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I am trying to recreate this figure (mainly the stacking of images) in Abobe Illustrator:

enter image description here

but using perspectives I get the following:

enter image description here

How do I go about either removing the vanishing points, or using another tool to create the effect in the first figure?

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You're way overthinking this; the example you've provided isn't a "true" perspective, it's an isometric projection. Each rhombus is the exact same size and angles for each of the corners. In a true perspective, the size and angles would be different for each.

You definitely don't need to use the Perspective Grid to produce that effect. Essentially, they've taken a square, rotated it 45°, then squashed it vertically (or stretched it horizontally if you want to look at it that way). Then a constant vertical translation is applied to each rhombus to provide the "stacking".

You can set this up quickly with perfect alignment using Illustrator's Transform effect on a rhombus like so:

Illustrator Screenshot

Expanding the effect will then allow you to recolor or place whatever image you need within the shapes:

Illustrator screenshot

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