Can we project a 4D object onto the 3D world?

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The LightCone in the "3D"

How does the light"cone" appear if we include the 3rd dimension of space into the picture?
 
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As a sphere growing from a point at present in both forward and backward time. The "cone" with a 3-D cross section is now a 4-D object, and you can't embed it in 3-space. That's why we do the 2-D cross section in diagrams.
 
What do you mean by "appear"? You can't draw 4 dimensions in 3 dimensional space so you can't "see" it in that sense.
 

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