Talk:Squircle

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An entry from Squircle appeared on Wikipedia's Main Page in the Did you know? column on 22 November 2006.

1 something must be wrong...
2 dinner plates?
3 Squircle or squirtle?
4 Squircle vs. Rounded square
5 An odd coincidence
6 Tell me more

[edit] something must be wrong...

Just glancing at the picture, you can see that the points on the "squircle" aren't equidistant from the circle and the square (and I don't think it's an optical illusion). Either the text is wrong, or the picture. One or the other.--345Kai 08:58, 22 November 2006 (UTC)

As far as I can see, the squircle illustrated is a superellipse, and yes, I agree with you that the construction doesn't produce the given curve, so I've removed it from the article. -- The Anome 11:03, 22 November 2006 (UTC)

After annoying some people, I think I have independently found where the problem is:

The squircle is a specific case (found by setting

n = 4

) of the class of shapes known as supercircles, which have the equation

$\left( x - a \right)^n + \left( y - b \right)^n = r^n$ .

Unfortunately, the taxonomy is not consistent - some authors refer to the class as supercircles and the specific case as a squircle, while others adopt the opposite naming convention. Supercircles in turn are a subgroup of the even more general superellipses, which have the equation

$\left|\frac{\left( x - a \right)}{r_a}\right|^n\! + \left|\frac{\left( y - b \right)}{r_b}\right|^n\! = 1$ ,

where

r a

and

r b

are the semimajor and semiminor axes. Superellipses were extensively studied and popularised by the Danish mathematician Piet Hein.

Please check the following link for a different definition where the square is inscribed inside the circle instead of the circle being inscribed inside the square. different squircle Is this the correct difinition or will both definitions work? It would be very nice to have matlab visually confirm either or both of these definitions with respect to the formula that purportedly defines a squircle. I don't have the technology to do this right now. I reformatted the comment immediately above so that it is indented the same amount throughout. I hope I did not cause irritation. Allan --68.73.227.43 15:15, 22 November 2006 (UTC)

You're correct - the description given there would appear to give the correct properties of the squircle, I must have got the square and circle the wrong way around. Modest Genius ^talk 21:09, 22 November 2006 (UTC)

[edit] dinner plates?

The reference provided for dinner plates is insufficient. It doesn't say whether or not those plates are actually mathematical "squircles" or just random shapes somewhere in between a square and a circle.--345Kai 09:03, 22 November 2006 (UTC)

The "cupboards" bit needs changing. Squircles packing may be less than, equal to or better than circle packing depending on the parameters. Rich Farmbrough, 10:53 22 November 2006 (GMT).

[edit] Squircle or squirtle?

This article doesn't seem to be consistent about the terms "squircle" vs. "squirtle". -Chinju 11:55, 22 November 2006 (UTC)

I think someone was having a joke at the mathematicians' expense: Squirtle is a Pokémon! I've changed all instances of 'squirtle' to 'squircle'. DAllardyce 11:59, 22 November 2006 (UTC)

This shape or something near it was a major element in 1960s design, and not just "modern" dinnerware and tumblers. A Scandinavian architect-designer made a brief splash with it as his "invention": his name is long gone from my fading brain. --Wetman 14:36, 22 November 2006 (UTC)

It's even in the article: Piet Hein ;) Modest Genius ^talk 21:10, 22 November 2006 (UTC)

[edit] Squircle vs. Rounded square

Why is the rounded square so difficult to generalize? What would its formula look like? Circeus 14:58, 22 November 2006 (UTC)

Because you would have to have at least 6 different defined curves (two of them being sets of parallel straight lines and 4 arcs of circles), each valid between different limits. I could scribble it down for you, but writing it out in math wikicode would take ages. As an example, one of the sections would be (x-r)^2+(y-r)^2=(r/4)^2, valid in the interval x>r AND y>r. there would be 4 of those, plus some straight lines. Modest Genius ^talk 21:16, 22 November 2006 (UTC)

[edit] An odd coincidence

Thank you for writing an interesting article, Keith. By coincidence I signed up as a Wikipedia user on November 22, 2006, the same day this article was featured on the main page. I didn't run across this article until today, though.

Here's what makes that simple coincidence seem like an odd coincidence to me. I won second prize in the math division of the International Science Fair in 1969 with a paper entitled "The Hypercircle". The "hypercircle" I wrote about was the locus of points satisfying the equation x⁴ + y⁴ = r⁴ in the Cartesian plane, where r is an arbitrary positive constant (the "hyperradius"). This corresponds exactly with the article's definition of a "squircle" centered on the origin (0, 0). Besides calculating the circumference of the "hypercircle" (using Simpson's rule and an adding machine – I didn't have access to a digital computer, since those were pretty hard to find in 1969) and the length of the "minor diameters", I also worked out the properties of "hypercircular functions", analogues of the familiar trigonometric functions. The "hypersine" was just the altitude of a right triangle with central angle θ and inscribed inside the "hypercircle", the "hypercosine" was the base of the same triangle, and so forth. Oh, yeah – I also expanded the "hypercircular" functions as power series, and tied them to a class of definite integrals whose exact form I can't recall right now, except that the integrand was the inverse of the square root of a polynomial. That paper has to be lying around here somewhere ... if I run across it I may add some of that stuff to this article (or maybe link to it somehow).

I do remember the judges asking me if I had considered redefining angular measure in terms of the arc length of this particular curve – that notion seemed foreign to me at the time, because I was unfamiliar with contour integrals. I also remember some discussion about whether I might have stolen the idea from somebody else. I was entirely unaware of Piet Hein's work with a "squircle" back then, although I had already read some of his little "Grooks", which remain among my favorite poems.

Well, I've rambled on too long already. Thanks again for making me smile! DavidCBryant 17:45, 28 November 2006 (UTC)

Wow, that IS an odd coincidence ;). Please feel free to add any of that work to the article! The idea of redefined sine and cosines is especially interesting, any idea what possible use these would be? Modest Genius ^talk 19:06, 28 November 2006 (UTC)

Would the redefined trig functions be called the squine and cosquine? Argyriou (talk) 00:33, 29 November 2006 (UTC)