer to draw: there are 49 continental US states but 254 Texas counties.

This was created with the following code.

    RelationGraph[MemberQ[#2["BorderingCounties"], #1] &, 
        EntityList[EntityClass["AdministrativeDivision", "USCountiesTexas"]]]

You can find El Paso county in the top left; it only borders one county just as Maine only borders one state.

Related posts

Shortest tours of Eurasia and Oceania

This is the final post in a series of three posts about shortest tours, solutions to the so-called traveling salesmen problem.

The first was a tour of Africa. Actually two tours, one for the continent and one for islands. See this post for the Mathematica code used to create the tours.

The second was about the Americas: one tour for the North American continent, one for islands, and one for South America.

This post will look at Eurasia and Oceania. As before, I limit the tours to sovereign states, though there are disputes over which regions are independent nations. I first tried to do separate tours of Europe and Asia, but this would require arbitrarily categorizing some countries as European or Asian. The distinction between Asia and Oceania is a little fuzzy too, but not as complicated.


Here’s a map of the tour of Oceania.

Here’s the order of the tour:

  1. Australia
  2. East Timor
  3. Indonesia
  4. Palau
  5. Papua New Guinea
  6. Micronesia
  7. Marshall Islands
  8. Nauru
  9. Solomon Islands
  10. Vanuatu
  11. Fiji
  12. Tuvalu
  13. Kiribati
  14. Samoa
  15. Tonga
  16. New Zealand

The total length of the tour is 28,528 kilometers or 17,727 miles.


Here’s a map of the the Eurasian tour.

Here’s the order of the tour:

  1. Iceland
  2. Norway
  3. Sweden
  4. Finland
  5. Estonia
  6. Latvia
  7. Lithuania
  8. Belarus
  9. Poland
  10. Czech Republic
  11. Slovakia
  12. Hungary
  13. Romania
  14. Moldova
  15. Ukraine
  16. Georgia
  17. Armenia
  18. Azerbaijan
  19. Turkmenistan
  20. Uzbekistan
  21. Afghanistan
  22. Pakistan
  23. Tajikistan
  24. Kyrgyzstan
  25. Kazakhstan
  26. Russia
  27. Mongolia
  28. China
  29. North Korea
  30. South Korea
  31. Japan
  32. Taiwan
  33. Philippines
  34. East Timor
  35. Indonesia
  36. Brunei
  37. Malaysia
  38. Singapore
  39. Cambodia
  40. Vietnam
  41. Laos
  42. Thailand
  43. Myanmar
  44. Bangladesh
  45. Bhutan
  46. Nepal
  47. India
  48. Sri Lanka
  49. Maldives
  50. Yemen
  51. Oman
  52. United Arab Emirates
  53. Qatar
  54. Bahrain
  55. Saudi Arabia
  56. Kuwait
  57. Iran
  58. Iraq
  59. Syria
  60. Lebanon
  61. Jordan
  62. Israel
  63. Cyprus
  64. Turkey
  65. Bulgaria
  66. North Macedonia
  67. Serbia
  68. Bosnia and Herzegovina
  69. Montenegro
  70. Albania
  71. Greece
  72. Malta
  73. Italy
  74. San Marino
  75. Croatia
  76. Slovenia
  77. Austria
  78. Liechtenstein
  79. Switzerland
  80. Monaco
  81. Andorra
  82. Spain
  83. Portugal
  84. France
  85. Belgium
  86. Luxembourg
  87. Germany
  88. Netherlands
  89. Denmark
  90. United Kingdom
  91. Algeria

The total length of the tour is 61,783 kilometers or 38,390 miles.