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Is there a mathematical reason why, on the US map, only three types of lines are needed to generate this map- vertical, horizontal and one diagonal? Does that

Is there a mathematical reason why, on the US map, only three types of lines are needed to generate this map- vertical, horizontal and one diagonal? Does that notion generalize to all border maps and if so why? Is it related how border maps can all be colored with only four colors?

trynewideas 4y ago
> Is there a mathematical reason why No. Here's another topological map of the contiguous US that's all rectangles, arranged in a triangle: https://www.reddit.com/r/MapPorn/comments/gy2jq6/the_topologists_map_of_the_contiguous_united/ https://www.reddit.com/r/MapPorn/comments/gy2jq6/the_topolog... I'm not sure how much I'd overthink this map, since it appears to have been created as a puzzle for children to help them understand the concept of topology: https://www.1001mathproblems.com/2015_11_01_archive.html https://www.1001mathproblems.com/2015_11_01_archive.html > Does that notion generalize to all border maps and if so why? No. > Is it related how border maps can all be colored with only four colors? If you're talking about the four color map theorem? Instinctively, my answer's no, but it might be a fun thing to prove.
- madcaptenor 4y ago
  For more maps like this, there are a bunch of quizzes on Sporcle that ask you to label the nodes of a graph, where the nodes are states/provinces/countries and the edges are borders: https://www.sporcle.com/playlists/Alcas/graph-y-geography https://www.sporcle.com/playlists/Alcas/graph-y-geography
Someone 4y ago
I would think horizontal and vertical line segments (or, by extension, straight line segments with two different orientations; take the map with only horizontal and vertical line segments and apply a shear mapping (https://en.wikipedia.org/wiki/Shear_mapping https://en.wikipedia.org/wiki/Shear_mapping) and a rotation) are sufficient. That’s basically what this is, if you draw it on a pixel grid.
klyrs 4y ago
> Is it related how border maps can all be colored with only four colors? Actually, that's not generally true. Michigan, for example, violates the axioms behind the four color theorem -- if regions are allowed to have multiple disconnected pieces, a map could theoretically require arbitrarily many colors. For a more pathological example, consider that US embassies are considered "US soil" in an island of their host nation. I'm rather curious about the chromatic number (that is, the smallest number of colors required) of the world map under that consideration.
- joshuaissac 4y ago
  Embassies are not considered to be foreign territory. The host country gives special privileges to the guest country in accordance with the Vienna Convention. But there can be an enclave of one country inside another (first-order enclave), as well as enclaves inside enclaves. Dahala Khagrabari was a third-order enclave, an enclave of India inside an enclave of Bangladesh inside an enclave of India in mainland Bangladesh. It was ceded to Bangladesh seven years ago as part of a border gore cleanup treaty. https://en.wikipedia.org/wiki/Dahala_Khagrabari https://en.wikipedia.org/wiki/Dahala_Khagrabari
  - jfengel 4y ago
    Don't ruin my joke. I used to date somebody who lived across from the Kenyan embassy in DC. The embassy had a few designated parking spaces, but only until 6:30 PM -- about the time I might arrive for a date. I liked to think that I was parking in Kenyan territory. I was not, for numerous reasons, but it made me happy anyway. Just be out by 6:30 AM. (It didn't become Kenya again until 9, but it was a travel lane during rush hour. That's why the spot was always open: the signs were so confusing that nobody wanted to park there first.)

trynewideas 4y ago

> Is there a mathematical reason why No. Here's another topological map of the contiguous US that's all rectangles, arranged in a triangle: https://www.reddit.com/r/MapPorn/comments/gy2jq6/the_topologists_map_of_the_contiguous_united/ https://www.reddit.com/r/MapPorn/comments/gy2jq6/the_topolog... I'm not sure how much I'd overthink this map, since it appears to have been created as a puzzle for children to help them understand the concept of topology: https://www.1001mathproblems.com/2015_11_01_archive.html https://www.1001mathproblems.com/2015_11_01_archive.html > Does that notion generalize to all border maps and if so why? No. > Is it related how border maps can all be colored with only four colors? If you're talking about the four color map theorem? Instinctively, my answer's no, but it might be a fun thing to prove.

madcaptenor 4y ago
For more maps like this, there are a bunch of quizzes on Sporcle that ask you to label the nodes of a graph, where the nodes are states/provinces/countries and the edges are borders: https://www.sporcle.com/playlists/Alcas/graph-y-geography https://www.sporcle.com/playlists/Alcas/graph-y-geography

Someone 4y ago

I would think horizontal and vertical line segments (or, by extension, straight line segments with two different orientations; take the map with only horizontal and vertical line segments and apply a shear mapping (https://en.wikipedia.org/wiki/Shear_mapping https://en.wikipedia.org/wiki/Shear_mapping) and a rotation) are sufficient. That’s basically what this is, if you draw it on a pixel grid.

klyrs 4y ago

> Is it related how border maps can all be colored with only four colors? Actually, that's not generally true. Michigan, for example, violates the axioms behind the four color theorem -- if regions are allowed to have multiple disconnected pieces, a map could theoretically require arbitrarily many colors. For a more pathological example, consider that US embassies are considered "US soil" in an island of their host nation. I'm rather curious about the chromatic number (that is, the smallest number of colors required) of the world map under that consideration.

joshuaissac 4y ago
Embassies are not considered to be foreign territory. The host country gives special privileges to the guest country in accordance with the Vienna Convention. But there can be an enclave of one country inside another (first-order enclave), as well as enclaves inside enclaves. Dahala Khagrabari was a third-order enclave, an enclave of India inside an enclave of Bangladesh inside an enclave of India in mainland Bangladesh. It was ceded to Bangladesh seven years ago as part of a border gore cleanup treaty. https://en.wikipedia.org/wiki/Dahala_Khagrabari https://en.wikipedia.org/wiki/Dahala_Khagrabari
- jfengel 4y ago
  Don't ruin my joke. I used to date somebody who lived across from the Kenyan embassy in DC. The embassy had a few designated parking spaces, but only until 6:30 PM -- about the time I might arrive for a date. I liked to think that I was parking in Kenyan territory. I was not, for numerous reasons, but it made me happy anyway. Just be out by 6:30 AM. (It didn't become Kenya again until 9, but it was a travel lane during rush hour. That's why the spot was always open: the signs were so confusing that nobody wanted to park there first.)