We show an ellipse, defined as the locus of points whose distances from two foci, given the box metric, always add up to a constant value.  In the diagram below, the distance between the two foci (symbolized by an o), which is 3, and the distances from both foci to any point on the figure, always add up to 16.

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. . . . . . .____________________ . . . . . . . .
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. . . . . .'. . . . . . . . . . . .|. . . . . . .
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. . . . .|. . . . . . . o . . . . .|. . . . . . .
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. . . . .|. . . . . o . . . . . . .|. . . . . . .
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. . . . ..____________________.,. . . . . . . . .
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The limitations of ASCII diagramming hinder visualization, I'm afraid. The two vertical sides are always 5 1/2 units from the nearer focus, and 7 1/2 units from the farther focus. The two horizontal sides are always 5 units from the nearer focus and 8 units from the farther focus.  The northwest and southeast sides of the figure are 45o diagonals 2 1/2 * sqrt (2) units long.  The northeast and southwest sides, which don't appear properly here, are also 45o diagonals, 1/2 * sqrt (2) units long.

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