The Ultimate Map Projection Showdown of Ultimate Destiny

I thought I'd do some polls of projections with similar purposes, so I'm starting with a poll comparing projections that aim to show the entire world without interrupting any land.

From left-to-right then top-to-bottom we have the CALM, Lee-Concialdi, Grieger Triptychial, Canters W23, Danseiji IV, and Dymaxion projections. Three conformal projections followed by three compromise projections

The CALM (Conformal Authagraph-Like Map) and the Lee-Concialdi projections on the first row, are both rearrangements of the Lee Tetrahedral projection with different centres to move the distortion to different places depending on which you prefer, I really like the half-hexagon layout of the Lee-Concialdi. The CALM is designed to have the same layout as the Authagraph projection, which I can't include here as they haven't published the equations so no one actually knows how to make it. Both of these are conformal.

The Grieger Triptychial is a rearrangement of the Peirce Quincuncial. It and the next two have very similar layouts, with the Grieger being conformal and the Canters and Danseiji being compromise projections. The other differences between these are mainly in where the distortions are, for example the Galapagos are massive on the Grieger but Antarctica has a weird shape on the Canters.

Last we have the Dymaxion, probably the most famous of these, based off the net of an icosahedron it has a pretty complex layout but a good balance of distortions for a compromise projection.

WATERMAN BUTTERFLY vs STEREOGRAPHIC

Waterman Butterfly Polyhedral Compromise Round 1: [Waterman Butterfly vs Ortelius Oval] Round 2: [Waterman Butterfly vs Cahill Conformal Butterfly] Round 3: [Waterman Butterfly vs Winkel-Tripel] Semi-Final: [Waterman Butterfly vs Peirce Quincuncial]

Stereographic Azimuthal Conformal Round 1: [Stereographic vs Eckert IV] Round 2: [Stereographic vs Azimuthal Equidistant] Round 3: [Stereographic vs Spilhaus-Adams] Semi-Final: [Stereographic vs Dymaxion]

Honestly two projections that I didn't think would get this far, I expected the final to be either the Peirce Quincuncial or Cahill Butterfly against one of the Lee variants! I find it interesting that both these projections are centred on 20°W rather than the prime meridian like most maps. Since we've already seen these projections for the four previous rounds I won't go into masses of detail here.

Waterman's Butterfly projection, invented in 1996 and inspired by Cahill's 1909 butterfly, is constructed by projecting the surface of the Earth onto a truncated octahedron and then unwrapping it. This results in a compromise projection with very low shape distortion and size distortion. It is also commonly shown with Antarctica placed in it's own little circle at the bottom.

In use since the Ancient Egyptians, each hemisphere of the stereographic projection can be constructed by projecting from a point on the surface of the Earth onto a plane tangent to the point on the globe opposite it like this. This results in a conformal projection that is the only one to represent all small circles as circles rather than ovals.

[Direct comparison on map-projections.net]

PEIRCE QUINCUNCIAL vs DYMAXION

Peirce Quincuncial Polyhedral Conformal Round 1: [Peirce Quincuncial vs Van der Grinten] Round 2: [Peirce Quincuncial vs Natural Earth II] Round 3: [Peirce Quincuncial vs Cahill-Keyes] Semi-Final: [Waterman Butterfly vs Peirce Quincuncial]

Dymaxion Polyhedral Compromise Round 1: [Dymaxion vs Stab-Werner] Round 2: [Dymaxion vs Lee Conformal Tetrahedron] Round 3: [Dymaxion vs Equidistant Conic] Semi-Final: [Stereographic vs Dymaxion]

And here we are with the boring loser poll, for losers. These projections got so far, but fell at the last hurdle. In this 3rd place playoff, we give them one last chance to redeem themselves.

Of the interesting polar aspect projections in this tournament, this is probably this simplest shaped one vs the most unique shaped one.

The square design of the Peirce Quincuncial can be seen as a trade-off for increased ease-of-use but with higher distortions in the oceans. While the Dymaxion's more complex icosahedral net reduces area distortion and allows for no land interruptions, but loses conformality and is perhaps more confusing to use.

[Direct comparison on map-projections.net]

Dymaxion Polyhedral Compromise Round 1: [Dymaxion vs Stab-Werner] Round 2: [Dymaxion vs Lee Conformal Tetrahedron] Round 3: [Dymaxion vs Equidistant Conic]

Stereographic Azimuthal Conformal Round 1: [Stereographic vs Eckert IV] Round 2: [Stereographic vs Azimuthal Equidistant] Round 3: [Stereographic vs Spilhaus-Adams]

The second semi-finals sees Buckminster Fuller's futurist map face off against the oldest projection in this tournament, in use since the Ancient Egyptians.

The Dymaxion projection was presented as a new way of looking at the world. Without the constraints of north and south it uses the net of an icosahedron as a basis to show all the continents without interruptions and how they're laid out compared to each other (the Peirce Quincuncial gets close but still splits Antarctica), because of this it's often used for projections that show human migration from Africa across the continents, and because of its accurate shapes and low size distortion it has become a popular choice for people's "favourite projection".

The Stereographic on the other hand has a very simple construction (projecting from a point on the surface to a plane opposite it like this for each hemisphere), and as such has found many uses over the past 2000+ years. Originally used for star charts, it was commonly used for world maps in the 16th and 17th centuries. As it is the only projection that shows all small circles on the globe as circles on the map, it is also often used for mapping circular features such as craters on the Moon, so they show as true circles rather than ovals.

[Direct comparison on map-projections.net]

Waterman Butterfly Polyhedral Compromise Round 1: [Waterman Butterfly vs Ortelius Oval] Round 2: [Waterman Butterfly vs Cahill Conformal Butterfly] Round 3: [Waterman Butterfly vs Winkel-Tripel]

Peirce Quincuncial Polyhedral Conformal Round 1: [Peirce Quincuncial vs Van der Grinten] Round 2: [Peirce Quincuncial vs Natural Earth II] Round 3: [Peirce Quincuncial vs Cahill-Keyes]

This first semi-final compares two projections with similar aims and a similar construction, will the Peirce Quincuncial repeat its victory over the Cahill-Keyes or will the popular butterfly layout propel Waterman's projection to the final!

Both these projections split the globe into octants, but go about flattening the octants slightly differently. Peirce represented each octant with a right-angle triangle, which makes the map into a simpler layout with less interruptions, but causes four points around the equator (the midpoints of the sides) to have large area distortion, though these points are all in the ocean.

Waterman on the other hand followed Cahill's lead and used equilateral triangles with the corners squashed a bit, while this does mean it isn't conformal and is more complex, it has slightly less size distortion. Waterman's projection is also often presented with Antarctica off in its own circle underneath the rest of the map.

[Direct comparison on map-projections.net]

Dymaxion Polyhedral Compromise Round 1: [Dymaxion vs Stab-Werner] Round 2: [Dymaxion vs Lee Conformal Tetrahedron] Round 3: [Dymaxion vs Equidistant Conic]

Stereographic Azimuthal Conformal Round 1: [Stereographic vs Eckert IV] Round 2: [Stereographic vs Azimuthal Equidistant] Round 3: [Stereographic vs Spilhaus-Adams]

The second semi-finals sees Buckminster Fuller's futurist map face off against the oldest projection in this tournament, in use since the Ancient Egyptians.

The Dymaxion projection was presented as a new way of looking at the world. Without the constraints of north and south it uses the net of an icosahedron as a basis to show all the continents without interruptions and how they're laid out compared to each other (the Peirce Quincuncial gets close but still splits Antarctica), because of this it's often used for projections that show human migration from Africa across the continents, and because of its accurate shapes and low size distortion it has become a popular choice for people's "favourite projection".

The Stereographic on the other hand has a very simple construction (projecting from a point on the surface to a plane opposite it like this for each hemisphere), and as such has found many uses over the past 2000+ years. Originally used for star charts, it was commonly used for world maps in the 16th and 17th centuries. As it is the only projection that shows all small circles on the globe as circles on the map, it is also often used for mapping circular features such as craters on the Moon, so they show as true circles rather than ovals.

[Direct comparison on map-projections.net]

Waterman Butterfly Polyhedral Compromise Round 1: [Waterman Butterfly vs Ortelius Oval] Round 2: [Waterman Butterfly vs Cahill Conformal Butterfly] Round 3: [Waterman Butterfly vs Winkel-Tripel]

Peirce Quincuncial Polyhedral Conformal Round 1: [Peirce Quincuncial vs Van der Grinten] Round 2: [Peirce Quincuncial vs Natural Earth II] Round 3: [Peirce Quincuncial vs Cahill-Keyes]

This first semi-final compares two projections with similar aims and a similar construction, will the Peirce Quincuncial repeat its victory over the Cahill-Keyes or will the popular butterfly layout propel Waterman's projection to the final!

Both these projections split the globe into octants, but go about flattening the octants slightly differently. Peirce represented each octant with a right-angle triangle, which makes the map into a simpler layout with less interruptions, but causes four points around the equator (the midpoints of the sides) to have large area distortion, though these points are all in the ocean.

Waterman on the other hand followed Cahill's lead and used equilateral triangles with the corners squashed a bit, while this does mean it isn't conformal and is more complex, it has slightly less size distortion. Waterman's projection is also often presented with Antarctica off in its own circle underneath the rest of the map.

[Direct comparison on map-projections.net]