Fredrik
Graduate Poster
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It's also possible that you're just overestimating the abilities of those who "can". No one can really visualize a 10-dimensional space for example.
Question for mathematicians/theoretical physicists
- Thread starter Elizabeth I
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When I said that the answer "is in fact a particular area, namely, the cross-sectional area of the hole", I didn't mean anything very deep. I just meant that the numbers come out the same.
But, still, there is a reason why they come out the same.
What about doing things in the opposite direction? Does it feel right to you that multiplying the height of a hole by its cross sectional area gives its volume? Or does that rule seem just as arbitrary to you as the one about squaring the diameter (in inches) and then dividing by 2 to get volume per height (in cubic meters per kilometer)?
Or, to change the subject slightly, how about distance divided by time? Are you happy thinking about 70 miles per hour as a speed? Or only as a description of how far you travel in a certain amount of time (and it's just a coincidence that this happens to be related to how fast you're going)?
And now for some nitpicking. Sorry, I can't help myself.
Should be 39.37 in/m (approximately), not 38.37.
But I guess you know this and it was just a typo, because the 78.74 is right.
There's a pi missing, or something like that. The area is actually about 56.75 square inches. (I'm assuming 8.5" is the diameter and a barrel is 42 US gallons, because then 70 barrels / 1000 ft. is right.)
Incidentally, try asking Google what is "70 oil barrels / 1000 feet in square inches". It will say, "56.59500 square inches". (Help, it's a conspiracy!)
Should be 39.37 in/m (approximately), not 38.37.
But I guess you know this and it was just a typo, because the 78.74 is right.
There's a pi missing, or something like that. The area is actually about 56.75 square inches. (I'm assuming 8.5" is the diameter and a barrel is 42 US gallons, because then 70 barrels / 1000 ft. is right.)
Incidentally, try asking Google what is "70 oil barrels / 1000 feet in square inches". It will say, "56.59500 square inches". (Help, it's a conspiracy!)
Elizabeth I
Philosopher
I read that one and found it incomprehensible, though fascinating. I was asking more about string theory being the same as the general theory of relativity, or gravity.
Elizabeth I
Philosopher
shadron
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Do you want to blow your mind about math in an easy but historical way, Elizabeth? Go look at this NOVA program: http://www.youtube.com/watch?v=h3GIhfyLXwc . It is the story of Archimedes and a copy of his Methods which seems to show the 2nd century BCE Greek discovering the basics of integral calculus. The text, known as the palimpsest (a text hidden by another) survived only because a monk in the 11th century poorly erased Archimedes writing to use the pages as a hymnal. To anyone who knows a little about math, the idea that Archmedes was doing this 1900 years before Newton and Leibnitz is simply flooring. What would the world be like today if that had gotten out of Syracuse then?
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godless dave
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It's not the same as General Relativity - it includes General Relativity. It tries to explain everything General Relativity explains plus some things General Relativity doesn't.
Have you considered therapy?
Sorry. Hoist by my own shortcut. The area is, as you say 56+in^2.
The reason for that slip up is as follows.
Volume of a 1 foot cylinder of 1inch diameter is 0.00097142barrels
For 1000 feet, that's 0.9714 which is near enough 1 for most quick calculation.
To get there, I (30 odd years ago) , calculated the volume per foot in cubic feet and converted to barrels by dividing by 5.6146
And there, I suppose, lies the source of my confusion.
To get a volume/1000ft, I square the diameter- not the radius. We are interested in the diameter of the drill bit and these are mostly standard sizes- 26", 17.5", 12.25", 8.5" etc. Over the years I have memorised the relevant squares- of the diameters- 8.5 equates to 72.25-and simply taken that number as being near enough the volume in barrels for 1000ft.
But -it is indeed NOT the area of the hole. It's the square of the diameter.
I've been using the approximation so long, I had forgotten that.
Although I've used cubic metres often over the years, I did so by mentally estimating the volume in barrels and then dividing by a constant. (6.2898, rounded to 6.3)
One day recently I realised this was a long road for a short cut and worked out the equivalent constant for converting standard diameters to cubic metres/km. That forced me to see the obvious- that I was calculating an area.
A 1 inch diameter hole contains .5067m3/km, so a quick & dirty conversion is diameter squared/1.97 (or even d^2/2)
I'd remind you, these are mental calculations. For accurate stuff we would use preprogrammed software, which has the effect of stopping one thinking about what is actually being calculated. I rarely carry a calculator as the environment simply wrecks electronics- hence the rules of thumb.
I'll go back and reread your other post now.
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Hard to answer. I KNOW that the volume is the area times the length- and I used that formula to work out the d^2/2 version, but yes, I'd have to say they FEEL about equally arbitrary.
Good question.
The answer may surprise you. It does me.
It depends on the number.
I think of 3 mph as a distance per time.
I think of 60 mph as a speed.
The difference is obviously to do with walking pace compared to driving speed. I was 26 when I first owned a car. Since then, I have fallen into the American habit of describing distances as times-Glasgow to Edinburgh is "an hour". No it isn't- it's 40 plus miles. If I think of it as a hike, I switch modes. I can damn near feel the "clunk" as my mind switches over. It feels like looking at a Neckar cube. You know that flip-flop feeling? How odd.
I don't automatically know where the slow class ends and the fast one starts. Depends on context. I used to run 10 miles in an hour (at my peak), but it doesn't feel like a speed. It feels like a distance over time.
This is funny as hell. You're messing with my mind!
When it comes to a second derivative- acceleration- I'm lost. I know very well what acceleration feels likeviscerally , but I have no feeling for what d/t/t means.
Thick as two short planks, basically. I only had one good maths teacher in high school and I think I broke her heart. She could see I wasn't totally dim, but we just weren't speaking the same language.
I strongly suspect language was critical. Like all competent mathematicians, she thought in concepts which were not necessarily verbal. I think in words.
I don't know how to think any other way.
To be honest, I put numbers in the "Invisible unicorn" category- useful in their way, but not something I actually believe in.
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Perhaps not "visualise", but they claim ability to "conceptualise"- to make physical sense of the equations. This is exactly what's happening when someone says a particular equation represents a one dimensional thing vibrating in a particular way and that this is somehow the same thing as a photon.
Could be.
But if I said thunder is caused by angels shifting coal you might be excused for thinking I was talking hooey. (It's how thunder was first explained to me).
I nurture similar suspicions about string theorists.
sol invictus
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But the difference is precisely what's at issue in this thread - mathematics. While string theory may not be the correct description of the world, it is certainly a consistent set of equations that model many aspects of the world extremely well (such as gravity).
It makes no difference what you think that means - it just is. And if you want to argue with it, you have to show the math is wrong (or that the results are inconsistent with experimental data). Anything else is pointless.
Elizabeth I
Philosopher
Sorry, you (or somebody else upthread) said that string theory was indistinguishable from general relativity. To my word-oriented brain (I'm beginning to think Soapy Sam has something here), that translates as "same as."
I just wanted to say that I am fascinated by how you think. It's also interesting to see how self-aware you are about it (the "clunk" of switching over that you mentioned, e.g.).
I recently attended a talk on math education where they said that rates are some of the most confusing things for people to learn, and where the biggest gains can be made with certain methods of visualizing what a rate (and what a change in rate!) means. Not sure if different teaching would have altered how you visualize things, but I still appreciate seeing how you think about them.
For me, for example, both 3 mph and 60 mph are intuitively rates. Although I know one is achievable walking and the other is not, they're both in the same mental bin.
(But then, I grew up in Southern California. We do like the cars.)
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