Talk:Archimedean spiral/Archive 1

This is an archive of past discussions about Archimedean spiral. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page.

Archive 1

Definition Locus vs. polar coordinates

I really think the locus definition should come first. It is the true origin of the curve (there were no functions back then) and it gives the reader a better sense of what it is. The polar coordinates definition is no more precise and, for most people, totally meaningless.

Rather, than just changing it back to my last edit we should talk about this. I mean it's like saying a circle is "r^2 = x^2 + y^2" that only works in ONE context: the cartesian plane-- the locus definition is useful in any context-- futurebird 04:00, 13 June 2006 (UTC)

Here is an even better one:

If a fly crawls radially outward along a uniformly spinning disk, the curve it traces with respect to a reference frame in which the disk is at rest is an Archimedean spiral (Steinhaus 1999, p. 137).

futurebird

Modern notation should come first, historical notes later. There's no point in writing up a mathematical article in words when there's a perfectly good 6-character equation that describes the same thing. As a historical note, it is interesting; that's why I think the second paragraph is the perfect spot for it. As for what lay readers will get the most out of, it's the picture, not the equation nor textual description. -- Xerxes 16:49, 13 June 2006 (UTC)

There is also a neuropsychiatric application -- the patient's drawing of this spiral is analyzed much as it their handwriting67.133.62.42 23:47, 10 May 2007 (UTC)

Mirror image of an Archimedean Spiral

"Taking the mirror image of this arm across the y-axis will yield the other arm."

If I flip the Archimedean_spiral.png across any single axis, the two arms will intersect regularly. If I flip it across both axes (or rotate it 180 degrees - same thing), they will nest, only intersecting at the origin. Maybe it's a terminology difference, but that's what comes to mind for me when I envision the other arms of a spiral. 206.124.146.40 (talk) 23:27, 23 December 2007 (UTC)

It depends on what the equations say when you plug in negative degrees, not on what's prettiest... AnonMoos (talk) 15:47, 28 December 2007 (UTC)

a and b

What are a and b supposed to represent? I know they are real numbers, but what do they actually represent in relation to the image of the spiral? —Preceding unsigned comment added by 77.97.89.135 (talk) 21:17, 22 January 2008 (UTC)

Rectangular equation

Out of curiosity, is it, in any way, shape or form, possible to express the Archimedean spiral in terms of a single equation with rectangular coordinates? Thanks if anyone can answer this. --70.124.85.24 (talk) 18:56, 1 March 2008 (UTC)

Basic substitution gives the following basic equation (first quadrant only):

sqrt(x²+y²)=arctan(y/x)

Where the arctangent function is not confined to giving values less than 2π. You could make it more complex to account for all four quadrants, and various constant parameters... AnonMoos (talk) 22:54, 1 March 2008 (UTC)

Superimposed Spirals

The main article could be improved if there were a discussion of the moire effect you get when superimposing two spirals on each other, where there is a displacement between their two points of origin, and one of the spirals has slightly different widths between the spiral arms? 216.99.201.235 (talk) 01:07, 15 June 2009 (UTC)

Not sure that has any strong association with Archimedean spirals (as opposed to other forms of spirals)... AnonMoos (talk) 02:26, 15 June 2009 (UTC)

clarkson formula

I still think that chapter should be removed and the Redirect was a very unfortunate decision. Now we have questionable section with an unconfirmed term in an article that was otherwise fine. Also note the google search results of LLC books, that were used as indirect evidence are not really usuable or reliable without checking first hand, since LLC does not only reprint older royalty free books, but it does also print books assembled out of selected WP material (a story regarding that was recently published in German news outlets). Hence there is a real danger of a circular referencing here and that WP promotes term that is unconfirmed was more or less non existant before. The redirect made that problem actually much worse, since Archimedan spiral has a much higher exposure and many uncritical readers not paying much attention to footnotes or the discussion page make take the name "clarkson formula" for granted.--Kmhkmh (talk) 13:11, 23 September 2010 (UTC)

If you think the closing admin (Timotheus Canens) made the wrong decision when closing Wikipedia:Articles for deletion/Clackson scroll formula then you should take this to Wikipedia:Deletion review. Gandalf61 (talk) 14:06, 23 September 2010 (UTC)

Well I I already asked him to take a second look for now. But independent of that that if the content is now here, it has an issue of this article as well. Or rather before the article had no quality issue, but now it has one.--Kmhkmh (talk) 14:41, 23 September 2010 (UTC)

P.S.: Ok the admin has answered and the redirect was chosen, because the content was already moved here, which I had non been aware of. So basically it is now a question of this article.--Kmhkmh (talk) 15:07, 23 September 2010 (UTC)

If the closing admin believed that the AfD consensus was in favour of deleting the Clackson scroll formula then their correct course of action was to remove the Clackson scroll section from this article (which you can still do yourself if you disagree with it) and then delete the original article. Creating a redirect was just a bad call on their part. After you questioned their closure, they started this discussion on my talk page. Gandalf61 (talk) 16:09, 23 September 2010 (UTC)

Due to the discussions above I removed the section below and I'll request a delettion of redirect. If anyone is interested he may to include the approximation in the article again, he may feel free to do so. However please do not readd any reference to Clackson or blacksmithing, unless you have a new reliable source that you've checked yourself. --Kmhkmh (talk) 20:06, 23 September 2010 (UTC)

Clackson scroll formula

The Clackson scroll formula,^[1]^{[unreliable source?]}^{[original research?]}

\ell =\pi sn^{2},\,

is used in blacksmithing to estimate the length c of stock required to produce a scroll in the form of an Archimedean spiral of n turns with a (on-center) spacing s between the turns.^{[citation needed]}

This estimate can be derived by thinking of the scroll as having a thickness of s and being wound tightly so that there is no space between the layers. When wound up the scroll is roughly a cylinder with a radius of sn, so its cross-sectional area is roughly πr² = πs²n². On the other hand, the scroll's cross-sectional area whether wound up or not is ℓs. Thus ℓs = πs²n², or ℓ = πsn².

construction

I added the construction from a historical American source (def. out of copyright worldwide) Math objects have a rich cultural history and aesthetic value, so I wanted to reflect that as well as practically describe how to construct one on paper. The method is not common knowledge, and therefore of value in a reference work. — Preceding unsigned comment added by 71.188.235.93 (talk) 15:14, 29 September 2011 (UTC)

That construction description appears to explain how to draw involute of circle, not archimedean spiral. Note the animated unwinding example in Involute. The contruction section should probably be removed.

Rcgldr (talk) 15:16, 26 December 2011 (UTC)

Hello all,

I just added the subsection "Separation distance" in which I state that the involute of a circle is a spiral with constant separation distance between successive coils, whereas the Archimedean spiral lacks this property.

But the two kinds of spirals differ very little in their outer regions, so there an "easy" construction of an involute of a circle as an approximation to a real Archimedean spiral can really be much more practicable than an "exact" construction of the Archimedean spiral with sophisticated and elaborated mechanisms with lots of possible inaccuracies. Problems of (practical) inaccuracy occur only for quite small radii.

-- Enyak (talk) 21:14, 27 February 2012 (UTC)

Spiral with constant separation distance of its turnings ?

The Archimedean spiral is often characterized as "spiral with constant separation distance of its turnings" , which is wrong, seen from a mathematical viewpoint. This is the reason for the changes I submitted. I have given an exact description of the constant distances which can be seen in the Archimedean spiral, but I would like to point out in addition, that these constant distances cannot be denominated as "constant distances between successive turnings of the spiral".

My message to Gandalf61:

Hello Gandalf61, I just noticed that you have deleted a part of my changes to this article which I find however quite important. I wanted to point at the often misleading characterization of the Archimedean spiral as a "spiral with constant separation distance of its turnings", which is wrong seen from a mathematical viewpoint. I have first edited the german article :

http://de.wikipedia.org/wiki/Archimedische_Spirale#.22Windungsabstand.22

and would like now to add a similar remark to the english article. I might however transfer it to a little subsection if this would help. -- Enyak (talk) 13:49, 27 February 2012 (UTC)

My reply: Once you had corrected the description of the spiral in the article, I couldn't see much point in re-stating the incorrect description and then just saying it was "somewhat misleading" with no further explanation. This will confuse a reader who has not come across the incorrect description in the first place. If you want to add this point to the article, I think you should (a) find a source (outside of Wikipedia) that uses the incorrect description and then (b) explain exactly why this description is not correct. Gandalf61 (talk) 15:42, 27 February 2012 (UTC)

It makes sense to reiterated that in more detail as the misleading description can found in (reputable) literature (see for instance A Primer on Logarithms, Nonplussed!: mathematical proof of implausible ideas, The universal book of mathematics, Chaos and fractals: new frontiers of science, Symmetry for some math books as well as in various science and engineering books [1], [2], [3]). In other words this "incorrect" or "different" use of the term distance in the context of spirals seems to be common enough to warrant a detailed explanation/clarification (as intended by enyak).--Kmhkmh (talk) 15:54, 27 February 2012 (UTC)

It should be handled in a similar manner as in the German article. Either not giving any information on the "distance" or "thickness" of the spiral arms at all or explaining in detail that the "(constant) distance of spiral arms" differs from the usual notion of distance between non intersecting differentiable planar curves or parallel curves (namely that the orthogonality for the tangents does not hold). Personally I'd prefer the latter as it provides more information to readers and clarifies the actual situation, hence avoiding potential misunderstandings. Since the math part of such a detailed explanation should be fairly evident to anybody with some background in undergraduate math I don't think a particular reference/citation for such an explanation (as presumably intended by enyak) is needed.--Kmhkmh (talk) 15:42, 27 February 2012 (UTC)

Hello Gandalf and Kmhkmh, I have now added the subsection "Separation distance" in which I explain the difference between the two viewpoints geometrically in a few words. Of course one might give a few links to sources with the discutable usage of the term "constant distance between coils". I would place these links after the word "widespread" in the sentence

The description of the Archimedean spiral as a spiral "with constant separation distance between successive coils" is fairly widespread (*links*) but somewhat misleading.

Since I am a beginner here, I need some time to learn about putting links in a text correctly ...

-- Enyak (talk) 18:14, 27 February 2012 (UTC)

I have added a source that uses the "constant separation" description, so you can see how it is done. Feel free to add more sources. Gandalf61 (talk) 09:13, 28 February 2012 (UTC)

Tkank you for adding the source and improving the wording. --Enyak (talk) 11:19, 28 February 2012 (UTC)

Comparison of Archimedean spiral vs. involute of circle

Made a direct comparison graphic... AnonMoos (talk) 14:46, 1 May 2012 (UTC)

Calculus

This page should have some discussion of the importance to calculus of this spiral. Specifically, this was Archimedes's only foray into differential calculus (the first by any human being) when he calculated the tangent to a given point on the spiral. From Men of Mathematics: "Anticipating Newton and Leibniz by more than 2000 years he invented the integral calculus and in one of his problems anticipated their invention of the differential calculus ... The problem in which he used the differential calculus was that of constructing a tangent at any given point of his spiral." — Preceding unsigned comment added by Rene42 (talk • contribs) 17:40, 7 December 2014 (UTC)

Length

I am wondering about the arc length of the Archimedean spiral, in particular, at $\theta =2\pi$ and $r=1$ . Why is there not a section about this? Is it ${\sqrt {1+(2*\pi )^{2}}}$ which is OEIS: A233700? John W. Nicholson (talk) 21:34, 10 March 2015 (UTC)

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^ S.G. Clackson, "The Trinity Church Screen", SCAT Report 1981

[1] S.G. Clackson, "The Trinity Church Screen", SCAT Report 1981

[1]