251–260 of 335 posts
Surprising fact about the sun: it actually produces less heat per unit volume than a compost pile - this makes sense when you consider that fusion events are very rare. The reason the sun is so hot is that it has an enormous ratio of volume to surface area. Heat emission due to radiative cooling is proportional to the surface area, but heat creation due to fusion is proportional to the volume.
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it's true but really what they're missing is the number and variety of toppings on these pizzas.
Variety maybe, but surely number is also proportional to area (and presumably it's worse because "crust width" doesn't scale with diameter?)
> 16. If you let a 100g strawberry that is 99% water by mass dehydrate such that the water now accounts for 98% of the total mass then its new mass is 50g: https://en.wikipedia.org/wiki/Potato_paradox I really like this one. It's a perfect combo of intuitive from one perspective and mind bending from another. > 18. A one-in-billion event will happen 8 times a month: https://gwern.net/Littlewood This one, on the other…
#16 is something video games taught me, particularly Path of Exile. In PoE resistance are a flat multiplier to incoming damage. Eg monster does 100 damage per attack and you have 60% resistance then you take 40 damage. The interesting thing is that the higher your resistances the more effective each additional percentage point of resistance is. Let's say a monster does 100 damage per attack. If you have 0% resistance…
Let's say you have two vehicles, both doing 10,000 miles per year. One gets 10 mpg and the other 50.
Would you rather upgrade the 10 mpg vehicle to 13 mpg, or the 50 mpg to 100 mpg?
Not only should you pick the former - you should pick the former even if you could upgrade the 50 mpg vehicle to run on nothing.
> 0% selected the right answer on this SAT question: Circle A has 1/3 the radius of circle B, and circle A rolls one trip around circle B. How many times will circle A revolve in total? That's fun. I of course immediately selected 3 which means I could have a bright career in test preparation ahead of me.
That one got me good, so my future at College Board is as bright as yours. However, I don't think the argument by demonstration in that video is particularly convincing. Instead, I think's its easier to note that that the _center_ of a circle of radius r travels 2 * pi * r distance over one rotation. In the problem, the center of the smaller circle has to travel further than the circumference of the bigger circle - i…
> 16. If you let a 100g strawberry that is 99% water by mass dehydrate such that the water now accounts for 98% of the total mass then its new mass is 50g: https://en.wikipedia.org/wiki/Potato_paradox I really like this one. It's a perfect combo of intuitive from one perspective and mind bending from another. > 18. A one-in-billion event will happen 8 times a month: https://gwern.net/Littlewood This one, on the other…
>> 18. A one-in-billion event will happen 8 times a month: https://gwern.net/Littlewood > This one, on the other hand, I don't like. Depending on a whole bunch of subjective definitions, a one-in-billion event can happen a million times a second or practically never or whatever else you choose. I think this is about events happening to people, the number of people alive (and assuming they all communicate "miracle" oc…
However, the law of truly large numbers, as you frame it, is something you experience firsthand working in high level severity hospital settings in large metro areas. There's a large enough hospital catchment area that you start to see, on a fairly regular basis, the medical outcomes of all the bizarre and unbelievable things that happen rarely to any given person. It gets to a point it's difficult to know how to explain because the details of each case would be potentially identifying given how strange they are. And yet something happens all the time. Maybe not that one thing, but something of similar impact. It gives you a distorted sense of risk.
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I'd deny that our objects do not live in a real space. Spacetime is real-valued. The issue isn't the reals, but that "solid object" isn't defined properly, ie., the sets under question don't have well-defined volumes. As soon as you fix that problem, via measure theory, the paradox resolves. You dont need to ditch real numbers.
> Spacetime is real-valued There's really no evidence for this, as far as we know the real numbers are a pure mathematical invention and don't have any physicality. Even if you want to say that spacetime is dense (i.e. infinitely divisible), there's an infinite number of fields like that, the real numbers are just a convenient superset. There's no evidence that spacetime is dense either, and many practical ways in wh…
But if we accept that the Planck length is the smallest possible length and the Planck time is the smallest possible time, then it seems logical that the universe is an integer lattice of these. ("Spacetime is not real-valued")
https://en.wikipedia.org/wiki/Planck_length https://en.wikipedia.org/wiki/Planck_units#Planck_time
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"Weight" historically referred to mass, in common speech dating back forever. It’s the Germanic word which has been used throughout the history of English, whereas "mass" comes from Latin via French, like 5–6 centuries ago. The two words are almost exact synonyms, in historical/colloquial use. Both historically and today, a "pound" (Roman libra) is a unit of mass. People use a pound-force as a unit of force only in s…
Actually in the past "weight" or the Latin "pondus" (=> pound) always referred correctly to what is now named "mass". When someone mentioned "weight" just in a qualitative way, as a burden, they might have thought at the force that presses someone down, but whenever they referred to weight in a quantitative way, they referred to the weight as measured with a weighing scale, which gives the ratio between the mass of t…
That’s the same thing I just said. Why add “actually” in front? Yes, weight was historically measured with balance scales.
I guess I should have been clearer that the term “mass” as used in physics only dates from 3 centuries ago (from Newton), and did not historically mean weight in Latin. (Mass comes from Latin via French for lump of dough.)
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Just to make this more concrete: 1. There is a countable model of real numbers. 2. There even is a countable model of the entire set theory.
> There is a countable model of real numbers. What exactly happens when you try to apply Cantor's diagonal argument to this model? I guess that at some step, you get an answer like "outside of the model, yes this exists, but inside the model the answer is no", but I would like to see it precisely, how exactly the in-model reasoning diverges from the outside-model reasoning.
> 16. If you let a 100g strawberry that is 99% water by mass dehydrate such that the water now accounts for 98% of the total mass then its new mass is 50g: https://en.wikipedia.org/wiki/Potato_paradox I really like this one. It's a perfect combo of intuitive from one perspective and mind bending from another. > 18. A one-in-billion event will happen 8 times a month: https://gwern.net/Littlewood This one, on the other…
Yes, it doesn't explicitly state the rate or distribution of events. But it is a good reminder of what happens when your whatevers/second are pretty high - see the famous "One in a million is next Tuesday" [1]. "Rare" is soon if you roll the dice fast enough. Any time your service has a high TPS, your API gets a lot of calls, a button in your app gets pressed a lot, ... this applies. Critically, "a lot" is defined re…
Indeed. One fun example is LHC[1], where the probability of a proton in a single bunch hitting a proton in the bunch going the opposite direction is on the order of 10^-21, yet due to huge number of protons per bunch and large number of bunches per second, it still results in ~10^9 collisions per second.
[1]: https://www.lhc-closer.es/taking_a_closer_look_at_lhc/0.lhc_...
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It's mean to be a thought experiment but some people (like Veritasium) can't get around this simple realization. Like, "I wonder what could be happening inside of a black hole?" "Oh we would never know, because if we send a camera it would break. Also suppose we MaaaAaAKKeee aa cCAaaMeerra with ThE STROooonngGEEstt Material In the UUunniveersssee, so it wouldn't break, you would find that there's no wifi in there so…
(Disclaimer: I am not a physicist.) I think it's actually a pretty reasonable description? Sounds like it's a paradox because we assume a rigid body, but nothing in real life is a perfect rigid body. So the situation simply becomes a bunch of particles following circular orbits. If you measure distance along one direction (while constantly accelerating in relativistic speed) you get one number; if you measure distanc…
Of course, it's exactly the opposite: it's those "ideal" concepts are imperfect approximations of the real things, omitting lots of details which sometimes are not that important but sometimes are absolutely crucial.
My personal favourite example is attaching a perfect source of voltage (zero internal resistance) to a perfect wire (zero resistance). You can't arrive to this scenario starting with the real world entities: both the battery and the wire will have non-zero resistances and depending on their proportions, you end up with approximating either one of those as zero, or none, but never both.