Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life, in Organisms, Cities, Economies, and Companies

by Geoffrey West

So when the time comes you stuff in as much and as many as you possibly can to fill the entire volume of the tub. Now, recall that ordinary volumes scale faster than areas, so if you were to double the size of your washing machine by doubling all of its lengths while keeping its shape the same, its volume would increase by a factor of eight (23) whereas all of its surface areas would increase by a factor of four (22). Naively, you might therefore conclude that because sheets are essentially all area and consequently two-dimensional (their thickness being negligible), you could accommodate four times as many sheets by doubling the size of your washing machine. However, if we stuff all of the sheets into the tub so that they completely fill its entire volume and because this volume has increased by a factor of eight, then it’s clear that you can actually accommodate eight times as many sheets, rather than just four times. In other words, the total effective area of two-dimensional sheets filling three-dimensional washing machines scales like a volume rather than an area, so in this sense, we have turned an area into a volume.