just as you extract a cube out of a square, when you extract a 3d object from an octagon then it would have 16 vertices. it has 16 vertices, 10 faces 24 edges.
i think you were asking about a tesseract it is not a 3d shape but is 4d. just as a cube for a square,tesseract is for cube.
it looks like a smaller cube inside a larger cube where all corners of larger one are connected to smaller one respectively.
making this simple
join two points = line
join two parallel lines = square
join two parallel squares= cube
join two parallel cubes = tesseract.
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