The most accurate method of showing Earth's surface in terms of preserving spatial relationships and area is a globe (or DSL model globe). While all map projections have limitations, a globe represents the Earth in three dimensions, accurately reflecting distances, areas, and shapes.
Here's a brief overview of the other options you listed:
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Equal Area Projection: This type of map (such as the Peters projection) preserves area but can distort shapes and angles. It's useful for comparing size but not for navigation.
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Mercator Projection: This projection is known for preserving angles, making it useful for navigation, but it significantly distorts areas, particularly near the poles.
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Robinson Projection: This projection is a compromise that attempts to minimize distortion in area, shape, distance, and direction, but it is not equal-area and does not preserve any of those properties perfectly.
Overall, for accuracy in representing the Earth's surface, a globe is the best option.