Dealing With Extreme Scales
Humans do not always have the best sense of scale. We have a very hard time dealing with the very large and the very small. For example, we often drastically underestimate the forces of mother nature. I had the privilege of strapping on my crampons and venturing onto the snowy glaciers of Mt. Rainier. It was a beautiful expedition, very well managed, but I did not get the privilege of summiting on that trip. Avalanche dangers turned us back just a about an hour above Camp Muir. Needless to day, there was the proper portion of grumbling as we all descended the mountain, defeated. Rubbing salt in our wounds, the weather actually looked wonderful for a summit attempt.
Then, we heard a crack, and a low rumbling sound. We all turned to gaze across the snowfield to watch the icefall that had just shook a small mountain of ice loose, sending it tumbling to the ravine below. As we watched, it dawned on each one of us: some of these ice blocks are the size of a small house, and mother nature cast them carelessly down, watching with little more than a mild interest as the blocks dash against each other and the rocks below.
After perhaps half a minute, the rumbling ceased. The air on the snowfield went quiet once again. We continued on our way just as before, but not once after that experience did I hear a complaint about the summit being out of our reach.
I believe that that expedition had about fifty years of cumulative mountaineering experience, and yet all of us were silenced by the awesome force of nature put before us. For world builders, the situation is even more dire. Many scenarios put before a world builder are simply outside of human comprehension. No experience in hurricane Katrina can give you the mildest sense of the torrential forces of Juptier’s Great Red Spot, the great cyclonic storm the width of three Earths. No experience with pressurized containers can capture the crushing pressures of Challenger Deep, where pressures are comparable to that inside the chamber of a handgun. We world builders must visualize the un-visualizable, using nothing but pathetically sterile numbers. Comparing 500MPa to 10MPa just doesn’t quite capture the gut wrenching feel of “this press, used to squeeze 4” thick aluminum slugs into the shape of a SCUBA tank, exerts roughly fifty times the force of a woman’s stiletto heel bearing down on your hand!”
To cope with this deficit, we need to understand how humans think about scale. While every person is different, scientific studies have shown some common features of how we deal with scale in our minds. For most people, our sense of scale is focused on a region of the brain very closely entwined with our sense of spatial awareness. No surprise, the number line is a highly effective way of conveying relationships between numbers.
As it turns out, the number line we were taught in school is a little different from what our brains usually use. The schoolhouse version of the number line is nice and linear. The distance between 1 and 2 is the same as the distance from 2 to 3. Unfortunately, this is not a very effective tool for real life applications
Try as we might, there just isn’t a good way to put all of those objects onto the same scale. If we measure it in tons, the small objects like mice and cats scrunch up near almost-zero, like shown above. If we measure it in pounds, we’d need to draw a number line all the way off your screen about a thousand feet to the right before we could fit the tank onto it!
What we have found is that most species arrive at a common conclusion: we don’t really need to know how big something is, but we really do need to know how big something is relative to something else. I don’t need to know the other guy is 152 pounds and I’m 197.5 pounds. What I need to know is that I’m substantially larger than the other guy, so I’m likely to fare well in a conflict.
The number line in our brain reflects this logic. Instead of each additional step to the right adding some size, each additional step will multiply that size by some amount. This is called a logarithmic scale, and it handles large differences in sizes far better.
When we look at how people perceive things, this scale shows up nearly every time. Without it, it would be nearly impossible to intuitively compare the mass of a human against that of a truck (which we do every day when we get into a truck and feel the suspension sway underneath us). So as a world builder, we can generally assume that our reader thinks this way. It’s simply universal enough. To give our readers a sense of scale, we simply have to manipulate how they use their intuitive logarithmic scale.
The first and foremost trick in a world builder’s tool chest is to re-frame the problem. One neat feature of a logarithmic scale is that we can choose any reference point we want and the relationships work out the same. First, a bit of math. If I have 1kg or 2kg, and my reference point for the logarithmic scale is 1kg, those correspond to +0 and +0.301. That is to say, log(1kg/1kg)=+0 and log(2kg/1kg)=+0.301. If instead, I make my reference point 1g, and look at 1000g and 2000g (the same masses), I get +3 and +3.301. I get those numbers using the same mathematics as before: log(1000g/1g)=+3, log(2000g/1g)=+3.301. Note that, while I offset both masses, the difference between them is the same on a logarithmic scale. +0 to +0.301 is the same difference as +3 to +3.301. The reference point is arbitrary for these comparisons! Nifty!
In the real world, the reference point is not so arbitrary. Our brains are constantly wiring themselves for optimal functionality. We can’t have a perfect mathematical logarithmic scale in our heads. Instead we have one built out of neurons and calibrated for getting the best answers for real life questions. The vast majority of what it has to deal with is centered around the size of our self. We think of distances based on how fast we walk. We think of weights based on how much we can lift. Our brains have adapted to that, so our internal logarithmic scale is structured to get the answer as correct as possible for these self-centric tasks. In fact, having a good sense of mass based on the mass of yourself is essential for walking normally! When we deal with problems outside of this region, such as regional politics, we typically re-adjust our reference point temporarily so that the problem remains centered in this human-sized region which has been so carefully groomed by our lifestyle.
One of the most famous examples of these is try to capture the essence of Bill Gates’ wealth. Instead of merely saying his wealth is $56 billion, which is well outside the scope of any human-sized problem, we re-frame it. Bill Gates made an average of $53/second. Now we can phrase this on a human scale: Bill Gates has made so much money that if there was a $100 bill on the ground in front of him, and it would take him 5 seconds to pick it up, he would actually lose money by doing so!
We can leverage this as a world builder. If we can offer a reference point to our readers which keeps things in the same relative scale as themselves with respect to the world around them, they don’t have to do all the hard arithmetic. You can do it for them. The most valuable list I have found is a series of Wikipedia articles on order of magnitude, including my all time favorite page, Orders of Magnitude (Energy)! Let’s say you’re world building takes you into the realm of nuclear power. You find out that 1 gram of U-235 release 88GJ of energy. Great! Is that a lot? A little? Re-frame the problem. A “Ton of Oil Equivalent (TOE)” is 42GJ. If you phrase it that way, 1 gram of Uranium releases the equivalent of just over 2 tons of oil! You can build the “feel” of that amount of energy because we have an emotional feel as to how much a ton of oil is. As long as other energy calculations are done in terms of tons of oil, they don’t ever have to really worry about how much a ton of oil is. The emotional feel is enough.
Want to use that uranium explosively? Compare against something which releases its energy faster. The Massive Ordinance Air Blast weapon releases 50GJ of energy into the combat environment. This means you could phrase your gram of Uranium as more power than is released by the MOAB fuel/air bomb, nicknamed either “Mother of All Bombs” by many. (Trivia: it was produced mostly for intimidation purposes to be the big brother to the BLU-82 “daisycutter” that we used in Vietnam to turn forest into nice clean helicopter landing sites).
There’s another trick which you can use if the sense of scale is too vast to find a good baseline: repetitions. Consider a phrase like “It was to an elephant as an elephant is to a mouse” to capture the scale of something massive. It takes a relationship that the reader is familiar with, and asks them to apply it to arrive at a scale much larger or smaller than before. This can be key when trying to capture concepts such as the vastness of space, to which there is no convenient unit. Not even the lightyear does the job any more.
To use this trick, you want a relationship which is really concrete, because you’re going to stress the reader asking to layer it upon itself. Either you want very few layers, such as in the elephant example with 2 layers, or you want very rigid reliable layers (such as the British approach to large numbers: million, thousand million, million million/billion). The latter is used in my favorite depiction of the size of the earth: the destruction of Earth in Hitchiker’s Guide to the Galaxy.
We can create an example to test these techniques. Consider a gravity slingshot. This is a procedure where a vehicle, such as the Voyager probe, gets close enough to a planet to steal some momentum. The result is a massive “free” boost in velocity. But its not free right? It has to take it from the planet. Are we dooming ourselves?
The Voyager is approximately 815kg. A planet like earth is around 5,972,000,000,000,000,000,000,000kg! Oh we can see just how difficult this sense of scale is. If we put that in a logarithmic scale based on a kg, those numbers are +2.9 and +24.8 on a log scale. They’re 21.9 apart. Now we can see just how many Voyagers could be gravitationally slingshotted before we could even measure the difference in the Earth’s velocity. In fact, we can now see that if we sent out a million-million (aka. a trillion in US terms) vehicles, those vehicles would together, only mass a +14.9 (+2.9 + +12). Those masses are still 9.9 apart on the log scale… meaning the masses of those trillion Voyages are still less than 0.0000000002 of the mass of the earth! How crazy is that? Here’s how crazy: let’s reframe the problem. If the Earth was the mass of a human being, and we put together a bag containing a trillion Voyagers, their combined mass would be as a small grain of sand or a particle of pollen against the human! Clearly we’re not going to push the Earth anywhere anytime soon!
In physics class, we learned every action had an equal and opposite reaction. If you were to jump up, the earth has to move “down.” So why don’t we feel the earth shake? Using the same kind of logic as before, a 100kg human would register at a +2 on a logarithmic scale referenced to the kilogram. The Earth weighs in at a +24.8 (as before). So there’s a +22.8 difference between your mass and that of the Earth, so the ratio between how much you move and how much the Earth moves is also a +22.8 on the logarithmic scale. Lets use repetition to cut that down. We know how small an ant is. We know how big an Elephant is. In fact, someone with too much free time did the math to find that it takes about 750,000,000 ants to match the mass of 1 “small” Asian elephant, +8.8 difference on the logarithmic scale. If we repeat that, we get +8.8, +17.6, +26.4 (just repeating addition). Thus the difference between you and the Earth (+22.8) is only 10,000 times smaller than the difference between an ant and an elephant, recursively 3 times (+26.4). So want to know how much your jumping affects the Earth? Imagine an ant colony holding a party, jumping up and down as obnoxiously as they can. This ant colony is riding on the back of an elephant. This elephant is not walking on the ground. No, it’s on the back of a mega-elephant so big that it makes the elephant look like an ant. The mega-elephant itself, is walking on the back of a mega-mega-elephant. That mega-mega elephant is the earth. So jump all you want, you won’t bother it.
Feel small yet?
-C
