Talk:Spiral of Theodorus

Einstein

Who is Einstien? For a moment I thought this was a misspelling of "Einstein", but the former spelling gets a lot more google hits than the latter. Michael Hardy (talk) 03:55, 1 May 2008 (UTC)

True. It was a mispelling of "Einstein." Don't know about the Google hits with "Einstien." --pbroks13^talk? 05:30, 2 May 2008 (UTC)

Self-contradictory

It says in the article that the distance between one level and the next approaches a constant value, π, which would imply that this spiral approximates an Archimedean spiral. But it also says that it approximates a logarithmic spiral. Which is it? It can't be both. —David Eppstein (talk) 02:57, 2 May 2008 (UTC)

I researched a little more, its the Archimedes spiral, i'll fix it now. --pbroks13^talk? 05:39, 2 May 2008 (UTC)

Einstein spiral?

Does anyone have reliable citations for this alternate name? Google sure doesn't. 183 hits for "Einstein spiral", most of which are trying to sell me an "Albert Einstein Spiral of Peace World War 4 T-Shirt" or a "365 Days of Baby Einstein (spiral bound book)". 70.20.160.187 (talk) 06:57, 11 January 2009 (UTC)

It's actually included already in one of the refs – this PDF. --Pbroks13^talk? 18:50, 11 January 2009 (UTC)

Unfortunately, that is not a reliable source (in the Wikipedia meaning of the term), because papers on the arXiv are not peer-reviewed. -- Jitse Niesen (talk) 20:29, 11 January 2009 (UTC)

Why no pages with right-angled-triangle-spirals, using higher valued roots?

Why no pages with right angled triangle spirals, using higher valued roots? Square root triangles have an adjacent side equal to (n)^(1/d), a hypotenuse equal to (n+1)^(1/d) and an opposite side equal to ((n+1)^(2/d)-(n)^(2/d))^(1/2) , where d=2. Simply increase the value of d, to obtain other roots. The cube root spiral, not only grows, with upright triangle numbers having positive values of (n), but also with negative values of (n).Cuberoottheo (talk) 11:01, 6 April 2014 (UTC) Plotting the curve for the difference between the hypotenuse and its adjacent, r(n)=(n+1)^(1/3)-(n)^(1/3), for negative as well as positive values of triangle number, shows that only one triangle zero is required. Unfortunately from the point of view of cutting out triangles, it also needs many fractional negative values, between triangle zero and triangle minus one. These tiny fractional negative triangles, have values like, -1,...,-4/5,-3/4,-2/3,-1/2,-1/3,-1/4,-1/5,...,0. Cuberoottheo (talk) 07:36, 9 April 2014 (UTC) For example, if (n = -1/2) then r(n) = (+1/2)^(1/3)+(1/2)^(1/3) = +1.58740105, as if the positive spiral is being squashed towards n = -1/2, and then having turned its triangles inside out, expands spiralling the other way around.Cuberoottheo (talk) 13:04, 9 April 2014 (UTC) Triangles, n = +8, opposite = 0.5716199, n = +7, opposite = 0.5836908, n = +6, opposite = 0.5978095, n = +5, opposite = 0.6147444, n = +4, opposite = 0.6357483, n = +3, opposite = 0.6631416, n = +2, opposite = 0.7019145, n = +1, opposite = 0.7664208, n = +0, opposite = 0.0, adjacent and hypotenuse = +1. Triangles with fractional negative values down to n = -1/3, hypotenuse = ((n+1)^(2/3)+(n)^(2/3))^(1/2), adjacent = (n+1)^(1/3), opposite = (n)^(1/3). Triangle n = -1/2, +1.2599192, +0.793700, -0.793700. Triangles from n = -2/3 down towards n = -1, hypotenuse = (n)^(1/3), adjacent = (n+1)^(1/3), opposite = ((n+1)^(2/3)+(n)^(2/3))^(1/2). Triangle n = -1, hypotenuse = (n)^(1/3) = -1, adjacent = +1, opposite = (n+1)^(1/3) = 0. Triangle n = -2, hypotenuse = (n)^(1/3) = -1.259921, adjacent = (n+1)^(1/3) = -1, opposite = ((n+1)^(2/3)-(n)^(2/3))^(1/2) = 0.7664208i. Triangle n = -3, opposite = 0.7019145i, triangle n = -4, opposite = 0.6631416i, having opposite sides with imaginary numbers, just means that they are upside down and at a right angle to the their triangle`s adjacent sides. The square of the adjacent side of a square root triangle, equals area(n), the cube of a cube root triangle`s adjacent side equals volume(n). The sum of the cube root spiral`s, triangle`s volumes(adjacent sides cubed), from n = +1 up to (+n), is given by the arithmetic progression v+2v+3v+...+(n)v = (n(n+1))v/2. The sum of the square root spiral`s, triangle`s, squared adjacent sides, is also given by the sum of an arithmetic progression, a+2a+3a+...+(n)a = (n(n+1))a/2 .Cuberoottheo (talk) 14:45, 12 April 2014 (UTC) Maybe there are no pages with right angled triangle spirals, using higher valued roots, because nobody else has written about the subject yet?Cuberoottheo (talk) 12:11, 13 April 2014 (UTC) The volume of a balloon depends on the number of puffs of air put into it, the radial difference equation r(n)=(n+1)^(1/3)-(n)^(1/3), suggests that as more puffs are put in, the smaller each additional radial increase becomes, (until the balloon pops). Rather like a cube root spiral, a balloon can be inflated from either its inside, or its out side, even if it was inside out before being inflated. .Cuberoottheo (talk) 12:35, 20 April 2014 (UTC)

Simple explanation please

I'm not a math guy.

So ... can we say simply what the point of this is?

Far as I can tell, the cool thing about this is that the length of the outside side of the triangle is always 1.

Or ... what's the cool thing about this spiral? Can someone say it in simple terms?

Thanks! — Preceding unsigned comment added by 184.96.108.119 (talk) 19:34, 26 June 2016 (UTC)

Some interesting aspects of the Theodorus Spiral

Firstly the spokes never overlap, which is incredible.

Secondly as windings go round the additional number of spokes is seemingly random, which is surprising. For example in the first 100,655 windings the additional number of spokes is 18, 179 times, 19, 3612 times, 20, 4928 times and 21, 3612 times. Read More — Preceding unsigned comment added by Kesbooks (talk • contribs) 03:58, 28 December 2016 (UTC)