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Setting the biggest square = 1, The areas of the black squares are 1
+ 1/4 + 1/16 + . . . . (1/4)^n

The sum from n = 0 to infinity this would be 1/(1 - 1/4) = 4/3

The negative space, the white diamonds, are the same as half the black squares. Their total is -2/3

So the black squares and white diamonds together are 4/3 - 2/3 which equals 2/3.

Thank you, Clive
Tooth, for giving me this.