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Beyond Four Corners, USA

Posted 6 years ago
11 Replies
40 Total Likes


I recently saw a TV show set at Four Corners USA, the point where Utah, Colorado, Arizona, and New Mexico meet:

enter image description here

It made me wonder how frequent 4 or more geographical borders meet at one point. According to Wikipedia, 4 borders meeting at a point is called a quadripoint, 5 borders meeting is called a quintipoint, and in general it's called a multipoint. The entry only lists one quintipoint and goes on to say

Perhaps a dozen quintipoints of various levels of geopolitical subdivisions are scattered around the world;

This piqued my interest to find all multipoints, using the Wolfram Language.


Before I give the details on how to detect multipoints, I'd like to showcase the results.


  • Since borders are not always precise (or even well defined at times), I allowed for an error up to ~100 meters when classifying points.
  • The polygons were obtained from the "Country" and "AdministrativeDivision" Entity types (about 40,000 in total).
  • There are a total of 724 quadripoints in this dataset.
  • There are a total of 13 quintipoints in this dataset.
  • There is 1 10-point in this dataset!
  • There are only 6 multipoints in the dataset whose regions don't share the same parent region.


With 724 quadripoints, there are too many to list here, but here are a few interesting ones.

  • The only countries to form a quadripoint are Namibia, Botswana, Zambia, and Zimbabwe.

enter image description here

  • There are a considerable amount of counties in Iowa and Texas that are apart of multiple quadripoints, i.e. more than one corner is a quadripoint. This is because they are roughly arranged in a rectangular grid.

enter image description here

  • There were only 6 quadripoints found whose partent regions differ:

enter image description here

  • Here's a visual summary of all quadripoints found (note the level 3 regions were heavily thickened to become visible):

enter image description here


Here are all 13 quintipoints found:

  • Saint Kitts and Nevis: Saint George Gingerland - Saint James Windward - Saint John Figtree - Saint Paul Charlestown - Saint Thomas Lowland
  • Boyaca, Colombia: Chinavita - Garagoa - Miraflores - Ramiriquí - Zetaquirá
  • Counties in Florida, USA: Glades - Hendry - Martin - Okeechobee - Palm Beach
  • Usulutan, El Salvador: California - Ozatlán - Santa Elena - Tecapán - Usulután
  • Arequipa, Arequipa, Peru: Alto Selva Alegre - Cayma - Chiguata - Miraflores - San Juan de Tarucani
  • Cuenca, Azuay, Ecuador: Chiquintad - Cuenca - Ricaurte - Sidcay - Sinicay
  • Pea Reang, Prey Vêng, Cambodia: Kampong Popil - Mesa Prachan - Prey Sralet - Reab - Roka
  • Rieti, Lazio, Italy: Borgo Velino - Castel Sant' Angelo - Cittaducale - Micigliano - Rieti
  • Cosenza, Calabria, Italy: Marano Marchesato - Marano Principato - Rende - San Fili - San Lucido
  • Napoli, Campania, Italy: Boscotrecase - Ercolano - Ottaviano - Torre Del Greco - Trecase
  • Savona, Liguria, Italy: Bardineto - Boissano - Giustenice - Loano - Pietra Ligure
  • Torino, Piemonte, Italy: Cuceglio - Mercenasco - Montalenghe - Scarmagno - Vialfrè
  • Viterbo, Lazio, Italy: Bolsena - Capodimonte - Gradoli - Montefiascone - San Lorenzo Nuovo

As you can see, Italy takes the cake with 6 quintipoints! Here's a visual of these quintipoints, along with the error allowing them to be classified as such:

enter image description here

A Near 6-point

Notice in the top right map above, it looks like there is room for one more region in Viterbo, Lazio, Italy, which would make it a 6-point. Here's the 6th region (Grotte Di Castro) in black:

enter image description here

It turns out Grotte Di Castro is about 700 meters from the quintipoint, making this only a near 6-point:

enter image description here


As mentioned in the Wikipedia entry, there is a 10-point in Italy at the summit of Mount Etna:

  • Catania, Sicily, Italy: Adrano - Belpasso - Biancavilla - Bronte - Castiglione Di Sicilia - Maletto - Nicolosi - Randazzo - Sant' Alfio - Zafferana Etnea

enter image description here

Allowing for more error

If we allow for more error, we can find near-multipoints - regions that almost have a multipoint, but clearly don't. For example, there is a near-quintipoint in Texas, USA:

enter image description here


The idea to find multipoints within a collection of regions is as follows:

  1. Obtain the Polygon for each region.
  2. For each pair of regions, if there's a vertex from one of the polygons which is "close" to the other, mark these regions as touching. RegionDistance can be used for this.
  3. The relation of touching between pairs forms an adjacency matrix. From this, form a Graph and use FindClique to find all multipoints in this collection.

Here is code that does just that:

discretize[Polygon[pts_?MatrixQ]] := 
    MeshRegion[pts, Polygon[Range[Length[pts]]]]
discretize[Polygon[pts_?(VectorQ[#, MatrixQ] &)]] :=
  With[{pts2 = Select[pts, Length[#] > 2 &]},
    MeshRegion[Join @@ pts2, 
      Polygon[Range[# + 1, #2] & @@@ Partition[Prepend[Accumulate[Map[Length, pts2]], 0], 2, 1]]]
discretize[expr_] :=
  With[{mr = DiscretizeGraphics[Graphics[expr]]},
    mr /; MeshRegionQ[mr]
discretize[_] = $Failed;

polyLookup = discretize /@ (Join[
  EntityValue["Country", "Polygon", "EntityAssociation"],
  EntityValue["AdministrativeDivision", "Polygon", "EntityAssociation"]
] /. GeoPosition -> Identity);

MultiPoints[divs_List, n_] /; Length[divs] < n = {};

MultiPoints[divs_List, n_] :=
    Block[{polys, disj, cands},
       polys = polyLookup /@ divs;
         disj = Boole[Outer[CoordinateNear, polys, polys]] - IdentityMatrix[Length[divs]];
          cands = FindClique[AdjacencyGraph[divs, disj], {n, Infinity}, All];

          resolveMultiPoints[cands, AssociationThread[divs, polys]]

         ) /; MatrixQ[disj, IntegerQ]

       ) /; VectorQ[polys, MeshRegionQ]
MultiPoints[___] = {};

$tol = 0.001;
CoordinateNear[mr1_, mr2_, tol_:$tol] :=
    With[{d = {{-tol, tol}, {-tol, tol}}},
         NoneTrue[Transpose[{d+RegionBounds[mr1], d+RegionBounds[mr2]}], #1[[2,1]] > #1[[1,2]] || #1[[1,1]] > #1[[2,2]]&],
         Min[RegionDistance[mr1, MeshCoordinates[mr2]]] < tol

resolveMultiPoints[{}, _] = {};
resolveMultiPoints[cands_List, passoc_?AssociationQ] :=
    Select[cands, MultiPointQ[#, passoc]&]

MultiPointQ[cands_, passoc_?AssociationQ, tol_:$tol] :=
    Block[{coords, mrs},
       mrs = passoc /@ cands;
       coords = Union @@ MeshCoordinates /@ mrs;

       Or @@ Thread[And @@ (Thread[RegionDistance[#, coords] < tol]& /@ mrs)]

Now here's all multipoints formed from countries:

enter image description here

Now to explore all cases, we can start off by looking for multipoints in all subdivisions of a given region, e.g. given Florida, find all multipoints within the counties of Florida. This can be achieved by building a hierarchical graph connecting countries and administrative divisions. Then for a given region, this graph can be used to find all subdivisions and the above code can be used to find the multipoints.

ad = AdministrativeDivisionData[];
pr = EntityValue["AdministrativeDivision", "ParentRegion"];

$ADNetwork = Graph[Join[
    Thread["NullPointer" -> EntityList["Country"]],
    DeleteCases[Thread[pr -> ad], Rule[_Missing, _]]

ChildrenMultiPoints[reg_] := ChildrenMultiPoints[reg, 4]

ChildrenMultiPoints[reg_, o___] :=
    MultiPoints[Rest[VertexOutComponent[$ADNetwork, reg, 1]], o]

enter image description here

Lastly, to cover all cases we need to consider sets of regions that have differing parent regions. To do this, for a given parent region $P$, first find all other regions $R_i$ (on the same level) that touch this region. Then simply run MultiPoints on all subdivisions in $P \cup R_i$. I omit this code here, as there were only 6 instances that came out of this case.

POSTED BY: Greg Hurst
11 Replies

enter image description here - another post of yours has been selected for the Staff Picks group, congratulations !

We are happy to see you at the top of the "Featured Contributor" board. Thank you for your wonderful contributions, and please keep them coming!

POSTED BY: Moderation Team

Just logged in to give it a thumbs up! Very nice! Great work!

POSTED BY: Sander Huisman
Posted 6 years ago


POSTED BY: Greg Hurst
Posted 6 years ago

I visited the four corners area of the USA a few months ago. It isn't close to anything, so you really have to want to see it to make the trip. Couples who go usually have their picture taken kissing at the four corner point. Native Americans sell their wares along the circumference of a large circle surrounding the four corners.

This is a great post and I really enjoyed it very much. Thanks for a terrific job!!!

POSTED BY: John Snyder

Wonderful! First class job! One of the best posts I've seen here!

POSTED BY: Rand Baldwin
Posted 6 years ago

Interesting and fun topic. Well explained and worked out. Thank you.

POSTED BY: Dave Middleton

Very nice post, with interesting results.

I've always loved finding geographic oddities. This was fun to read, thanks Chip.

POSTED BY: Douglas Smith

I just run into this post again and again was amazed how cool this is, thank you! I am trying to reproduce the 10-point map in Italy, but I do not think I understand how to get geo-polygons for them. Any advice @Greg Hurst ?

POSTED BY: Vitaliy Kaurov
Posted 9 months ago

I think the issue is that ChildrenMultiPoints needed a second argument. I edited the post to give a default value of 4. The following now returns a 10-point and a quadripoint:

ChildrenMultiPoints[Entity["AdministrativeDivision", {"Catania", "Sicily", "Italy"}]]

enter image description here

Does this work for you now?

POSTED BY: Greg Hurst

Terrific contribution!

I have an interest in multipoints since there is one near where I live where six borders meet; if we discard one of them (which is a mere boundary of a property), it is definitely a quintipoint, although in administrative terms the units (parishes and precinct) are no longer officially in use. This brings up two important points if we want to assess the incidence of multipoints: (i) administrative borders change, so we should not forget to consider older maps; (ii) the lowest level of administrative units to consider must be property borders. It would, for example, be interesting to run this for cadastral maps. Even if we ignore property borders, administrative subdivision of land varies between countries, which affects the incidence of multipoints.

An unrelated feature is the geographic nature of multipoint: it is of course not a coincidence that many are located on mountain tops. At least at a lower administrative level, lakes are obvious places for many borders to join (quadripoint in lakes are common on Swedish cadastral maps). From the land-surveyors perspective, prominent points in the terrain were preferentially chosen, which is why islands in lakes and mountains were often used where several properties met. When I check cadastral maps of Lake Sommen (county of Östergötland), there are 2 quintipoints and 4 quadripoints on islands. There is also 1 quintipoint in the middle of a narrow strait. For comparison, there are 3 quintipoints and 6 quadripoints located in water. In total: 6 quintipoints and 10 quadripoints in a single lake.

POSTED BY: Per Milberg
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