Re: Looking for two mathematical characters

From: Philippe Verdy (
Date: Mon Jun 16 2003 - 05:43:22 EDT

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    From: "Patrick Andries" <>
    > I'm looking for two mathematical characters.
    > 2) An angle operator (combining mark ?) looking like this _| , where
    > a )
    > n| a ) n occurrences of a
    > a Ż means a )
    > n|
    > obviously a Ż should all be written on a single line.

    Suppose we note the operator with a functional syntax:
    fxp(a, n) = a Ż

    This function can also be written using the functional notation
    of the power binary operation:

    pow(a, n) = a

    So that:

    fxp(a, n) = pow(a, fx(n))
              = pow(a, pow(a, pow(a, ...))) ;(with n applications of pow)

    So we have, for n > 1:

    log fxp(a, n) = a . pow(a, pow(a, ...)); (with n-1 applications of pow)
                   = a . fxp(a, n - 1)

    And thus fxp(a, n) = pow(a, a . fxp(a, n - 1))

    Or using the operator notation:
                     n - 1|
      n| ( a . aŻŻŻŻŻ )
    a Ż = a

    It seems clear that the fx(n) function introduced above acts as a
    functor, which could be noted as a unary operator _|, that modifies
    the argument n that follows it. I do not see it as a combining
    character, even if its glyph appears on the right side of its
    parameter n, as in fact it also has a left side where the
    exponent operator is used.

    It cannot be rendered the same way as textual characters because it
    implies a begin and end, and thus some sort of parenthesing, that
    requires a complex mathematical layout.

    If it was used with a linear text layout, one would write it like:

    a _| n, or: a _| (n - 1)

    which could be then possibly rendered as the following in HTML
    to remove the implied parentheses:

    a<sup>_| n</sup>, or: a<sup>_| n - 1</sup>

    With some rich layout (MathML-like) markup, it would be something like:


    The placement of the operator is then not relevant for the Unicode
    semantic, and it is clearly not combining the same way as combining
    diacritics on letters.

    Also, the existing "power-n" characters are unified in Unicode using
    a compatibility decomposition into "<sup>n</sup>" where "<sup>" is
    considered as a layout markup used to replace a semantic binary
    operator "^" that should be present if such markup is removed.

    So the question is only its identification as a character "_|"
    that could be encoded for MathML usage, or in linear text.
    The exact rendering in a 2D layout would put it on the right and
    below a full expression, but in a linear text, we cannot encode
    2D properties.

    If there such character "_|" in Unicode ? Yes.
    With mathematical properties? Yes.
    With the correct semantic? No.

    The existing semantic of this mathematical character means "not",
    and it is a unary character that will not be layout in MathML
    correctly by extending its glyph below and to the right of its
    arbitrarily long second argument.

    What could be its representatie glyph in the Unicode chart?
    it would be similar to representatie glyphs used for mathematical
    radicals or summations, ignoring the specific placement and
    layout constraints, so I think it should be something like "_|"
    aligned on the descent line of Latin/Greek letters, probably with
    a "m"-width, and an ascent line similar to the lowercase letters
    l and k. This glyph would be appropriate for use in linear plain
    text, and additional markup could give it a more specific layout
    while preserving visually its "mathematical-combining" semantic
    usable for generic notations.

    If this character is introduced for a notation, I think that
    other characters should be introduced as well for several types
    of surroundings: above, below, left, right, and their
    combinations. The isolated left and right are already encoded
    as a single mathemetical character "|".

    Other surrounding notations exist in some languages such as in
    Egyptian hieroglyphic "cartouches" (sorry this is the French word,
    there may exist another appropriate word in English), or in musical
    notations, using special BEGIN and END character pairs, acting as
    special parentheses (and that could be *rendered* in a 1D linear layout
    with "[" and "]" characters, possibly with an additional notation
    like "[Rahmes](cartouche)".)

    The specific layout of your proposed mathemetical operator cannot
    be specified by Unicode: it must go to the markup language
    specification, and Unicode will only show a representation
    appropriate for usage and encoding in plain text, and will leave
    the specific layout specification to MathML, if it accepts this
    character with its additional layout properties.

    -- Philippe.

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