To visualize 4-dimensional objects, pick a speed to be 1 (a speed your visual cortex can handle), and picture the object passing through 3-dimensional space at that speed. At each time, picture a cross-section of the object, with the cross-section changing at the chosen speed.
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Easy: tesseract, 4-simplex, 4-orthoplex. Hard: Those same shapes but at inconvenient angles, other 4-polytopes, the 3-sphere.
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