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A piece of clothing requires Tzitzit to be placed on it if it has four corners. The Sifrei (234) states:

כסותך, פרט לטגא ולתובלא ולתיבלטירים לתקרקים לבורסים לבורדסים לפי שאינם מרובעים:‏
"Thy clothing" (Deut. 22:12): to exclude a Taga, a Tuvla, a Tiblatirin, a Tikrakim, a Borsim and a Bordsim [I assume those are types of clothing] for they aren't square.

The Beit Yosef (OC 10) quotes Mahari Abuhov similarly:

אלו הארבע כנפים לדברי הכל צריך שיהיו מרובעות לא שיהיו עגולות
All agree these four corners must be square not circular.

Does anyone discuss how sharp a corner it has to be in order to be obligated in Tzitzit? What if it is an obtuse angle? What if it is slightly rounded so as not to be pointy?

(Note I'm asking the question from the positive side: one can say a bracha on tzitzit of what kind of corner? not: what kind of corner does one no longer need to worry about putting tzitzit on?)

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There's probably some better way to formulate the question math-wise. Any ideas how to quantify corner sharpness mathematically? –  Double AA Feb 10 '13 at 19:58
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A side note about the clothing: טגא appears to be the Roman toga (which actually was shaped like a half-moon), תובלא possibly the tebenna (another kind of toga), תיבלטירים possibly a corruption for פיבלטירים, i.e. fibulatorium (an outer cloak fastened at the shoulder with a brooch), תקרקים unclear, but some suggest saga nervica, another type of traveling cloak, בורסים the birrus (a thick woolen cloak) and בורדסים a Brundisian cloak. Apparently the Sifre is discouraging Jews from adopting Hellenistic dress; see, e.g., here: books.google.ca/… –  Noam Sienna Feb 11 '13 at 1:49
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Beur Halacha (10 "velo") questions this. Aruch Hashulchan (10:4) writes that the roundness needs to extend for 3 tefachim. The Yavetz writes that it needs to extend 3 etzbaos on each side, and others say as long as it appears to the eye as round it is not obligates in tzitzis (last two opinions brought in Piskey Teshuvos 10:9).

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What does roundness's extending for a certain length mean? –  msh210 Feb 10 '13 at 20:39
    
@msh210 I assume that it means that the adjoining edges of the garment are visibly non-perpendicular when evaluated at points more than a given distance from the axis of symmetry. –  Fred Feb 10 '13 at 23:07
    
@Fred Hmmm I kinda of thought it meant length of the curve between the points where it is straight (ie between points where second derivative is zero). –  Double AA Feb 11 '13 at 0:18
    
@DoubleAA If that's true, then the necessary distance from one "straight" point to another changes with the shape of the curve. –  Fred Feb 11 '13 at 3:49
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@DoubleAA My guess is as good as anyone else's. My answer is just a translation of the sources that speak about the question. –  Michoel Feb 11 '13 at 5:21
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