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I need a good quality rasterized world map, Winkel-Tripel projection, in which all countries have different (random?) colors. No borders, text, grid or anything else on the map.

Is GIS the right way to go? I found a lot of map data on naturalearthdata.com. Can I create this map with QGIS? Do I need ArcGIS?

I'm completely new to GIS; it looks like an exciting field. I'd appreciate any help to get started.

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You can find a shapefile (vector) for countries as of 2010 from http://epp.eurostat.ec.europa.eu. It think that it would be easier to change the symbology of a vector file than deal with a raster. Arcmap will project on the fly and you can set the map to a winkel projection (projected->world or world-sphere->winkel).

A quick world map projected in winkel projection

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There's an svg of this projection on wikipedia. All country codes in main map group and easily editable in illustrator

svg tripel winkel projection

https://commons.wikimedia.org/wiki/File:Perlshaper_Winkel-Tripel_example1.svg#/media/File:Perlshaper_Winkel-Tripel_example1.svg

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