A different puzzle of the same kind (depicted in the animation) uses four equal quadrilaterals and a small square, which form a larger square. When the quadrilaterals are rotated about their centers they fill the space of the small square, although the total area of the figure seems unchanged. The apparent paradox is explained by the fact that the side of the new large square is a little smaller than the original one. If a
is the side of the large square and θ is the angle between two opposing sides in each quadrilateral, then the quotient between the two areas is given by sec2
θ − 1. For θ = 5°
, this is approximately 1.00765, which corresponds to a difference of about 0.8%.
Sam Loyd’s paradoxial dissection. In the “larger” rearrangement, the gaps between the figures have a combined unit square more area than their square gaps counterparts, creating an illusion that the figures there take up more space than those in the square figure. In the “smaller” rearrangement, the gaps take up one fewer unit squares than in the square.