Talk:Vesica piscis

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[edit] Use

When and by whom was this symbol used historically? --FOo 06:07, 18 Jul 2004 (UTC)

Pythagoreans and early Christians, and, according to the originator of the article, 'Pagans' (though this is an increadibly vague term). CheeseDreams 18:39, 31 Oct 2004 (UTC)

[edit] Shape

Please note that the Vesica Piscis is the fish shape NOT the overlapping circles. Vesica Piscis is latin for fish face.CheeseDreams 18:40, 31 Oct 2004 (UTC)

Please see the new diagram for explanation. CheeseDreams 21:30, 3 Nov 2004 (UTC)

I'm not sure. I've seen vesica piscis used to refer to the almond-shaped (or, if you're Pagan, vulva-shaped) region formed by the overlapping circles, without the "fish tails" found on the ichthus. See, e.g.: [1] [2] [3]—FOo 23:37, 3 Nov 2004 (UTC)
There is an easy way to settle this issue. If someone could measure the width of the whole shape (including fish tails) and the height, and also measure the width of the shorter tailless shape. And then inform what the height-width ratio is of each of the two shapes. It is quite late in UTC land, so I, myself, will not check it till later.CheeseDreams 00:24, 4 Nov 2004 (UTC)

Here is a detailed link. It is briefly mentioned on this website, and at the bottom of the page on this website. CheeseDreams 19:45, 4 Nov 2004 (UTC)

[edit] Merge to Fish shaped religious symbol thing

I would like to propose merging this article with that of Ichthys. The object in question is the same, and they are really just uses of the same thing by different peoples. I think there should be a section on "Christian use of THAT SHAPE" and "Pre-Christian use of THAT SHAPE" in the surviving article, and then any other bits that are in each article as well. However, I do not know what the resulting article ought to be called, maybe Fish shaped religious symbol thing CheeseDreams 20:25, 2 Nov 2004 (UTC)

I'd say they're more like twins, raised apart. They have different identities, different religions, and despite a family resemblance they don't look quite the same any more (one being geometric, the other freehand). Even the lack of a common name underscores how forced and difficult it would be to put them together. Leave them with their own pages, each with a link to his "brother". Tverbeek 02:06, 15 Dec 2004 (UTC)

[edit] Formula wanted

Any mathematicians out there that can conjure up a formula for: (a) Area of overlap when both circles are equal size? (b) Area of overlap when the circles are unequal size? 172.152.244.122 02:04, 10 February 2006 (UTC)

See Mrs. Miniver's problem for a little discussion... AnonMoos 15:57, 24 February 2006 (UTC)

[edit] "Pythagorean theorem"

Dear "Noe", the Pythagorean theorem has a somewhat tangential relationship to explaining the fact under discussion (since one must FIRST do a fair amount of geometry before being able to apply the Pythagorean theorem) -- but the equilateral triangles can be DIRECTLY SEEN in the construction of the Vesica Piscis (as shown in the diagram). The sqare root of three comes in from the well-known property that if the side of an equilateral triangle is 1, then the height of the triangle -- from center of base to vertex -- is half the square root of three. You could indeed derive this property by means of the Pythagorean theorem, but only if you constructed the triangles first. AnonMoos 02:34, 8 May 2006 (UTC)

I teach an international group of children in an IB school, and though it may depend on different math teaching traditions in different countries, I find that the Pythagorean theorem is known to nearly everyone, but the height of an equilateral triangle is not.--Niels Ø 10:15, 10 May 2006 (UTC)
That's nice -- you can't apply the Pythagorean theorem unless you hjave FIRST constructed the triangles, can you now? AnonMoos 11:29, 10 May 2006 (UTC)
I don't know what the point of that was meant to be, but here is my point (again): I think the explanation I shortened down is too long for the present purposes, and it does not reach the desired conclusion, unless one consideres finding the height of an equilateral triangle to be trivial. So, either the explanation should be finished, making it even longer, or it should be replaced by something shorter and less complicated, that doesn't pretend to be a complete proof. I tried to do the latter, and I still think that is the best choice. The pair of facts that (i) the Pythagorean theorem is involved, while (ii) the Pythagoreans believed in a different result, seems mildly amusing to me, which is why I mentioned the Pythagorean theorem here, but it may really be irrelevant here.--Niels Ø 15:48, 10 May 2006 (UTC)
I really don't understand how your Triangophobia is relevant to the article, but I do know that the fact that the centers of the two circles and the two circle intersection points are at the vertices of two back-to-back equilateral triangles (as shown in red in the construction diagram) is a hightly relevant fact about the geometry of the Vesica Piscis -- and that your edit suppresses all mention of this relevant geometric fact, and replaces it with a vague hand-waving invocation of the Pythagorean theorem (which is utterly useless to explain anything unless you construct the triangles first). AnonMoos 04:36, 11 May 2006 (UTC)

Things that I fail to understand are happening in the section ==Mystical and religious significance==. What does lengthy geometrical arguments and lists of rational approximants have to do with that section? Please explain!--Niels Ø 18:19, 15 May 2006 (UTC)

At least he's not deleting relevant information, the way you have always done... AnonMoos 03:42, 16 May 2006 (UTC)
Now, please be civil and get your facts right. I deleted it from a section where it was irrelevant, and when it was pointed out - correctly - that it was relevant, I added it to the section where it was relevant. Is that a problem? Your using the word "always" suggests you have checked my contributions list, but your accusation tell me you haven't. Your using the word "he" disguises the fact it's yourself you are talking about.--Niels Ø 07:21, 16 May 2006 (UTC)
I'm sorry, but in every single one of your edits, as of 03:42, 16 May 2006, you deleted relevant information FROM THE ARTICLE AS A WHOLE. AnonMoos 16:41, 16 May 2006 (UTC)

What useful information did I remove in the edit [4]? Listing other rational approximations to \sqrt{3} is hardly relevant here. Anyway, I'm glad the two of us are no longer alone in this field, and I warmly welcome user:The Anomes action (see following section in this discussion).--Niels Ø 20:22, 16 May 2006 (UTC)

I'm the one who added the list of rational approximations, replacing this sentence:
The fraction 265:153 is a ratio of whole numbers under 1000 which approximates the square root of 3 (though 362:209 and 989:571 are actually closer approximations).
— which made the choice of approximations seem arbitrary. On the contrary, for any irrational number there is a well-defined sequence of best rational approximations, and you can always do better with big enough integers. I could have expressed that better. The symbolic significance of 153 is interesting and (though I'm not religious) I think it ought to be restored somehow. I wonder whether Dr.I.J.Matrix is aware of it. —Tamfang 05:03, 18 May 2006 (UTC)
I actually agree a little with Noe on this one -- some editor(s) seemed to claim that 265:153 was a remarkably close approximation to the square root of 3, so other editor(s) added the cautionary note that 362:209 and 989:571 were even better, but there's no real reason to go into a whole discussion of fractional rational approximations here... AnonMoos 15:51, 21 May 2006 (UTC)
And (not surprisingly) I agree with AnonMoos here... I'm very interested in rational approximants; in fact, I have written a (so far unpublished) article about them, continued fractions, and related topics. It's just not relevant here. Anyway, 265:153 is a so-called best approximant, as all fractions with smaller denominator (or with smaller numerator) are poorer approximations. Comparing it to other approximants involving numbers less than 1000 is only relevant in the context of a decimal system, where limiting numbers to 3 decimal digits imposes that limit.--Niels Ø 16:41, 21 May 2006 (UTC)

[edit] The "measure of the fish"

I have moved the section below out of the article because of concerns about verifiability.

From the article:

Mystical and religious significance
It has been the subject of mystical speculation at several periods of history, perhaps first among the Pythagoreans, who considered it a holy figure. The mathematical ratio of its width (measured to the endpoints of the "body", not including the "tail") to its height was reportedly believed by them to be 265:153. This ratio, equal to 1.73203, was thought of as a holy number, called the measure of the fish. The geometric ratio of these dimensions is actually the square root of 3, or 1.73205... (since if you draw straight lines connecting the centers of the two circles with each other, and with the two points where the circles intersect, then you get two equilateral triangles joined along an edge, as shown in light red in the diagram).
Rational approximations to the square root of 3, improving as larger integers are used, include:
1:2, 3:5, 4:7, 11:19, 15:26, 41:71, 56:97, 153:265, 209:362, 571:989 . . .
The number 153 appears in the Gospel of John as the exact number of fish Jesus caused to be caught in a miraculous catch of fish, which is thought by some to be a coded reference to Pythagorean beliefs.

This section contains a number of unsourced speculations:

  • "...subject of mystical speculation at several periods of history..." -- which periods of history? Cite please?
  • "...perhaps first among the Pythagoreans, who considered it a holy figure." -- cite please?
  • "...was reportedly believed by them to be 265:153." -- "reportedly"? Reported where?
  • "...a holy number, called the measure of the fish" -- cite please?
  • "...is thought by some to be a coded reference to Pythagorean beliefs." -- thought by whom? Cite please?

Can someone please provide verifiable cites from mainstream sources for these assertions before restoring this material? And, before you ask, please can you give something other than a website that does not appear to me to provide cites for any of these assertions:

-- The Anome 07:20, 16 May 2006 (UTC)

You could read "Musings on the Vesica Piscis" - an article in "Nexus Network Journal" (ISSN: 1590-5869) Volume 6, Number 2 (a maths + architecture journal). If you want an earlier reference, you could read Porphyry and Plato which touch briefly upon it (e.g. in Timaeus), Porphyry recounting that Pythagoras caught 153 fish in a single catch of the net from the side of a boat. "On the Measurement of the Cycle" (Archimedes) also mentions "the measure of the fish" in this 153/vesica piscis context. The significantly more modern "City of Revelation" by Mitchell covers this topic, and goes into detail on the gematria and isopsephia aspects. Clinkophonist 17:15, 20 May 2006 (UTC)

...So I suppose I should add the section back. Clinkophonist 17:18, 20 May 2006 (UTC)

Thanks -- I read the Mitchell book a long time ago (but I don't have it now), and I knew that many of these claims date back at least to late 19th-century / early 20th century books (and certainly weren't invented on some website), but I didn't really have specific references. AnonMoos 15:47, 21 May 2006 (UTC)
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