Talk:Plastic ratio

Question about the origin of the name[edit]

Is there a source for golden ratio being the origin of the name? Septentrionalis 05:14, 1 December 2006 (UTC)[reply]

Why is it called the "plastic" constant/number ? Fathead99 (talk) 17:17, 12 February 2008 (UTC)[reply]

I've added some more on this. —David Eppstein (talk) 18:00, 12 February 2008 (UTC)[reply]

Solution to the other equations[edit]

How was it discovered that the Plastic Number is also a solution to all those other equations? —The preceding unsigned comment was added by Vjasper (talk • contribs) 23:04, 8 January 2007 (UTC).[reply]

Most of them are the result of multiplying the original equation through by another polynomial and then simplifying; doing this as often as necessary.

For example, multiplying by x²,

x^{5}=x^{3}+x^{2}=x^{2}+x+1\;

This one can also be gotten by multiplication by x² - 1. Some of them are wrong; and they are all questionable as bordering on OR. Septentrionalis PMAnderson 23:28, 8 January 2007 (UTC)[reply]

Ummm .. I agree they are not all particularly interesting, but none of them are actually wrong - they can all be easily derived from x³=x+1, as follows:

(1)\ x^{3}=x+1\

(2)\ x^{5}=x^{3}+x^{2}{\mbox{  from (1)}}

(3)\ \Rightarrow x^{5}=x^{2}+x+1{\mbox{  from (1) and (2)}}

(4)\ x^{4}=x^{2}+x{\mbox{ from (1)}}

(5)\ \Rightarrow x^{5}=x^{4}+1{\mbox{ from (3) and (4)}}

(6)\ x^{6}=x^{4}+x^{3}{\mbox{ from (1)}}

(7)\ \Rightarrow x^{6}=x^{4}+x+1{\mbox{ from (1) and (6)}}

(8)\ \Rightarrow x^{6}=x^{2}+2x+1{\mbox{ from (4) and (7)}}

(9)\ x^{7}=x^{5}+x^{4}{\mbox{ from (1)}}

(10)\ x^{4}=x^{5}-1{\mbox{ from (5)}}

(11)\ \Rightarrow x^{7}=2x^{5}-1{\mbox{ from (9) and (10)}}

(12)\ x^{8}=x^{6}+x^{5}{\mbox{ from (1)}}

(13)\ \Rightarrow x^{8}=x^{4}+x^{3}+x^{2}+x+1{\mbox{ from (3), (6) and (12)}}

(14)\ x^{9}=x^{7}+x^{6}{\mbox{ from (1)}}

(15)\ x^{7}=x^{4}+x^{3}+x^{2}{\mbox{ from (3)}}

(16)\ \Rightarrow x^{9}=x^{6}+x^{4}+x^{2}+x+1{\mbox{ from (1), (14) and (15)}}

(17)\ x^{12}=2x^{10}-x^{5}{\mbox{ from (11)}}

(18)\ \Rightarrow x^{12}=2x^{10}-x^{4}-1{\mbox{ from (5) and (17)}}

(19)\ x^{14}=x^{10}+2x^{9}+x^{8}{\mbox{ from (8)}}

(20)\ x^{10}=x^{9}+x^{5}{\mbox{ from (5)}}

(21)\ x^{10}=x^{9}+x^{4}+1{\mbox{ from (5) and (20)}}

(22)\ x^{14}=3x^{9}+x^{8}+x^{4}+1{\mbox{ from (19) and (21)}}

(23)\ x^{9}=x^{8}+x^{4}{\mbox{ from (5)}}

(24)\ \Rightarrow x^{14}=4x^{9}+1{\mbox{ from (22) and (23)}}

Not saying these derivations need to appear in the article - just saying that all these relations are mathematically correct. Gandalf61 10:54, 9 January 2007 (UTC)[reply]

Very clever. Thanks for the explanation. Vjasper 01:38, 11 January 2007 (UTC)[reply]

The plastic number satisfies a polynomial equation p(x) = 0 iff p is a multiple of its minimal polynomial, namely x³ − x − 1. That's trivial algebra. As far as I can see, the whole section is patently useless, and should be deleted, unless there is some evidence that these particular multiples of x³ − x − 1 are not just random examples but have significance of their own, which I doubt. — Emil J. 11:53, 17 February 2009 (UTC)[reply]

I agree. The remark "it is also a solution of the polynomial equation p(x) = 0 for every polynomial p that is a multiple of x³ − x − 1, but not for any other polynomials with integer coefficients" is, as EmilJ says, trivial. If no one objects in the next week, I will remove it. --macrakis (talk) 07:21, 31 December 2009 (UTC)[reply]

I'm not objecting, but I'd like to point out that I said no such thing. My comment, as well as the preceding discussion, concerned an older version of the article, and the offending section was deleted some time ago. In fact, it was replaced with the very remark you quote, so that's supposed to be the solution, not the problem. — Emil J. 13:52, 4 January 2010 (UTC)[reply]

Understood. But is it necessary to say at all? One could just as well say in the i article that i is a root of not just x²+1, but of all its multiples, but not for any other polynomials with integer coefficients. As you say, that is trivial algebra. When you say "I'm not objecting", are you agreeing that that statement can be removed? --macrakis (talk) 16:33, 4 January 2010 (UTC)[reply]

I agree that there is no point in stating the obvious in such a hairy way. The only useful information in that sentence is that the minimal polynomial of ρ is x³ − x − 1, and this, too, follows immediately from the definition of ρ and the fact that ρ is not an integer (1 < ρ < 2), both of which are stated earlier in the article. Thus, I don't see any particular reason to retain the statement, I won't shed tears for it if it is removed from the article. OTOH, it does not really do any harm there, either. So, count me as indifferent to its fate. — Emil J. 18:16, 4 January 2010 (UTC)[reply]

Request to review edit by 165.234.104.24, 27 March 07[edit]

Could someone knowledgable in this subject please review the edit made by 165.234.104.24 on 27 March? This IP address has made a considerable number of vandalism edits on other articles, and I am reluctant to allow this edit to stand. -- Arwel (talk) 19:55, 10 April 2007 (UTC)[reply]

Looks ok to me. The new polynomial listed as having the plastic number as a root, x^7-2x^4-1, equals (x^3-x-1)(x^4+x^2-x-1), where the desired root occurs due to the left factor. —David Eppstein 20:38, 10 April 2007 (UTC)[reply]

OK, thanks. -- Arwel (talk) 21:45, 10 April 2007 (UTC)[reply]

External Link--Ian Stewart article[edit]

The external link, "Tales of a Neglected Number" written by Ian Stewart, from the June 1996 Scientific American, contains the line "In 1991 Steven Arno of the Supercomputing Research Center in Bowie, MD proved that Perrin pseudoprimes must have at least 15 digits." Obviously, the statement is false, as Perrin pseudoprimes with as few as 6 digits were discovered in 1982 by William Adams and Daniel Shanks. I recommend that the line above be removed from Wikipedia's version of Ian Stewart's article.75.165.242.83 (talk) 05:11, 8 January 2012 (UTC)[reply]

Wikipedia does not have a version of this article — it's on somebody else's web site that we have no control over. But probably we should cite it properly rather than linking to a copyright violation. —David Eppstein (talk) 05:46, 8 January 2012 (UTC)[reply]

Characteristic ratios?[edit]

How are the characteristic ratios of 3/4 and 1/7 determined?--Wikimedes (talk) 08:02, 4 August 2012 (UTC)[reply]

Good question. What does that sentence even mean? —Tamfang (talk) 08:40, 4 August 2012 (UTC)[reply]

It appears to be a close paraphrase from the Padovan (2002) reference. What it means, I don't know.--Wikimedes (talk) 16:56, 4 August 2012 (UTC)[reply]

Ten years later, we are no wiser :( —Tamfang (talk) 08:02, 11 March 2023 (UTC)[reply]

Problems with the infobox[edit]

Tracked in Phabricator
Task T50725

When I mouse over the "continued fraction" section of the infobox my cursor doesn't seem to notice any of the hyperlinks (it doesn't turn into a hand). Can anyone reproduce? Pokajanje|Talk 03:25, 22 May 2013 (UTC)[reply]

Yes, it behaves as you say in my browser. That’s most weird.—Emil J. 12:04, 22 May 2013 (UTC)[reply]

It's fixed now. I wonder why that was? Pokajanje|Talk 03:13, 23 May 2013 (UTC)[reply]

I forgot this discussion was in two places. See WP:Village pump (technical)#Odd link not clickable and follow the bug in the "tracked" template to the right → Technical 13 (talk) 09:54, 23 May 2013 (UTC)[reply]

How to denote[edit]

I notice that while the page denotes the plastic number as $\rho$ , some of the sources denote it as P or something else. Which way should the plastic number be denoted? 50.90.50.100 (talk) 22:31, 13 May 2015 (UTC)[reply]

I've seen it mostly denoted as

\psi

in the geometrical literature, and I feel tempted to replace it as such in the entire article. No identd (talk) 02:51, 8 November 2017 (UTC)[reply]

Square dissection[edit]

'There are precisely three ways of partitioning a square into three similar rectangles' What about splitting it into three squares, where two of the squares have a side length of zero? --Paul Murray (talk) 07:40, 20 December 2017 (UTC)[reply]

I suspect that would make those a degenerate rectangle or a degenerate square, no? --No identd (talk) 21:00, 21 December 2017 (UTC)[reply]

2nd formula[edit]

I've just removed this from the lead (following the first formula):

2\rho ={\frac {{\sqrt[{3}]{12{\sqrt {3}}+4{\sqrt {23}}}}+{\sqrt[{3}]{12{\sqrt {3}}-4{\sqrt {23}}}}}{\sqrt {3}}}

The second formula is interesting as in the case of an Equilateral triangle $12{\sqrt {3}}$ is the condition where the Perimeter of the triangle equals its Area.

I do not know if this formula should appear - or if there is a good source to support it. But there are several problems with the text, which need to be fixed if it is reinserted:

interesting - weasel word
case of - presumes a context in which an equilateral triangle is a "case", but no such context is given
Equilateral, Perimeter, Area - idiosyncratic capitalization
$12{\sqrt {3}}$ is the condition - how is a number a condition? I suppose what is meant is that an equilateral triangle has perimeter=area if some characteristic of the triangle equals this number, but which characteristic is that? (A good guess would be that this number is the area and the perimeter, but don't leave us guessing!)
So, the number $12{\sqrt {3}}$ shows up in a certain triangle, but what, then, about other elements in the formula such as $4{\sqrt {23}}$ , or what about the formula as a whole, or the plastic number?
Even if the above is fixed, it may not be clear that this information is significant enough to make it into the article, let alone into the lead.

--Nø (talk) 15:20, 19 December 2019 (UTC)[reply]

Short description[edit]

@Tamfang: re "or is there an esoteric meaning of integer"? See algebraic integer. But I agree that your short description is better per WP:SDNOTDEF, because less likely to confuse readers who are just using short descriptions to disambiguate searches rather than to provide the most accurate definition in the fewest characters. —David Eppstein (talk) 06:07, 3 February 2024 (UTC)[reply]