There is an interesting fact that bubbles to the surface when you examine all manner of phenomena. As 3D objects get larger their surface area grows more slowly than their volume. This first affected my life when I was trying to eat some of Old El Paso’s new Fiery Soft Tacos (they were not that fiery; they have not sponsored this post) and they could barely fit anything in them. I was left with more taco filling than I could wrap and was deeply disappointed.
While this example is about Old El Paso’s teeny tiny tacos, the maths works the same for anything you would want to fold up and store food in. If you double the radius of a tortilla, it takes four times the amount of flour to make it (it’s proportional to r^2 and (2r)^2 = 4r^2) but the amount it can hold increases by eight times (it’s proportional to r^3 and (2r)^3 = 8r^3). This means that the ratio of the flour used to food encased improves as you increase the size of the taco. If Old El Paso had used the same amount of ingredients and instead just made it into fewer but larger tacos, my misery could have been prevented.
The same relationship holds for the surface of the Earth versus its volume. If you were to split the surface of the Earth up into 100m^2 blocks so that people could build houses on them that were 4m tall we could fit (surface area of Earth = 510 million square kilometres) / (100m^2 = 1/10,000 square kilometres)= 5.1 trillion houses. This is a shockingly large number and may seem unreasonable, but it is important to remember that most of the worlds land is ocean, used for agriculture, or not developed at all.
However, if we were able to turn the whole volume into houses, we would be able to build a whopping (volume of Earth = 1 quadrillion cubic kilometres) / (400m^3 = 1/2,500,000 cubic kilometres) = 2.5 sextillion houses.
This is not a number most people are able to understand intuitively, but written out in full that is 2,500,000,000,000,000,000,000 houses, enough for every person currently on Earth to to have 350 billion each. This is an absurd number and shows just how incomprehensibly large the planet we are standing on really is.
There are many other examples of where this matters in nature, why don’t dragonflies sink when they step on water? Because they are small enough that the surface tension of the water is powerful enough to support the weight, the surface tension scales with x^2 (because it is a surface phenomenon), while the gravity working against the surface tension scales with x^3 (because it is based on the volume of the organism). Just like with the houses, once things get large enough effects that scale cubically (with x^3) dominate and things that were important as smaller scales become irrelevant (it would be absurd for something the size of an elephant to stand on the surface of water just due to water tension).
I think this is the kind of mathematical exploration that should be done more in classrooms, while an enormous amount of time is dedicated to the matching of functions to shapes on a graph or algebraic expressions, understanding the physical manifestations is what allows one to have a real mathematical understanding of the world.
For a great animated explanation of this phenomena, it is explored well in the Life and Size series by Kurzgesagt.
This is beautiful
Amazin’