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Each independent coordinate adds one dimension to the description.

x

y

90 degrees

2 numbers: x, y 3 numbers: x, y, z 4 numbers: x, y, z, w

FIGURE 2 / Flat drawings of shapes. Gold edges stand for the fourth direction of a hypercube. A flat page cannot show all its true angles. [34,36]

Does a longer line gain another dimension?

A dimension adds an independent coordinate. In Euclidean space, a new axis can be chosen perpendicular to the existing axes. [34,36]

A finite square already requires two coordinates; a solid cube requires three. Infinite extension is unnecessary.

Higher dimensions are mathematically consistent. Their physical existence requires evidence beyond a projected drawing. [6,10,36]

Making a line longer does not issue it a new direction.