[math-fun] Microwave ovens
Why do microwave ovens alternate between rotating the food-carousel clockwise and rotating it counterclockwise (or, as the Brits would say, anticlockwise)? Jim Propp
perhaps this is like certain kinds of biodynamic/macrobiotic cooking. an acqaintance of mine claimed that one should stir everything an equal number of times clockwise and counterclockwise, to avoid a buildup of yin or yang. Yang, in this case, probably refers to Yang-Lee quantum field theories (clearly chiral ones). Cris On Jan 26, 2014, at 7:28 PM, James Propp <jamespropp@gmail.com> wrote:
Why do microwave ovens alternate between rotating the food-carousel clockwise and rotating it counterclockwise (or, as the Brits would say, anticlockwise)?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
It's so the cord won't get all twisted up. Brent :-) On 1/26/2014 6:28 PM, James Propp wrote:
Why do microwave ovens alternate between rotating the food-carousel clockwise and rotating it counterclockwise (or, as the Brits would say, anticlockwise)?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Most certainly a kind of motor is used where the initial internal position (and possibly the initial current) determines the direction. A similar thing used to be (long ago) a problem with tractors when the motor was idling: it could start moving backward when put into move, potentially injuring/killing the people accompanying it (usually walking behind). Best, jj * James Propp <jamespropp@gmail.com> [Jan 27. 2014 10:12]:
Why do microwave ovens alternate between rotating the food-carousel clockwise and rotating it counterclockwise (or, as the Brits would say, anticlockwise)?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On 1/27/2014 1:22 AM, Joerg Arndt wrote:
Most certainly a kind of motor is used where the initial internal position (and possibly the initial current) determines the direction.
But I've noticed that mine seems to be completely consistent. It reverses every time.
A similar thing used to be (long ago) a problem with tractors when the motor was idling: it could start moving backward when put into move, potentially injuring/killing the people accompanying it (usually walking behind).
A two-stroke engine will also run backwards. Even though the ignition timing is poor for running backward it can get started that way by kicking back when starting. Once at the start of a motorcycle race I saw kid have his bike lurch backwards as everyone else leaped forward. A look of puzzlement came over his face as he tried easing out the clutch again and the bike started to back up. Then he threw down the bike as if it had betrayed him. Brent Meeker
Best, jj
* James Propp <jamespropp@gmail.com> [Jan 27. 2014 10:12]:
Why do microwave ovens alternate between rotating the food-carousel clockwise and rotating it counterclockwise (or, as the Brits would say, anticlockwise)?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Those little 2-stroke glowplug model airplane engines are completely symmetrical: since they don't have any electrical ignition, there's no way the piston & combustion chamber can know which direction the propeller is going. More than once when I was trying to start mine, it kicked back & ran backwards, which didn't do good things to the airplane it was attached to... BTW, Shannon was an avid unicyclist, who I believe cycled the hallways at MIT. (Most) Unicycles are also completely symmetrical; I assume that Shannon was good enough to unicycle backwards. At 10:29 AM 1/27/2014, meekerdb wrote: A two-stroke engine will also run backwards.
I've noticed that when I put my cup of water+teabags in the microwave to boil, it almost always end up with the cup handle pointed toward the *back* of the microwave -- the most inconvenient position for removing it. (Oh, the perversity of inanimate objects!) (I often vary the time that it's in there. The "teacup" is a modern 2-cup pyrex measuring cup, so the handle is a significant fraction of the cup's weight.) --Dan On 2014-01-26, at 6:28 PM, James Propp wrote:
Why do microwave ovens alternate between rotating the food-carousel clockwise and rotating it counterclockwise (or, as the Brits would say, anticlockwise)?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
My microwave completes a single rotation in 20 seconds, regardless (I think) of the inserted object's mass distribution. That makes a final cup handle position very predictable, based on insertion angle and length of time. On Jan 27, 2014, at 3:58 PM, Dan Asimov <dasimov@earthlink.net> wrote:
I've noticed that when I put my cup of water+teabags in the microwave to boil, it almost always end up with the cup handle pointed toward the *back* of the microwave -- the most inconvenient position for removing it. (Oh, the perversity of inanimate objects!)
(I often vary the time that it's in there. The "teacup" is a modern 2-cup pyrex measuring cup, so the handle is a significant fraction of the cup's weight.)
My carousel turns one revolution in 7sec. So when I heat my coffee cup I use a multiple of 7 so that it ends up in front where I put it. Brent On 1/27/2014 12:58 PM, Dan Asimov wrote:
I've noticed that when I put my cup of water+teabags in the microwave to boil, it almost always end up with the cup handle pointed toward the *back* of the microwave -- the most inconvenient position for removing it. (Oh, the perversity of inanimate objects!)
(I often vary the time that it's in there. The "teacup" is a modern 2-cup pyrex measuring cup, so the handle is a significant fraction of the cup's weight.)
--Dan
On 2014-01-26, at 6:28 PM, James Propp wrote:
Why do microwave ovens alternate between rotating the food-carousel clockwise and rotating it counterclockwise (or, as the Brits would say, anticlockwise)?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Remember the New Yorker cartoon of the guy saying to his microwave, No, I don't want to play a game of chess, I just want you to reheat the pizza? Maybe if you would play chess more often with yours, Dan, it would leave the handle at the front. On Mon, Jan 27, 2014 at 4:32 PM, meekerdb <meekerdb@verizon.net> wrote:
My carousel turns one revolution in 7sec. So when I heat my coffee cup I use a multiple of 7 so that it ends up in front where I put it.
Brent
On 1/27/2014 12:58 PM, Dan Asimov wrote:
I've noticed that when I put my cup of water+teabags in the microwave to boil, it almost always end up with the cup handle pointed toward the *back* of the microwave -- the most inconvenient position for removing it. (Oh, the perversity of inanimate objects!)
(I often vary the time that it's in there. The "teacup" is a modern 2-cup pyrex measuring cup, so the handle is a significant fraction of the cup's weight.)
--Dan
On 2014-01-26, at 6:28 PM, James Propp wrote:
Why do microwave ovens alternate between rotating the
food-carousel clockwise and rotating it counterclockwise (or, as the Brits would say, anticlockwise)?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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-- Dear Friends, I have now retired from AT&T. New coordinates: Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
Excellent suggestion, Neil. (Can't imagine why I didn't think of that myself.) --Dan On 2014-01-27, at 3:25 PM, Neil Sloane wrote:
Remember the New Yorker cartoon of the guy saying to his microwave, No, I don't want to play a game of chess, I just want you to reheat the pizza? Maybe if you would play chess more often with yours, Dan, it would leave the handle at the front.
Just so long as you don't play global thermonuclear war. https://www.youtube.com/watch?v=ecPeSmF_ikc On Tue, Jan 28, 2014 at 12:25 PM, Neil Sloane <njasloane@gmail.com> wrote:
Remember the New Yorker cartoon of the guy saying to his microwave, No, I don't want to play a game of chess, I just want you to reheat the pizza? Maybe if you would play chess more often with yours, Dan, it would leave the handle at the front.
On Mon, Jan 27, 2014 at 4:32 PM, meekerdb <meekerdb@verizon.net> wrote:
My carousel turns one revolution in 7sec. So when I heat my coffee cup I use a multiple of 7 so that it ends up in front where I put it.
Brent
On 1/27/2014 12:58 PM, Dan Asimov wrote:
I've noticed that when I put my cup of water+teabags in the microwave to boil, it almost always end up with the cup handle pointed toward the *back* of the microwave -- the most inconvenient position for removing it. (Oh, the perversity of inanimate objects!)
(I often vary the time that it's in there. The "teacup" is a modern 2-cup pyrex measuring cup, so the handle is a significant fraction of the cup's weight.)
--Dan
On 2014-01-26, at 6:28 PM, James Propp wrote:
Why do microwave ovens alternate between rotating the
food-carousel clockwise and rotating it counterclockwise (or, as the Brits would say, anticlockwise)?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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-- Dear Friends, I have now retired from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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For integer n >= 0, we know F(n) := Sum_{1 <= k <= n} k^2 = n(n+1)(2n+1)/6 (= n^3/3 + n^2/2 + n/6) . It's also well known that if F(n) is an integer square, the only integer solutions are (n, F(n)) in {(0, 0), (1, 1), (24, 70^2)}. QUESTION: --------- What are the rational solutions p/q, s/t to the equation F(p/q) = (s/t)^2 ??? Or in other words, solutions p, q, s, t to the Diophantine equation (2p^3 + 3p^2 q + p q^2) t^2 = 6q^3 s^2 --Dan
On 1/26/2015 11:29 PM, Daniel Asimov wrote:
For integer n >= 0, we know
F(n) := Sum_{1 <= k <= n} k^2 = n(n+1)(2n+1)/6 (= n^3/3 + n^2/2 + n/6) .
It's also well known that if F(n) is an integer square, the only integer solutions are (n, F(n)) in {(0, 0), (1, 1), (24, 70^2)}.
Those are the solutions with n >= 0; there's also F(-1) = 0.
QUESTION: ---------
What are the rational solutions p/q, s/t to the equation
F(p/q) = (s/t)^2 ???
With a little assistance from PARI/GP, I find that the elliptic curve k^2 = n^3/3 + n^2/2 + n/6 is equivalent to the minimal model y^2 = x^3 - 36x via the substitution k = y/72, n = (x - 6)/12. This is curve 576h2 in Cremona's tables. It has rank 1 (with (x, y) = (-3, 9) as a generator, corresponding to (n, k) = (-3/4, 1/8)) and a four-element torsion subgroup (consisting of the points with y = 0 and the point at infinity). So there are infinitely many rational points on the curve; some others with small denominators are (n, k) = (1/2, 1/2), (-2/3, 1/9) and (-49/50, 7/125). -- Fred W. Helenius fredh@ix.netcom.com
Thank you, Fred! Very interesting! (It's probably high time I learned how to use PARI.) --Dan
On Jan 27, 2015, at 5:16 PM, Fred W. Helenius <fredh@ix.netcom.com> wrote:
On 1/26/2015 11:29 PM, Daniel Asimov wrote:
For integer n >= 0, we know
F(n) := Sum_{1 <= k <= n} k^2 = n(n+1)(2n+1)/6 (= n^3/3 + n^2/2 + n/6) .
It's also well known that if F(n) is an integer square, the only integer solutions are (n, F(n)) in {(0, 0), (1, 1), (24, 70^2)}.
Those are the solutions with n >= 0; there's also F(-1) = 0.
QUESTION: ---------
What are the rational solutions p/q, s/t to the equation
F(p/q) = (s/t)^2 ???
With a little assistance from PARI/GP, I find that the elliptic curve k^2 = n^3/3 + n^2/2 + n/6 is equivalent to the minimal model y^2 = x^3 - 36x via the substitution k = y/72, n = (x - 6)/12. This is curve 576h2 in Cremona's tables. It has rank 1 (with (x, y) = (-3, 9) as a generator, corresponding to (n, k) = (-3/4, 1/8)) and a four-element torsion subgroup (consisting of the points with y = 0 and the point at infinity). So there are infinitely many rational points on the curve; some others with small denominators are (n, k) = (1/2, 1/2), (-2/3, 1/9) and (-49/50, 7/125).
-- Fred W. Helenius fredh@ix.netcom.com
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Consider the tricylinder T -- the intersection of three cylinders each of unit radius in 3-space with perpendicular axes. Let S denote its surface bd(T). PUZZLE: Find the solid angle in closed form at a point of S common to the 3 cylinders. --Dan
Perhaps I should define my terms. The solid angle at a point on a convex surface S is the limit of the solid angle subtended by the surface in an epsilon-neighborhood N(eps) of the point, as epsilon -> 0. And that is the area on the unit sphere of the set of all perpendicular unit vectors to the supporting hyperplanes* to S on N(eps) (when the vectors are translated to the origin). --Dan ________________________________________________________ * < http://en.wikipedia.org/wiki/Supporting_hyperplane >
On Feb 2, 2015, at 10:27 AM, Daniel Asimov <asimov@msri.org> wrote:
Consider the tricylinder T -- the intersection of three cylinders each of unit radius in 3-space with perpendicular axes. Let S denote its surface bd(T).
PUZZLE: Find the solid angle in closed form at a point of S common to the 3 cylinders.
--Dan
Thanks! That may well be the problem. As of now we don't even play once a week. --Dan
Neil Sloane <njasloane@gmail.com> wrote:
Remember the New Yorker cartoon of the guy saying to his microwave, No, I don't want to play a game of chess, I just want you to reheat the pizza? Maybe if you would play chess more often with yours, Dan, it would leave the handle at the front.
Yes, and there is something funny about a cup of coffee with cream, if you rotate the cup the cream will mix itself with the coffee (basic physics). So you can mix cream and coffee without using a spoon, ... If you explain that to an ordinary person, they usually look at you with strange eyes. I made the experiment many times just to see their faces. ha ha ha. Best regards, and have a mice coffee. Simon Plouffe
I usually put milk and sugar into my coffee cup first, and stir my drink using the turbulence that results from the final step: adding the coffee itself. And yes, I too get strange looks from people when I explain what I am doing. Jim Propp On Monday, January 27, 2014, Simon Plouffe <simon.plouffe@gmail.com> wrote:
Yes, and there is something funny about a cup of coffee with cream, if you rotate the cup the cream will mix itself with the coffee (basic physics). So you can mix cream and coffee without using a spoon, ...
If you explain that to an ordinary person, they usually look at you with strange eyes. I made the experiment many times just to see their faces. ha ha ha.
Best regards, and have a mice coffee.
Simon Plouffe
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On 1/27/2014 6:06 PM, James Propp wrote:
I usually put milk and sugar into my coffee cup first, and stir my drink using the turbulence that results from the final step: adding the coffee itself.
And yes, I too get strange looks from people when I explain what I am doing.
Jim Propp
I always do it that way. But if you put mice in first they'll stir it for you. Brent
On Monday, January 27, 2014, Simon Plouffe <simon.plouffe@gmail.com> wrote:
Yes, and there is something funny about a cup of coffee with cream, if you rotate the cup the cream will mix itself with the coffee (basic physics). So you can mix cream and coffee without using a spoon, ...
If you explain that to an ordinary person, they usually look at you with strange eyes. I made the experiment many times just to see their faces. ha ha ha.
Best regards, and have a mice coffee.
Simon Plouffe
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OK, well, now I can tell people that Jim Propp does it my way, so it must be close to optimal. The Propp method is also admirably suited for use with the Keurig machine. My wife, however, claims to have the optimal solution, which is to drink coffee black (omit the "pollutants"). However, my considered opinion is that the purpose of coffee is to warm and flavor a nice cup of cream.
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun- bounces@mailman.xmission.com] On Behalf Of James Propp Sent: Monday, January 27, 2014 9:07 PM To: math-fun Subject: Re: [math-fun] Microwave ovens
I usually put milk and sugar into my coffee cup first, and stir my drink using the turbulence that results from the final step: adding the coffee itself.
And yes, I too get strange looks from people when I explain what I am doing.
Jim Propp
On Monday, January 27, 2014, Simon Plouffe <simon.plouffe@gmail.com> wrote:
Yes, and there is something funny about a cup of coffee with cream, if you rotate the cup the cream will mix itself with the coffee (basic physics). So you can mix cream and coffee without using a spoon, ...
If you explain that to an ordinary person, they usually look at you with strange eyes. I made the experiment many times just to see their faces. ha ha ha.
Best regards, and have a mice coffee.
Simon Plouffe
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Jim's approach of adding tea (or, I assume, coffee) to the milk, according to the Richard and Louise Guy, is best because it minimizes the slight bit of curdling that would occur if you went the other way around. The fact that the tea is both hot and acidic is the reason. David On Wed, Jan 29, 2014 at 2:07 AM, David Wilson <davidwwilson@comcast.net> wrote:
OK, well, now I can tell people that Jim Propp does it my way, so it must be close to optimal. The Propp method is also admirably suited for use with the Keurig machine. My wife, however, claims to have the optimal solution, which is to drink coffee black (omit the "pollutants"). However, my considered opinion is that the purpose of coffee is to warm and flavor a nice cup of cream.
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun- bounces@mailman.xmission.com] On Behalf Of James Propp Sent: Monday, January 27, 2014 9:07 PM To: math-fun Subject: Re: [math-fun] Microwave ovens
I usually put milk and sugar into my coffee cup first, and stir my drink using the turbulence that results from the final step: adding the coffee itself.
And yes, I too get strange looks from people when I explain what I am doing.
Jim Propp
On Monday, January 27, 2014, Simon Plouffe <simon.plouffe@gmail.com> wrote:
Yes, and there is something funny about a cup of coffee with cream, if you rotate the cup the cream will mix itself with the coffee (basic physics). So you can mix cream and coffee without using a spoon, ...
If you explain that to an ordinary person, they usually look at you with strange eyes. I made the experiment many times just to see their faces. ha ha ha.
Best regards, and have a mice coffee.
Simon Plouffe
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participants (14)
-
Cris Moore -
Dan Asimov -
Daniel Asimov -
David Wilson -
David Wolfe -
Fred W. Helenius -
Hans Havermann -
Henry Baker -
James Propp -
Joerg Arndt -
meekerdb -
Mike Stay -
Neil Sloane -
Simon Plouffe