The fluid measurements analogy doesn't work though because volumes don't behave the same way as areas with fixed aspect ratios. Moreover, the reason why √2 is s

The fluid measurements analogy doesn't work though because volumes don't behave the same way as areas with fixed aspect ratios. Moreover, the reason why √2 is such a useful ratio is because it maintains the same aspect ratio when the sheet is doubled or halved. That's why his argument was nonsensical.