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I've noticed that the \intercal symbol is popular for denoting the transpose of a matrix, which is all good and fine (and something I've adopted as I agree, it looks slick).

But if this symbol were intended for matrix transposes I can't help but suspect it would have won the name \transpose, which it doesn't have.

So I'm curious what this symbol actually is intended for, what it means, where the name comes from, what it evokes etc.? I've look long and hard on-line fueled by the passion of failure in a sense. A classic example of useless I found here:

https://www.tutorialspoint.com/tex_commands/intercal.htm

Well that explains it! Not!

On-line searches are confounded by the programming language of the same name. And I have found the symbol on any number of mathematical symbol lists but never explained beyond its name. The intercal symbol.

So I'm curious. What exactly is the \intercal symbol? What does the name mean? What is its history and where is it used (other than for matrix transposes)?

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    I'm not sure this is on topic, U+224b has full name INTERCALATE and has been in Unicode since (at least) 1.1 so was probably inherited from one of the code sets that were lumped in to the standard, the name intercal comes from iso879 isoamsb, the original sgml math character entity set. various on line dictionaries have meanings for intercalate in calendars and chemistry but I don't see any of them used with the T like symbol in this slot, so I have no idea. I would guess it's just used for transpose as a substitute bold sans serif T – David Carlisle Jun 7 '18 at 0:58
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    not sure this question is entirely on topic here, but identifying the origin of the name probably is. i have always assumed it is short for "intercalation", the insertion of something into an existing structure. that term (though i have never seen the symbol used in that context) was important in the creation of tables of the values of functions, a task now made pretty much obsolete by the application of computer algorithms. the term is also used in other disciplines -- chemistry, geology, medicine, ... – barbara beeton Jun 7 '18 at 1:04
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    Comments appreciated. Have researched intercalation now but still found no even part way clear description of that operator is does, and whence the symbol originated. That it's already in Unicode is noted and thanks for that, likewise that it's in a TeX package(s) and not a primitive. That said if it is genuinely considered off-topic by enough I'm happy to remove it of course, but put forth that the reason I landed here (that this symbol comes highly recommended on many LaTeX pages as a transpose symbol) puts it into the TeX context for some (many?) of us I guess. – Bernd Wechner Jun 7 '18 at 3:08
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    As an aside it is grouped here: unicode.org/charts/PDF/U2200.pdf with apparent logical operators XOR, NAND and NOR. THis symbol remains a curio for me. – Bernd Wechner Jun 7 '18 at 3:11
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    See The Comprehensive LaTeX Symbol List and search for intercal. There's a note below “Table 51: AMS Binary Operators” about the symbol. Elsewhere in the document too, it occurs as a binary operator, but not everywhere. – ShreevatsaR Jun 7 '18 at 3:26
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The earliest mention I can find is the SGML standard (1986) which lists intcal in its additional mathematical symbols list under "binary and large operators" the definition being (in full)

<!ENTITY intcal SDATA "[intcal]"--/intercal B: intercal-->

which means the the entity &intcal; could be used to make an unspecified system specific character (with as for all characters in this standard no encoding or pictorial clue what the character should look like). Note however the section heading and the /B in the entity definition comment both imply that the intended use is as a binary operator (which would imply that using it as a funky T for matrix transpose is a mis-use)

Unicode 1.0 added INTERCALATE at U+22BA citing SGML intcal as a prior character

https://www.unicode.org/versions/Unicode1.0.0/ch06.pdf

In MathML we assigned alias &intercal; to this and defined it (as required by XML which does not have SDATA entities) to U+22BA so MathML defines

<!ENTITY intcal           "&#x022BA;" ><!--INTERCALATE -->
<!ENTITY intercal         "&#x022BA;" ><!--INTERCALATE -->

so both of names &intcal; and &intercal; work in MathML, and so in HTML since version 5, which incorporated the MathML entities.

Separately the AMS (AMS this time standing for American Mathematical Society, not the AMS in ISOAMSB which stands for ISO Additional Mathematical Symbols set B) produced the TeX AMS Fonts and assigned the character the name \intercal as a binary operator with definition (in its later latex2e form)

\DeclareMathSymbol{\intercal}     {\mathbin}{AMSa}{"7C}

The unicode math package keeps the \intercal name defining it, as in MathML, to be U+22BA, again as a binary operator

\UnicodeMathSymbol{"022BA}{\intercal                 }{\mathbin}{intercal}%

So U+22BA is intercal and looks like ⊺ which is sort of like a squashed dropped sans serif T, not to be confused with T (U+0054) which looks like T, or top (U+22A4) which looks like ⊤.

As the editor of at least some of the specifications mentioned above I can confirm that the name has just been inherited without any actual knowledge of what the character is supposed to be used for.

Wikipedia suggests "intercalate" may be used in

timekeeping

Intercalation or embolism in timekeeping is the insertion of a leap day, week, or month into some calendar years to make the calendar follow the seasons or moon phases. Lunisolar calendars may require intercalations of both days and months.

Chemistry

In chemistry, intercalation is the reversible inclusion or insertion of a molecule (or ion) into materials with layered structures.

University Admin

Intercalation, in the context of university administration, is a period when a student is allowed to officially take time away from studying for an academic degree.

But nowhere do I see any use of a symbol resembling T

  • 4
    Excellent and much appreciate research! I fall just shy of calling at an answer to the actual question preferring honestly to leave it flagged unanswered until (as a community) we have uncovered some actual intended use for the symbol I guess. As some have noted this is indeed rather peripheral to TeX if not entirely off topic, but if we stand any hope of someone showing up some day with reference on the intended use and origin of the symbol then it's worth keeping this pen unanswered I think. There is a delicious hint here: proofwiki.org/wiki/Symbols:T – Bernd Wechner Jun 8 '18 at 0:12

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