Re: [math-fun] Spirography
No. The gears shown in the picture, gosper.org/Moregears.png while as they are both can always be pulled apart orthogonally and don't work physically due to very slight intersection, can probably be modified so that they work physically and in some phases can't be pulled apart orthogonally. Combining this with the multilayer gear idea (so that you're always in such a phase), it would be possible to make a pair of gears which can't ever be separated orthogonally. They could probably still be separated without using the third dimension, though. Assuming that those modifications work. I might give it a try. Julian On Wed, Jun 18, 2014 at 4:54 PM, rwg <rwg@sdf.org> wrote: -------- Original Message -------- Subject: Re: [math-fun] Spirography Date: 2014-06-18 13:38 From: Allan Wechsler <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Reply-To: math-fun <math-fun@mailman.xmission.com> Gosper, is Julian all pessimistic because the classic spirograph has the rotor rolling around the inside of the stator? Does the problem persist if the rotor rolls around the outside? On Wed, Jun 18, 2014 at 4:23 PM, Steve Witham <sw@tiac.net> wrote: Date: Mon, 16 Jun 2014 11:54:40 -0400 [in issue 29]
From: James Propp <jamespropp@gmail.com>
Has anyone designed a mechanism that permits the spirographer to focus on circumferential force?
The knockoff product I have, has holes in both parts, suitable for pushing pins through, so in theory you only have to push on the pen tip. Rather than a wax tray the first thing I'd think of would be corrugated cardboard & the second, cork.
I'm imagining something like a latchable/unlatchable zipper.
That's a fascinating idea in itself. It could be used in funicular
railways, for instance. Although the zipper tooth design is asymmetrical, I have a coat that unzips from both ends. --Steve
Hello, here are some drawings made with epicycloids, http://www.plouffe.fr/nknmod M/ as you may know, epicycloids are gears in some way. the formula is simple n*2^n mod M where M is equal to various values, the drawings are mysterious and still unsolved as far as I am concern, I have lots of others at http://www.plouffe.fr/graphes/ but not arranged in a web page yet. for M = 131 and k=2 the graph is a complete graph, but for M = 1285 the period is 16*1285 = 20560 and really not like the case M= 131 and this is the whole point : why is it like that and why does it varies so much for no apparent reason. Another direction on the thing, you may try with M = n*257, nice too. ! Best regards, Simon plouffe 2014-06-19 12:58 GMT+02:00 Bill Gosper <billgosper@gmail.com>:
No. The gears shown in the picture, gosper.org/Moregears.png while as they are both can always be pulled apart orthogonally and don't work physically due to very slight intersection, can probably be modified so that they work physically and in some phases can't be pulled apart orthogonally. Combining this with the multilayer gear idea (so that you're always in such a phase), it would be possible to make a pair of gears which can't ever be separated orthogonally. They could probably still be separated without using the third dimension, though. Assuming that those modifications work. I might give it a try.
Julian
On Wed, Jun 18, 2014 at 4:54 PM, rwg <rwg@sdf.org> wrote:
-------- Original Message -------- Subject: Re: [math-fun] Spirography Date: 2014-06-18 13:38 From: Allan Wechsler <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Reply-To: math-fun <math-fun@mailman.xmission.com>
Gosper, is Julian all pessimistic because the classic spirograph has the rotor rolling around the inside of the stator? Does the problem persist if the rotor rolls around the outside?
On Wed, Jun 18, 2014 at 4:23 PM, Steve Witham <sw@tiac.net> wrote:
Date: Mon, 16 Jun 2014 11:54:40 -0400 [in issue 29]
From: James Propp <jamespropp@gmail.com>
Has anyone designed a mechanism that permits the spirographer to focus on circumferential force?
The knockoff product I have, has holes in both parts, suitable for pushing pins through, so in theory you only have to push on the pen tip. Rather than a wax tray the first thing I'd think of would be corrugated cardboard & the second, cork.
I'm imagining something like a latchable/unlatchable zipper.
That's a fascinating idea in itself. It could be used in funicular
railways, for instance.
Although the zipper tooth design is asymmetrical, I have a coat that unzips from both ends.
--Steve _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I wasn't able to get that link, or any of several variants I tried, to work. (Something to do with the /nknmod M/, presumably.) --Dan On Jun 19, 2014, at 4:13 AM, Simon Plouffe <simon.plouffe@gmail.com> wrote:
Hello,
here are some drawings made with epicycloids, http://www.plouffe.fr/nknmod M/
. . .
http://www.plouffe.fr/nknmod%20M/ On Thu, Jun 19, 2014 at 6:56 AM, Dan Asimov <dasimov@earthlink.net> wrote:
I wasn't able to get that link, or any of several variants I tried, to work.
(Something to do with the /nknmod M/, presumably.)
--Dan
On Jun 19, 2014, at 4:13 AM, Simon Plouffe <simon.plouffe@gmail.com> wrote:
Hello,
here are some drawings made with epicycloids, http://www.plouffe.fr/nknmod M/
. . .
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
Hello, no you are right it is the link http://www.plouffe.fr/nknmod%20M/ sorry there is a space between the d and the 2, which in web language is %20, does this count as a <word> in the computer language statistics ? best regards and bonne soirée en ce mois de juin si merveilleux. Simon Plouffe
Thanks, I think. --Dan On Jun 19, 2014, at 10:24 AM, Mike Stay <metaweta@gmail.com> wrote:
http://www.plouffe.fr/nknmod%20M/
On Thu, Jun 19, 2014 at 6:56 AM, Dan Asimov <dasimov@earthlink.net> wrote:
I wasn't able to get that link, or any of several variants I tried, to work.
(Something to do with the /nknmod M/, presumably.)
Therefore you are! On Thu, Jun 19, 2014 at 10:55 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Thanks, I think.
--Dan
On Jun 19, 2014, at 10:24 AM, Mike Stay <metaweta@gmail.com> wrote:
http://www.plouffe.fr/nknmod%20M/
On Thu, Jun 19, 2014 at 6:56 AM, Dan Asimov <dasimov@earthlink.net> wrote:
I wasn't able to get that link, or any of several variants I tried, to work.
(Something to do with the /nknmod M/, presumably.)
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
Mike -- I appreciate your belief in my existence. But there's a logical flaw in "I think, therefore I am": Since the premise assumes the existence of an "I" -- using "I" as its subject -- the conclusion is already present in the premise, and so the conclusion adds nothing. --Dan On Jun 19, 2014, at 1:25 PM, Mike Stay <metaweta@gmail.com> wrote:
Therefore you are!
On Thu, Jun 19, 2014 at 10:55 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Thanks, I think.
I always liked the contrapositive: I am not, therefore I don't think. Tom Dan Asimov writes:
Mike -- I appreciate your belief in my existence. But there's a logical flaw in "I think, therefore I am":
Since the premise assumes the existence of an "I" -- using "I" as its subject -- the conclusion is already present in the premise, and so the conclusion adds nothing.
--Dan
On Jun 19, 2014, at 1:25 PM, Mike Stay <metaweta@gmail.com> wrote:
Therefore you are!
On Thu, Jun 19, 2014 at 10:55 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Thanks, I think.
That is likely to be true for all sufficiently large values of t. Jim Propp On Friday, June 20, 2014, Tom Karzes <karzes@sonic.net> wrote:
I always liked the contrapositive:
I am not, therefore I don't think.
Tom
Dan Asimov writes:
Mike -- I appreciate your belief in my existence. But there's a logical flaw in "I think, therefore I am":
Since the premise assumes the existence of an "I" -- using "I" as its subject -- the conclusion is already present in the premise, and so the conclusion adds nothing.
--Dan
On Jun 19, 2014, at 1:25 PM, Mike Stay <metaweta@gmail.com <javascript:;>> wrote:
Therefore you are!
On Thu, Jun 19, 2014 at 10:55 AM, Dan Asimov <dasimov@earthlink.net <javascript:;>> wrote:
Thanks, I think.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Speaking of likeable comtrapositives, here's my all-time favorite (which I learned from Michael Larsen, who may have originated it): If the matrix M^{-1} is not invertible, then the matrix M is not invertible. :-) Jim Propp On Friday, June 20, 2014, Tom Karzes <karzes@sonic.net> wrote:
I always liked the contrapositive:
I am not, therefore I don't think.
Tom
Dan Asimov writes:
Mike -- I appreciate your belief in my existence. But there's a logical flaw in "I think, therefore I am":
Since the premise assumes the existence of an "I" -- using "I" as its subject -- the conclusion is already present in the premise, and so the conclusion adds nothing.
--Dan
On Jun 19, 2014, at 1:25 PM, Mike Stay <metaweta@gmail.com <javascript:;>> wrote:
Therefore you are!
On Thu, Jun 19, 2014 at 10:55 AM, Dan Asimov <dasimov@earthlink.net <javascript:;>> wrote:
Thanks, I think.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Dan, I disagree. The subject of a sentence doesn't have to exist and isn't assumed to exist. Of course, if the sentence is true, and says that the subject does something that requires existence, then the conclusion is warranted. In fact, try "I don't exist." --ms On 2014-06-20 00:45, Dan Asimov wrote:
Mike -- I appreciate your belief in my existence. But there's a logical flaw in "I think, therefore I am":
Since the premise assumes the existence of an "I" -- using "I" as its subject -- the conclusion is already present in the premise, and so the conclusion adds nothing.
--Dan
On Jun 19, 2014, at 1:25 PM, Mike Stay <metaweta@gmail.com> wrote:
Therefore you are!
On Thu, Jun 19, 2014 at 10:55 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Thanks, I think.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Unicorns don't exist. Charles Greathouse Analyst/Programmer Case Western Reserve University On Fri, Jun 20, 2014 at 10:03 AM, Mike Speciner <ms@alum.mit.edu> wrote:
Dan, I disagree. The subject of a sentence doesn't have to exist and isn't assumed to exist. Of course, if the sentence is true, and says that the subject does something that requires existence, then the conclusion is warranted. In fact, try "I don't exist."
--ms
On 2014-06-20 00:45, Dan Asimov wrote:
Mike -- I appreciate your belief in my existence. But there's a logical flaw in "I think, therefore I am":
Since the premise assumes the existence of an "I" -- using "I" as its subject -- the conclusion is already present in the premise, and so the conclusion adds nothing.
--Dan
On Jun 19, 2014, at 1:25 PM, Mike Stay <metaweta@gmail.com> wrote:
Therefore you are!
On Thu, Jun 19, 2014 at 10:55 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Thanks, I think.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
If that makes sense (and of course it does), then "Unicorns" is being used with two different meanings (hypothetical horselike creatures with certain qualities, and real ones with the same qualities). --Dan On Jun 20, 2014, at 9:57 AM, Charles Greathouse <charles.greathouse@case.edu> wrote:
Unicorns don't exist.
Doesn't the same apply, mutatis mutandis, to "I think, therefore I am"? Or was your point just that the RHS is merely a specialization of the LHS? Charles Greathouse Analyst/Programmer Case Western Reserve University On Fri, Jun 20, 2014 at 1:32 PM, Dan Asimov <dasimov@earthlink.net> wrote:
If that makes sense (and of course it does), then "Unicorns" is being used with two different meanings (hypothetical horselike creatures with certain qualities, and real ones with the same qualities).
--Dan
On Jun 20, 2014, at 9:57 AM, Charles Greathouse < charles.greathouse@case.edu> wrote:
Unicorns don't exist.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I don't believe self-interlocking gears can be made to work under the following conditions: 1. It "works" in 2D 2. Both gears turn at a constant velocity 3. Interlocking is by a single tooth, when centered on the line between the circles of the gears, being gripped by the two surrounding teeth of the other gear because its width is greater than the gap from the gripping teeth. 4. Both gears are the same size. The reasoning is as follows. For each gear, draw a circle through the "fattest" part of each gear, where the highest percentage of the circle is covered by the teeth, and such that the circles so drawn intersect. The "occupancy" of a gear on a circle is equal to the fraction of the circle covered by the teeth. Consider one of the two points at which those circles intersect. In order for the gears to mesh and spin without clashing, with both spinning at a constant rate, the occupancy of one gear plus the occupancy of the other must sum to 1 or less. Now consider the case when one tooth from either gear is centered on the axis between the two centers. If the occupancy is 1 or less, then the chord from where the edges of the tooth intersect that gear's circle, must be shorter than the chord from where the edges of the gap from the other gear intersect *that* gear's circle. Thus, that single tooth cannot be gripped by the gap from the other gear. This argument might be able to be generalized for the case that the gears are different sizes. So either there's got to be something going on in 3D, or the gears need to stutter, or the interlocking has to be something more complicated then just "grabbing" of a single tooth. -tom On Thu, Jun 19, 2014 at 3:58 AM, Bill Gosper <billgosper@gmail.com> wrote:
No. The gears shown in the picture, gosper.org/Moregears.png while as they are both can always be pulled apart orthogonally and don't work physically due to very slight intersection, can probably be modified so that they work physically and in some phases can't be pulled apart orthogonally. Combining this with the multilayer gear idea (so that you're always in such a phase), it would be possible to make a pair of gears which can't ever be separated orthogonally. They could probably still be separated without using the third dimension, though. Assuming that those modifications work. I might give it a try.
Julian
On Wed, Jun 18, 2014 at 4:54 PM, rwg <rwg@sdf.org> wrote:
-------- Original Message -------- Subject: Re: [math-fun] Spirography Date: 2014-06-18 13:38 From: Allan Wechsler <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Reply-To: math-fun <math-fun@mailman.xmission.com>
Gosper, is Julian all pessimistic because the classic spirograph has the rotor rolling around the inside of the stator? Does the problem persist if the rotor rolls around the outside?
On Wed, Jun 18, 2014 at 4:23 PM, Steve Witham <sw@tiac.net> wrote:
Date: Mon, 16 Jun 2014 11:54:40 -0400 [in issue 29]
From: James Propp <jamespropp@gmail.com>
Has anyone designed a mechanism that permits the spirographer to focus on circumferential force?
The knockoff product I have, has holes in both parts, suitable for pushing pins through, so in theory you only have to push on the pen tip. Rather than a wax tray the first thing I'd think of would be corrugated cardboard & the second, cork.
I'm imagining something like a latchable/unlatchable zipper.
That's a fascinating idea in itself. It could be used in funicular
railways, for instance.
Although the zipper tooth design is asymmetrical, I have a coat that unzips from both ends.
--Steve _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- -- http://cube20.org/ -- http://golly.sf.net/ --
participants (9)
-
Bill Gosper -
Charles Greathouse -
Dan Asimov -
James Propp -
Mike Speciner -
Mike Stay -
Simon Plouffe -
Tom Karzes -
Tom Rokicki