As a followup to my general post about sentence spacing, here are some brief notes about managing sentence spacing in LaTeX, followed by a not-so-brief explanation of what LaTeX is doing.

By default, LaTeX adds slightly more space between sentences than it does between words.  The space between sentences is about 33 percent wider than the space between words.

You can disable this extra space entirely with the \frenchspacing directive.  From the point LaTeX encounters that directive until the end of the document (or until it reaches a \nofrenchspacing directive), it will use the same amount of space between sentences as between words.

LaTeX tries to figure out where sentences end and apply the extra space on its own.  It usually does a good job, but sometimes it guesses wrong.  Fortunately, it has options for you to adjust its word and sentence spacing yourself, so you can almost always get it to do what you really want.

§ LaTeX’s Basic Rules

The basic rules are these:

1. A sentence-ending punctuation character (a period, exclamation point, or question mark) followed by a space is considered to signal the end of a sentence.
2. There can be any number of right parentheses, quote marks, and right brackets between the punctuation and the space.
3. However, if the character immediately before the punctuation is a capital letter, LaTeX will not end a sentence there.

So these would all be seen by LaTeX as sentence breaks and would get extra space added during text layout:

…thank you. You're…
…the best.  No one…
…else is.         Except…
``Stop!'' The word…
…really right?)  Anyway…

These, however, would not be treated as the ends of sentences:

A Ph.D. in what…
She was SHOUTING.  I didn't like it…

§ Non-sentence Punctation

If you have some punctuation that makes LaTeX think there’s a sentence where there isn’t, you have two options available to you.

A lot of the time, the false sentences come from things like abbreviated titles, e.g. Mr. Rogers.  In those cases, you would probably prefer to tightly bind the two words together.  For that, you can use a tilde to add a nonbreaking space, which LaTeX also calls a tie:

Mr.~Rogers

LaTeX will not break a line at a nonbreaking space, nor will it stretch nonbreaking spaces when justifying lines.

In other cases, you might want LaTeX to treat the space after the punctuation as a normal space that can be wrapped and stretched as needed.  There are two ways to do that.

One option is to use the \@ macro between the punctuation and the space.  This interrupts LaTeX’s sentence-ending calculations and causes it to treat the subsequent space like a normal inter-word space.

It was David vs.\@ Goliath all over again. 
``What are you doing?\@'' she asked. 
``What are you doing?''\@ she asked. 
There are many options, e.g.\@ a polar bear. 

Alternately, you can use \ (a backslash followed by a space) to explicitly tell LaTeX to use a normal space at that location, regardless of what its calculations say:

It was David vs.\ Goliath all over again. 
``What are you doing?''\ she asked. 
There are many options, e.g.\ a polar bear. 

The second option is, as you can see, a little more concise.

§ Unrecognized Sentence Punctuation

Conversely, sometimes you have sentences that end with a capital letter right before the period (or other end-of-sentence punctuation).  In that case, you can put \@ just before the punctuation to make sure LaTeX adds the extra space you want between that sentence and the next one:

He had a PhD\@. That worried me. 

As in the previous examples, \@ interrupts LaTeX’s special spacing calculations.  When placed before sentence punctuation, it causes LaTeX to ignore the immediately-preceding character, which means its special “capital before punctuation” rule never comes into play.

§ Sentences Without Recognized Punctuation

On rare occasions, I’ve run into cases where I’d like to have sentence spacing after characters that LaTeX doesn’t normally recognize as sentence-ending.

For example:

I thought---  No, that's not right. 

The best approach I’ve found for this is:

I thought---\spacefactor3000{}  No, that's not right. 

The short explanation for that wordy construct is that it’s forcing LaTeX’s layout engine to apply end-of-sentence semantics at the macro location.  (It might or might not help to know that \@ is basically equivalent to \spacefactor1000{}.)

§ More Details than You Probably Want

The above should be sufficient if you just want to know how to get or suppress LaTeX’s end-of-sentence spacing as needed.  But if you want to know what’s going on under the hood, feel free to read on.

# Space Factors

Every character in TeX has a numeric “space factor” assigned.  That space factor primarily affects the rate at which space after the character is allowed to grow or shrink as TeX adjusts the width of a line to make it justified.  When TeX expands a space, it does it in proportion to the space factor divided by 1000.

Most characters have a space factor of 1000.  That means most inter-word spaces use a proportion of 1 for expansion or, in other words, TeX will expand or shrink all of the spaces by the same amount all the time.

Some characters have a slightly larger space factor.  Commas have a space factor of 1250, for instance, and semicolons have a space factor of 1500.  So let’s say there’s a comma in a line of text and TeX wants to make the line wider.  For every 1 point of space that TeX adds to the “normal” spaces in the line (the ones with a space factor of 1000), it will add 1.25 points to the space after the comma.  (When shrinking space, TeX uses the inverse of that proportion.  So if the normal spaces were decreased by 1 point, the space after the comma would only be decreased by 1/1.25 or 0.8 points.)

# Widening a Space Based on the Space Factor

In addition to these rules about growing and shrinking spaces, TeX has another rule, which is that if a character’s space factor is greater than or equal to 2000, it automatically adds an extra amount to the width of the following space.  That extra amount is defined by the font, but most fonts are pretty similar to the default Computer Modern.  Ten-point Computer Modern uses a width of 3.3333 points for normal spaces and adds an extra 1.1111 points when the “extra space” rule is triggered.

The three standard end-of-sentence punctuation marks—period, question mark, and exclamation point—all have space factors of 3000.  Colons have a space factor of 2000.  So all four of those characters will trigger the addition of extra width to any space that directly follows them.  Spaces after end-of-sentence punctuation will grow three times faster than normal spaces and shrink at one third their rate.  Spaces after colons will grow twice as fast and shrink at half the rate of normal spaces.

# Skipping Some Characters’ Space Factors

A few other punctuation characters have space factors of zero.  These are all characters that sometimes appear between sentence punctuation and the space after the sentence.  They include the right parenthesis, single quote mark, and right bracket.

When TeX encounters a character with a space factor of zero, it carries over the space factor from the previous character.  This allows the spacing algorithm to effectively ignore some characters.  As an example, consider the string “a.) ”.  The “a” has a space factor of 1000.  The “.” has a space factor of 3000.  The “)” has a space factor of zero, which means TeX will carry over the previous value of 3000.  When it finally reaches the space, TeX will use the 3000 value to add extra space and grow the space at three times the rate of a normal space.

# Capital Letters

The final rule TeX has is that if a character has a space factor less than 1000 (but greater than zero) and the next character has a space factor greater than 1000, that next character’s space factor is reduced to just 1000.

All capital letters hava a space factor of 999.  That means any of the special-spacing characters effectively lose their special space factor after a capital letter.  The spaces after “PhD.”, “NASA:”, and “FOMO;” will be normal spaces and will not have any extra width added to them.

# Setting the Space Factor Explicitly

You can set a space factor explicitly at any point with the \spacefactor command, followed by the new value.  As usual in TeX, you can have an explicit assignment (\spacefactor=1000) or allow TeX to figure it out implicitly (\spacefactor 1000 or \spacefactor1000).  Also as usual, the command will consume any space characters after it, so you need an empty statement after it if you want to preserve the space (\spacefactor1000{}).

# Putting it All Together

All of that should explain why the LaTeX macro \@ is equivalent to \spacefactor1000{}.  When placed between a capital letter and a period, it forces the pre-period space factor to 1000, which in turn allows the period to trigger the usual extra space and altered growth and shrinking behaviors.  When placed between a period and a space, it forces the space to see a space factor of 1000 and be treated as a normal space.

The “\ ” (backslash space) macro always inserts a space with a space factor of 1000, which is why it’s equivalent to (and can be seen as a shorthand for) “\@ ” (backslash at space).

§ Reference Material

The bulk of this information can be found in TeX by Topic, chapter 20.  See also the LaTeX2e reference on \spacefactor, an answer to “Is it possible to have non-french spacing without extra stretch?” on the TeX Stack Exchange, and a similar answer to “What is the proper use of \@ (i.e., backslash-at)?”.

There is an ongoing debate in some parts of the Internet about how much space should go after the end of a sentence.  Practically every publisher and quite a lot of other people will say there should be one space.  A minority of people—principally Gen X and older, I suspect—will say there should be two spaces.

I have opinions.

In short:  Two spaces are too much.  One space is okay, but it can feel a little crowded.  Adding just a little extra space—about 33 percent more—after a sentence gives a nice balance.  This blog does the latter.

§ A Brief History of Sentence Spacing

In the earliest days of the printing press, the standard among typesetters was to have about three times more space between sentences than between words.  But even their inter-word spaces were much larger than we use today.  In numeric terms, there was a space of one em between sentences, and one-third of an em between words.  An “em”, for our purposes, is simply a unit of measurement in printing that scales in proportion to the size of the type.  For example, text set at 72 points will have an em size of one inch, while 36 point text will have an em size of half an inch.  This proportionality makes the em a convenient reference for the sizes of things in typeset text.

Here’s how those earliest spacing conventions might look with some sample text:

As Gregor Samsa awoke one morning from uneasy dreams, he found himself transformed in his bed into a gigantic insect. When he lifted his head a little, he could see his dome-like brown belly. The bed quilt could hardly keep in position and was about to slide off completely. His numerous legs, which were pitifully thin compared to the rest of his bulk, waved helplessly before his eyes. It was no dream. His room, a regular human bedroom, only rather too small, lay quiet between the four familiar walls.

When typewriters, with their fixed character widths, became common, people tried to adapt printing rules that had been developed for proportional typefaces.  Typewriters’ fixed-width spaces were a little larger than the printers’ third-of-an-em proportional-type spaces.  Typing conventions eventually settled on a single fixed-width space between words and two fixed-width spaces between sentences.

If it were typed according to these rules, our sample text would look something like this:

As Gregor Samsa awoke one morning from uneasy dreams, he found himself transformed in his bed into a gigantic insect.  When he lifted his head a little, he could see his dome-like brown belly.  The bed quilt could hardly keep in position and was about to slide off completely.  His numerous legs, which were pitifully thin compared to the rest of his bulk, waved helplessly before his eyes.  It was no dream.  His room, a regular human bedroom, only rather too small, lay quiet between the four familiar walls. 

During the twentieth century, publishers gradually began to close up spacing in their printing, both between words and between sentences.  In the first part of the century, inter-word spacing remained roughly a third of an em, but inter-sentence spacing shrank to half an em.  That gave something along these lines:

As Gregor Samsa awoke one morning from uneasy dreams, he found himself transformed in his bed into a gigantic insect. When he lifted his head a little, he could see his dome-like brown belly. The bed quilt could hardly keep in position and was about to slide off completely. His numerous legs, which were pitifully thin compared to the rest of his bulk, waved helplessly before his eyes. It was no dream. His room, a regular human bedroom, only rather too small, lay quiet between the four familiar walls.

During the second half of the twentieth century, professional publishers continued to close up inter-word and inter-sentence spacing.  By the end of the century, most publications used just a quarter of an em between both words and sentences.  I have seen speculation that the convergence between inter-word and inter-sentence spacing grew out of increasing use of automated typesetting.  (“Two Spaces - an Old Typists' Habit?” gives a brief history of automated typesetting and makes a plausible case for how the extra inter-sentence space was lost.)

This sort of spacing is what you’re almost certainly used to seeing everywhere, but here’s what it looks like with our sample text:

As Gregor Samsa awoke one morning from uneasy dreams, he found himself transformed in his bed into a gigantic insect. When he lifted his head a little, he could see his dome-like brown belly. The bed quilt could hardly keep in position and was about to slide off completely. His numerous legs, which were pitifully thin compared to the rest of his bulk, waved helplessly before his eyes. It was no dream. His room, a regular human bedroom, only rather too small, lay quiet between the four familiar walls.

Now, well into the twenty-first century, pretty much every newspaper, magazine, and book publisher uses the same amount of space between sentences as between words, and that space is generally about a quarter of an em.  Every major style guide recommends typing a single space after a sentence.  The only significant exceptions, as far as I can tell, are scientific journals that use LaTeX for their typesetting (and which haven’t explicitly turned off LaTeX’s extra inter-sentence spacing).

§ Readability Studies

Practically every study done on the readability effects on end-of-sentence spacing has been inconclusive.  The one exception is “Are two spaces better than one?” by Johnson, Bui, and Schmitt.  That study purported to find that putting two spaces after sentences improved readability exclusively for people who themselves put two spaces after sentences when writing.  The findings are even weaker than that summary implies, however, for reasons that are covered well by Matthew Butterick.  In short, the test conditions were well outside normal reading conditions and there were unexplored statistical differences between the overall reading abilities of one- and two-space-using readers.

In other words, no one has really demonstrated a tangible benefit to any variation of end-of-sentence spacing.

§ Aesthetics

So that leaves just the question of aesthetics.  What looks good?

As noted previously, nearly every newspaper, magazine, and book you read uses the same amount of space at the end of a sentence as between words.  Nearly every website you read is the same, unless the website author has gone out of their way to do something different.  Everyone reads things with just one space between sentences all the time, so there’s the aesthetics of familiarity, if nothing else.

My personal view is a little more nuanced than just one space versus two.  Using a single space is servicable and two spaces are probably a bad idea, but I think I can do better than either.

Let’s look again at our example text with one space after each sentence:

As Gregor Samsa awoke one morning from uneasy dreams, he found himself transformed in his bed into a gigantic insect. When he lifted his head a little, he could see his dome-like brown belly. The bed quilt could hardly keep in position and was about to slide off completely. His numerous legs, which were pitifully thin compared to the rest of his bulk, waved helplessly before his eyes. It was no dream. His room, a regular human bedroom, only rather too small, lay quiet between the four familiar walls.

I find that readable, but it also feels a bit crowded.  The sentences seem to run together without any room to breathe.

Let’s put two spaces after each sentence and see how that looks:

As Gregor Samsa awoke one morning from uneasy dreams, he found himself transformed in his bed into a gigantic insect.  When he lifted his head a little, he could see his dome-like brown belly.  The bed quilt could hardly keep in position and was about to slide off completely.  His numerous legs, which were pitifully thin compared to the rest of his bulk, waved helplessly before his eyes.  It was no dream.  His room, a regular human bedroom, only rather too small, lay quiet between the four familiar walls.

To me, that feels like too much space.  As I read the paragraph, the extra space between sentences feels almost like an interruption each time.

I mentioned earlier that some publishers still use LaTeX’s inter-sentence spacing.  In particular, LaTeX allocates 33 percent more space at the end of a sentence than it does between words.  (For very specific numbers, see this Stack Exchange answer on the topic.)  If a standard inter-word space is now a quarter of an em, that should lead to inter-sentence spacing of a third of an em.  Here’s what that looks like:

As Gregor Samsa awoke one morning from uneasy dreams, he found himself transformed in his bed into a gigantic insect. When he lifted his head a little, he could see his dome-like brown belly. The bed quilt could hardly keep in position and was about to slide off completely. His numerous legs, which were pitifully thin compared to the rest of his bulk, waved helplessly before his eyes. It was no dream. His room, a regular human bedroom, only rather too small, lay quiet between the four familiar walls.

Personally, I like that balance of spacing.  There’s a very slight extra bit of space between sentences, but not enough that it really draws attention to itself.  This is often the sort of spacing I use when I’m writing longer-form things in HTML that don’t need to adhere to some surrounding style conventions.  For reference, I’m using Unicode character U+2004 THREE-PER-EM SPACE between sentences.  It can be represented in HTML using the XML entity  .

But I actually do something slightly different on this blog.  Because I’m picky about a variety of minutiae, I want this blog to look “right” (according to me) in all sorts of environments, including in text-mode web browsers (and, I guess, in other browsers that use monospaced fonts).  So this blog ends sentences with a combination of a regular space character and U+2009 THIN SPACE (or  ).  In a monospaced font, those will be shown as two full-width spaces.  In proportional fonts, the thin space is typically somewhere between a sixth and a fifth of an em.  In the font I use, it’s closer to a fifth.  Consequently, in most browsers those two spaces will add up to just a tiny bit more than my preferred total of a third of an em between sentences.

That combination of spaces looks like this (and also like every regular bit of text on this blog):

As Gregor Samsa awoke one morning from uneasy dreams, he found himself transformed in his bed into a gigantic insect.  When he lifted his head a little, he could see his dome-like brown belly.  The bed quilt could hardly keep in position and was about to slide off completely.  His numerous legs, which were pitifully thin compared to the rest of his bulk, waved helplessly before his eyes.  It was no dream.  His room, a regular human bedroom, only rather too small, lay quiet between the four familiar walls.

§ External Links

The Chicago Manual of Style, a fairly conservative style guide, has a blog article about their one-space recommendation, as well as their own history of sentence spacing in publishing.

The Associated Press Stylebook doesn’t have anything online that’s publicly accessible, but a Journalist’s Resource basic guide to the AP style summarizes the guidance as, “Use only one space after the end of a sentence. Period.”

The Modern Language Association, whose style guide is used in many academic settings, says to use one space after a period unless specifically directed otherwise by an instructor.

The American Psychological Association, one of the last holdouts for using two spaces after a sentence, updated their guidelines in 2019 to recommend a single space.

Microsoft Word started flagging two-space sentences as errors in 2020.

Matthew Butterick, a well-regarded typographer online, calls the use of two spaces a “typewriter habit”—a holdover from the days of typewriters that no longer serves any useful purpose.

It’s a little difficult to find a comprehensive online LaTeX reference about its sentence spacing.  It doesn’t help that many LaTeX people seem a bit bitter about the ubiquity of using a single space to separate sentences.  (For example.)  Probably some of the most thorough coverage is in (some of) the answers to the TeX Stack Exchange question “Double space between sentences”.

§ In Conclusion

Don’t type two spaces after sentences in a medium with a proportionally-spaced font.  The closest typographic convention to this practice hasn’t been in use professionally for over a century, and it doesn’t look great alongside modern fonts and typography.

If your only choices are between using two standard spaces and a single standard space after your sentences, use the single space.

In many environments, the choice between one and two spaces is a false dichotomy.  If you can, try adding just a little extra space between your sentences.  I find that sentences look good with about 33 percent more space between them than between the words they contain.  I probably wouldn’t add more than about 50 percent of my inter-word spacing to the ends of my sentences.

A little while ago, I finally got around to revamping my website generation.  I’d been using Blosxom for nearly two decades, but I’d also been meaning to move to something a bit newer for a while.  So now I’m using Hugo.

Most of my stuff ported over without too much difficulty.  I’d been using Markdown with Blosxom, so it translated pretty cleanly to Hugo, with the just the addition of some front matter to each file.  Some of the Blosxom plugins I used had their own de facto front matter, so it was just a matter of changing it into YAML.

Overall, Hugo is very nice.  Its server mode and draft posts are particularly useful as I write things.  There were many times I wanted to preview a post in Blosxom and didn’t have a good way to do so.  Hugo solves that problem.  Draft posts let me work on things in the same repository as my published work without making things live until they’re ready.

Because Hugo facilitates working with the website as a whole much more readily, I also finally developed a proper deployment mechanism.  It’s just a Makefile with a few lines to generate intermediate files, run hugo to generate the HTML, and then scp the files to my webhost.  But having that automation makes it far easier to update things.  I could have done something similar with Blosxom, but Hugo has better affordances for it.

While I was at it, I also reworked my website theme.  I’ve had a lot more design practice since I put together the old theme.  While I might not say I’m good at this yet, I think I’ve at least gotten better.  You can compare, say, the old version and new version of my ebook post to see some of the difference.

I doubt anyone outside of me really care about the particulars of this website design.  But in case you do, the theme is available on GitLab.

X11 has a useful feature called the compose key.  After pressing the compose key, you can type a sequence of keys on the keyboard to get various Unicode characters.  For example, Compose o c generates a copyright symbol (©).  This allows for typing a lot of extended characters with a US keyboard map (without the need to have dead keys or other non-US layout features).

There are, however, a lot of key combinations.  See the X11R7.7 documentation for its full list.  So this page focuses on the patterns in the key combinations and restricts itself to the ones I use more often.

Note: I use the XKB configuration option compose:prsc on my systems to map my Print Screen key to the compose key.  How I do that varies.  On some systems, I have Option "XkbOptions" "compose:prsc" in the InputClass section for my keyboard in an xorg.conf file.  On other systems, where it isn’t as easy to modify the system config files, I run setxkbmap -option compose:prsc automatically when I log in.

You can see what other keys are available to map as a compose key with:

grep "compose:" /usr/share/X11/xkb/rules/base.lst

Anyway, on to the list.

§ Quote Marks

The less-than and greater-than keys (or left and right angle brackets, if you prefer) can be combined with single and double quotes to make curly left and right quotes.  They can be typed in either order; both < ' and ' < result in a left single quote mark.

A comma can be used instead of an angle bracket to produce the low version of the quote symbol, for languages that use it to open quotes.

	<	>	,
'	‘	’	‚
"	“	”	„

Guillemets are the result of doubling angle brackets.  Use a leading period for single guillemets.

Combination	Result	Unicode Name
< <	«	U+00AB	LEFT-POINTING DOUBLE ANGLE QUOTATION MARK
> >	»	U+00BB	RIGHT-POINTING DOUBLE ANGLE QUOTATION MARK
. <	‹	U+2039	SINGLE LEFT-POINTING ANGLE QUOTATION MARK
. >	›	U+203A	SINGLE RIGHT-POINTING ANGLE QUOTATION MARK

§ Accented Characters

Most letters can be combined with another character to add an accent.  The letter and accent character can be typed in either order; both ' a and a ' will give “á”.  Typing the letter last is preferred, since there are some multi-accent combinations that only make sense if you type all of the accents first.

The standard accent characters are:

Character	Accent Type	Example	Note
' (single quote)	acute accent	' a → á
` (backtick)	grave accent	` a → à
^ (caret)	circumflex	^ a → â	> (greater than sign) also works: > a → â
~ (tilde)	tilde	~ a → ã
" (double quote)	diaeresis; umlaut	" a → ä
* (asterisk)	ring	* a → å	Lowercase o also works, but only if it precedes the letter to be modified: o a → å
, (comma)	cedilla	, c → ç	Also works as an ogonek: , a → ą
/ (forward slash)	slash	/ o → ø
- (dash)	macron	- a → ā	_ (underscore) also works: _ a → ā
. (period)	dot	. e → ė	Also removes the dot from a lowercase i: . i → ı

In similar ways, the < (less than symbol) combines as a caron (e.g. < e → ě); - (dash) combines as a stroke (e.g. - d → đ); and = (equals sign) combines as a double acute accent (e.g. = o → ő).  I don’t really use those, though.

There are some patterns for characters with multiple additions (e.g. * ' A → Ǻ, LATIN CAPITAL LETTER A WITH RING ABOVE AND ACUTE), but (1) I don’t use them often enough to need to remember the key combinations, and (2) many of them rely on keyboard layouts that can directly generate precomposed characters with one of the two modifiers.

§ Prime Symbols

I use these with some regularity, but they’re unfortunately not available as compose character combinations.  In some media, the HTML codes suffice.  In others, I can sometimes type the codepoint in hex.

Symbol	Character	Codepoint	HTML
(Single) Prime	′	U+2032	′
Double Prime	″	U+2033	″
Triple Prime	‴	U+2034	‴
Quadruple Prime	⁗	U+2057	⁗

In many programs, I can press Ctrl-Shift-U, followed by the Unicode codepoint, followed by the space or enter key.  So I can get a prime symbol with Ctrl-Shift-U 2 0 3 2 Enter.  That works for my terminal program and Firefox, at least.

In Emacs, I have to use C-x 8 Ret 2 0 3 2 Ret (et al.).

Regardless, I have to actually remember the codepoint, which is less convenient than the mnemonics afforded by compose key combinations.

§ Fractions

Many fractions can be created by typing two digits in sequence.  The first digit will be the numerator and the second will be the denominator.

All denominators from two to ten can be used with a numerator of one:

• 1 2 → ½
• 1 3 → ⅓
• 1 4 → ¼
• 1 5 → ⅕
• 1 6 → ⅙
• 1 7 → ⅐
• 1 8 → ⅛
• 1 9 → ⅑
• 1 1 0 → ⅒

For non-unit numerators, all multiples of ⅙ and ⅛ are available, in their simplified forms (which means all multiples of ⅓ and ¼ are also available).

Multiples of ⅙:

• 1 6 → ⅙
• 1 3 → ⅓
• 1 2 → ½
• 2 3 → ⅔
• 5 6 → ⅚

Multiples of ⅛:

• 1 8 → ⅛
• 1 4 → ¼
• 3 8 → ⅜
• 1 2 → ½
• 5 8 → ⅝
• 3 4 → ¾
• 7 8 → ⅞

And, weirdly, 0 3 composes to ↉.

§ Subscripts and Superscripts

All of the digits, as well as the characters plus sign (+), equals sign (=), left parenthesis, and right parenthesis can be superscripted or subscripted by prefixing them with a caret or underscore, respectively.  The letters i and n can be superscripted with the sequences ^ _ i and ^ _ n, respectively.

For example:

• ^ 2 → ²
• _ 8 → ₈
• _ ) → ₎
• ^ _ i → ⁱ

§ Math Symbols

A number of math symbols are available.  Unfortunately, the Unicode character U+2212 MINUS SIGN (−) is not available, even though it’s a better choice than the plain dash in mathematical contexts.  The same techniques described in “Prime Symbols” above can be used to insert a minus sign by its codepoint.  You can also use − in HTML.

Most math symbol combinations can be typed in either order, but the inclusive inequalities (≤ and ≥) need the equals sign to come after the less-than or greater-than character..

Combination	Result	Unicode Name
x x	×	U+00D7	MULTIPLICATION SIGN
: - - :	÷	U+00F7	DIVISION SIGN
- , , -	¬	U+00AC	NOT SIGN
+ - - +	±	U+00B1	PLUS-MINUS SIGN
/ = = /	≠	U+2260	NOT EQUAL TO
< =	≤	U+2264	LESS-THAN OR EQUAL TO
> =	≥	U+2265	GREATER-THAN OR EQUAL TO

§ Circled Numbers

Any one- or two digit number can be put in a circle by surrounding it with parentheses.  The same works for single upper- and lowercase letters.

For example:

• ( 1 ) → ①
• ( 4 2 ) → ㊷
• ( S ) → Ⓢ

Note: For the copyright and registered copyright symbols, use o c and o r (or their variants; see “Other Characters” below).  The simple circled letters are different codepoints.

§ Currency

Key combinations for currency symbols often cover every ordering and capitalization combination.  They include:

Combination	Result	Unicode Name
C E	₠	U+20A0	EURO-CURRENCY SIGN
C = = C c = = c E = = E e = = e	€	U+20AC	EURO SIGN
C | | C c | | c	¢	U+00A2	CENT SIGN
L - - L l - - l	£	U+00A3	POUND SIGN
Y = = Y y = = y Y - - Y y - - y	¥	U+00A5	YEN SIGN
O X O x o X o x X O X o x O x o	¤	U+00A4	CURRENCY SIGN

§ Whitespace

Two space characters turn into a nonbreaking space.  (Or you can use   in HTML.)

A space and a period turn into U+2008 PUNCTUATION SPACE.  The Unicode specification says this is a “space equal to narrow punctuation of a font”, or about the width of a period.

Using space characters other than the normal space is, of course, risky, since many programs won’t indicate that the alternate characters are any different from the normal ones.

§ Other Characters

Combination	Result	Unicode Name		Note
- - -	—	U+2014	EM DASH
- - .	–	U+2013	EN DASH
. .	…	U+2026	HORIZONTAL ELLIPSIS
- >	→	U+2192	RIGHTWARDS ARROW
< -	←	U+2190	LEFTWARDS ARROW
o o 0 * * 0	°	U+00B0	DEGREE SIGN	I find o o the easiest to type.
. ^ ^ . . -	·	U+00B7	MIDDLE DOT	You can think of this as either a raised dot (combining a period with a caret) or a smaller bullet (using a dash with a period instead of the bullet combination's equals sign with a period).
. =	•	U+2022	BULLET
? ?	¿	U+00BF	INVERTED QUESTION MARK
! ?	‽	U+203D	INTERROBANG
? !	⸘	U+2E18	INVERTED INTERROBANG
s o o s S O O S	§	U+00A7	SECTION SIGN	Either order; upper- or lowercase
P P	¶	U+00B6	PILCROW SIGN
t m T M T m t M	™	U+2122	TRADE MARK SIGN	Any mix of capitalization
s m S M S m s M	℠	U+2120	SERVICE MARK	Any mix of capitalization
o c O C O c o C	©	U+00A9	COPYRIGHT SIGN	Any mix of capitalization.  C O and C o also work, but combinations starting with lowercase c don't (they give different characters).  So I prefer to always start with the o.
o r O R O r o R	®	U+00AE	REGISTERED SIGN	Any mix of capitalization.  R O also works, but none of the other capitalization mixes make characters with the "r" first.  So, just as with the copyright symbol, I prefer to always start with the o.
T H	Þ	U+00DE	LATIN CAPITAL LETTER THORN
t h	þ	U+00FE	LATIN SMALL LETTER THORN
D H	Ð	U+00D0	LATIN CAPITAL LETTER ETH
d h	ð	U+00F0	LATIN SMALL LETTER ETH
S S	ẞ	U+1E9E	LATIN CAPITAL LETTER SHARP S
s s	ß	U+00DF	LATIN SMALL LETTER SHARP S

A couple of months ago I got a FitBit Luxe.  I’ve long used the pedometer on my phone for step tracking, but it’s not always accurate and I don’t always have it on me.  (Plus, I really wanted longitudinal heart rate data, which isn’t feasible with something like a phone.)

To “celebrate” having an always-on health tracker, I’m going to start on a project I came across a while back but for which I hadn’t put together the pieces of participation until now: taking the One Ring to Mordor with Frodo.

§ Walking to Mordor and Back

This is most directly influenced by Nerd Fitness’s “Walking to Mordor and Back” Google spreadsheet.  It’s a fitness motivation tool that, in turn is based on the Éowyn Challenge.  The Éowyn Challenge takes all the traveling various characters do in The Lord of the Rings and breaks it into segments, as shown on their “Walk to Rivendell and Beyond” page.  The goal of the Éowyn Challenge is to pick a segment and track your personal progress in traveling the same distance as the characters.  Each segment has a breakdown by day and mile of where the characters went and what they did.

Nerd Fitness has taken all of Frodo’s segments from the Éowyn Challenge (Hobbiton to Mount Doom back to Hobbiton and then to the Grey Havens) and put them into a spreadsheet for ease of tracking.  They omit all the mile-by-mile details in favor of looking at the overall distance traveled—a bit over 3,600 miles.

§ My Plan

Nerd Fitness’s spreadsheet is designed around having a group of participants and adding their miles to get a total.  (Which is not how group travel works, but we’ll ignore that for now.)  I’m just going to track myself and see how long it takes me to hit all the landmarks myself.  I expect it’ll be significantly longer than it took Frodo.

I have an IFTTT rule to add each day’s fitness summary to a Google spreadsheet.  So I added a sheet to my copy of the “Walking to Mordor and Back” document; that sheet just imports all the lines from my FitBit spreadsheet and calculated a running mileage total.  I’ve modified the main sheet of the “Walking to Mordor and Back” to automatically pull data from that secondary sheet.  It’ll update the mile total and arrival dates all on its own.

With all of that automated, I just have to wear the FitBit and live as usual.  As long as I remember to check on the spreadsheet periodically, I’ll be able to see how I’m doing.

I’m starting the mile accumulation today, January 1.  Let’s see how long it takes me to finish!