The \lowercase trick

Question

We have a number of questions or rather answers using (and explaining) the \lowercase “trick”.

\begingroup
\lccode`\~=`\.
\lowercase{\endgroup\def~}#1{foo with #1}

However, after reading on uppercase and lowercase codes I started wondering how it is possible that it is working. Quoting TeX by Topic:

To each of the character codes correspond an uppercase code and a lowercase code [...]. These can be assigned by

\uccode number equals number

and

\lccode number equals number .

In IniTeX codes `a..`z, `A..`Z have uppercase code `A..`Z and lowercase code `a..`z. All other character codes have both uppercase and lowercase code zero.

The commands \uppercase{...} and \lowercase{...} go through their argument lists, replacing all character codes of explicit character tokens by their uppercase and lowercase code respectively if these are non-zero, without changing the category codes.

The last sentence made me thinking: how can the “trick” work if both ~ and the symbol that gets a definition have lowercase code=0? My naive expectation would be that

\lccode`\~=`\.

gives ~ the same lowercase code it had before and is equivalent to \lccode`~=0. But to me this would imply that the “trick” shouldn't work? How come it does work?

\documentclass{article}
\begin{document}

\verb+\the\lccode`. += \the\lccode`. \par
\verb+\the\lccode`~ += \the\lccode`~

\begingroup
\lccode`\~=`\.
\lowercase{\endgroup\def~}#1{foo with #1}

\catcode`\.=13

.{bar}

\end{document}

Answer 1

Your intuition is wrong. In TeX's syntax,

`\~

is a number, precisely the ASCII code of the character appearing after the backquote (it can also be a length one control sequence, meaning just the same, to accommodate cases like %). So the assignment

\lccode`\~=`\.

is just the same as

\lccode 126=46

so the lowercase counterpart of the tilde becomes the period, as far as \lowercase is concerned.

If you write

\lccode`\~=\lccode`\.

then you'd assign the tilde the lccode 0, but it's very different.

---

Some more words may be good for the beginner. One can think to \lccode as an array of length 256, with its index starting at 0. The command

\lccode <number1> = <number2>

stores <number2> in the slot numbered <number1>. When \lowercase does its job and finds a character with charcode x, it looks in slot x of the \lccode array; if the numbers stored there is 0, \lowercase does nothing; otherwise it finds a number y > 0 and replaces the charcode x with the charcode y (not changing the catcode).

As usual in TeX, the same primitive that performs the assignment can be used for retrieving the value in the slot. Thus \the\lccode`\A will print 97 (under normal settings) and

\count255=\lccode`A

would store 97 in the count register 255. One might even say

\count\lccode`A=42

in order to store 42 in count register 97. But this doesn't seem good usage of \lccode. ;-)

There are several other arrays besides \lccode and \uccode: \sfcode (space factor codes), \mathcode (math codes), \delcode (delimiter code) and \catcode (category code).