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More tinycoon maths!
14 years ago
So far I've been keeping up the rate of losing 2 pounds (~1 kilogram) a week, and as of today I have lost 60 pounds so far. :) Let's do some more microphile math to celebrate!
As we discussed last time, the two sizes I'm going to use for this are 10 inches (25,4 centimeters) and 3 inches (7,6 centimeters), but now my weights at those sizes would be 240 grams (8,45 ounces) and 6,45 grams (0,23 ounces) respectively.
Strength
A creature's mass and weight increases or decreases directly with its volume, which in turn changes with the cube of that creatures change in height or length. However, its strength is largely based on the cross-section of its bones and muscles, which as an area function will change only with the square of height or length. This is the square cube law, and it is why macros don't actually work in real life, with limited exceptions such as provided by water bouyancy (for whales) or epochs of increased atmospheric oxygenation (the time of the dinosaurs). It is also why creatures like ants (which are actually fairly inefficient) are so seemingly mighty - with even less weight to carry around compared to their diminished strength, they can lift and move things out of proportion to their size. Naturally, I have to wonder how strong I would be as a tinycoon!
I haven't done any bench presses in years, so I have no idea what I would be able to power lift in that way. However, I can do pushups without instantly collapsing, so let's use that as a stand-in benchmark. When I do a pushup I am lifting about 140 pounds (63,6 kilos) off the ground with my arms, or a little over 58% of my body weight; you never lift full weight with a pushup because of how you're balanced on your toes and so distributing your mass. That's not the maximum amount I can lift, or I wouldn't be able to do more than one of them, but it's the best value I have at the moment for these purposes. So, proceeding!
At 10 inches tall, my strength would decrease as such:
140 pounds * (10 in / 77 in)^2 = 2,36 pounds * 454,5 grams/pound = 1073 grams
In other words, I could lift about 4,5 times my full bodyweight at that height for as long / as many times as I could do pushups at my real height! Between the metal and the liquid contents a standard soda can is about 370 grams, and the average shoe is a pound or less, so I could heft such things all day long if sent to fetch them or put them away for my master. Awkwardness aside I could carry both my master's shoes at once! I could also heft a female ferret, or a smaller male ferret. :3
At 3 inches tall, the same strength measure would change as follows:
140 pounds * (3 in / 77 in)^2 * 454,5 grams/pound = 96,6 grams (about 3,4 ounces)
That's even more of a disparity. Now I am repeatedly lifting 15 times my bodyweight! I wouldn't be able to easily lift or carry a can of soda, but I could drag it around on a sledge. More in my range would be small phones, switchblades, quarter pound hamburgers (the loss in weight while cooking being made up for by the bun and condiments), and so on. Again, my actual lifting strength would be greater than this, because this is based off of the amount of weight I am effectively "hefting" in a pushup at my real height, but I don't have any way right now of measuring what my actual limits are; it's safe to say, though, that this is probably the comfortable limit of what I could do for a short span of time at most without immediately straining myself too badly. I could definitely still press hard enough to give a meaningful foot massage, but it's a plain fact that I'd be a more "useful" slave at the 10 inch size most artists tend to draw me at.
Jumping
Jumping is a direct function of strength. You are exerting muscle power against the force of gravity to lift your body weight off the ground into the air. Because your mass (and thus weight) decreases with the cube of height but strength decreases only with the square of height, it is reasonably close to accurate to say that jumping distance should increase linearly proportionate to your new height as you shrink. Shrinking to 10 times smaller than usual while keeping the same proportionate strength and build will make you 100 times weaker but also 1000 times lighter, so you should be able to jump 10 times farther relative to your new size than you could at your original height. This is why things like fleas can jump so high - and they're actually pretty inefficient at it!
So. I'm not really anything to write home about in the athletic department. I can consistently clear about 3 feet (36 inches) off the ground at my true height, which distance is under a meter and less than half my height. But this is going to look a lot more impressive once I turn into a tinycoon! Let's see how much better.
At 10 inches, this is going to end up with a pretty cool result:
36 inches / 77 inches tall = 0,4675 factor of my height
0,4675 * 10 inches = 4,675 inches * (77/10) = 36 inches!
So not only is 10 inches the height at which my weight in pounds translates directly into the same number in grams, but it is also a height at which my jumping capability is exactly the same as my full size human incarnation! This isn't necessarily a surprise, either. After all, cats are on a similar scale, and can jump the same amount or even farther with their four legs. That's how they get up on your kitchen counters!
At 3 inches, though, I have to accept that my jumping distance is curtailed... or do I?
0,4675 * 3 inches = 1.4 inches * (77/3) = still 36 inches!
Eat it, fleas. I'm just a pitiful humanoid and I'm managing better than them! Of course, they have armored exoskeletons and really not the best kind of body for this sort of thing, but they can still manage a couple feet up onto your couch. Also, I am not accounting for air pressure and air density, which might start to actually be a factor at this height, but barring that, I can jump the same distance as at full size due to my massive increase in relative strength. At least until I start trying to carry heavy objects with me!
The conclusion, of course, is that your jumping distance is pretty much a constant! It doesn't matter how much you shrink, you're going to go just as far. If you want to jump farther, you need to get objectively stronger. By the same token, the standard trope of the hapless micro faced with the daunting obstacle of human-scale stairs is not actually much of an issue as people would have you believe – physics is your friend as a micro, and you can clear stairs just by jumping!
More thought experiments in this vein will come once I reach another milestone. :)
As we discussed last time, the two sizes I'm going to use for this are 10 inches (25,4 centimeters) and 3 inches (7,6 centimeters), but now my weights at those sizes would be 240 grams (8,45 ounces) and 6,45 grams (0,23 ounces) respectively.
Strength
A creature's mass and weight increases or decreases directly with its volume, which in turn changes with the cube of that creatures change in height or length. However, its strength is largely based on the cross-section of its bones and muscles, which as an area function will change only with the square of height or length. This is the square cube law, and it is why macros don't actually work in real life, with limited exceptions such as provided by water bouyancy (for whales) or epochs of increased atmospheric oxygenation (the time of the dinosaurs). It is also why creatures like ants (which are actually fairly inefficient) are so seemingly mighty - with even less weight to carry around compared to their diminished strength, they can lift and move things out of proportion to their size. Naturally, I have to wonder how strong I would be as a tinycoon!
I haven't done any bench presses in years, so I have no idea what I would be able to power lift in that way. However, I can do pushups without instantly collapsing, so let's use that as a stand-in benchmark. When I do a pushup I am lifting about 140 pounds (63,6 kilos) off the ground with my arms, or a little over 58% of my body weight; you never lift full weight with a pushup because of how you're balanced on your toes and so distributing your mass. That's not the maximum amount I can lift, or I wouldn't be able to do more than one of them, but it's the best value I have at the moment for these purposes. So, proceeding!
At 10 inches tall, my strength would decrease as such:
140 pounds * (10 in / 77 in)^2 = 2,36 pounds * 454,5 grams/pound = 1073 grams
In other words, I could lift about 4,5 times my full bodyweight at that height for as long / as many times as I could do pushups at my real height! Between the metal and the liquid contents a standard soda can is about 370 grams, and the average shoe is a pound or less, so I could heft such things all day long if sent to fetch them or put them away for my master. Awkwardness aside I could carry both my master's shoes at once! I could also heft a female ferret, or a smaller male ferret. :3
At 3 inches tall, the same strength measure would change as follows:
140 pounds * (3 in / 77 in)^2 * 454,5 grams/pound = 96,6 grams (about 3,4 ounces)
That's even more of a disparity. Now I am repeatedly lifting 15 times my bodyweight! I wouldn't be able to easily lift or carry a can of soda, but I could drag it around on a sledge. More in my range would be small phones, switchblades, quarter pound hamburgers (the loss in weight while cooking being made up for by the bun and condiments), and so on. Again, my actual lifting strength would be greater than this, because this is based off of the amount of weight I am effectively "hefting" in a pushup at my real height, but I don't have any way right now of measuring what my actual limits are; it's safe to say, though, that this is probably the comfortable limit of what I could do for a short span of time at most without immediately straining myself too badly. I could definitely still press hard enough to give a meaningful foot massage, but it's a plain fact that I'd be a more "useful" slave at the 10 inch size most artists tend to draw me at.
Jumping
Jumping is a direct function of strength. You are exerting muscle power against the force of gravity to lift your body weight off the ground into the air. Because your mass (and thus weight) decreases with the cube of height but strength decreases only with the square of height, it is reasonably close to accurate to say that jumping distance should increase linearly proportionate to your new height as you shrink. Shrinking to 10 times smaller than usual while keeping the same proportionate strength and build will make you 100 times weaker but also 1000 times lighter, so you should be able to jump 10 times farther relative to your new size than you could at your original height. This is why things like fleas can jump so high - and they're actually pretty inefficient at it!
So. I'm not really anything to write home about in the athletic department. I can consistently clear about 3 feet (36 inches) off the ground at my true height, which distance is under a meter and less than half my height. But this is going to look a lot more impressive once I turn into a tinycoon! Let's see how much better.
At 10 inches, this is going to end up with a pretty cool result:
36 inches / 77 inches tall = 0,4675 factor of my height
0,4675 * 10 inches = 4,675 inches * (77/10) = 36 inches!
So not only is 10 inches the height at which my weight in pounds translates directly into the same number in grams, but it is also a height at which my jumping capability is exactly the same as my full size human incarnation! This isn't necessarily a surprise, either. After all, cats are on a similar scale, and can jump the same amount or even farther with their four legs. That's how they get up on your kitchen counters!
At 3 inches, though, I have to accept that my jumping distance is curtailed... or do I?
0,4675 * 3 inches = 1.4 inches * (77/3) = still 36 inches!
Eat it, fleas. I'm just a pitiful humanoid and I'm managing better than them! Of course, they have armored exoskeletons and really not the best kind of body for this sort of thing, but they can still manage a couple feet up onto your couch. Also, I am not accounting for air pressure and air density, which might start to actually be a factor at this height, but barring that, I can jump the same distance as at full size due to my massive increase in relative strength. At least until I start trying to carry heavy objects with me!
The conclusion, of course, is that your jumping distance is pretty much a constant! It doesn't matter how much you shrink, you're going to go just as far. If you want to jump farther, you need to get objectively stronger. By the same token, the standard trope of the hapless micro faced with the daunting obstacle of human-scale stairs is not actually much of an issue as people would have you believe – physics is your friend as a micro, and you can clear stairs just by jumping!
More thought experiments in this vein will come once I reach another milestone. :)
If I remember right it was about lower body mass being less effected by gravity. Though you would fall at the same speed as say, a bowling ball, The impending contact with the floor would be less harsh for a micro. A normal human could get hurt jumping from a high distance, but a micro could be hoisted up on high and dropped without a problem.
That's some pretty neat info to have, Do you know if there are any differences in how one uses oxygen at different sizes? Would be interesting to find out if micros can hold their breath a lot longer.
Definitely do keep these coming -- it's been too long since I've done anything math-related and I really REALLY should get back into it.