Which dot character to use in which context?

Question

Wikipedia lists several dot characters in Unicode. These are the ones that are ambiguous to me:

• interpunct, middle dot (·) · · U+00B7 "midpoint (in typography)"
• dot operator (⋅) ⋅ ⋅ U+22C5
• bullet operator (∙) ∙ U+2219 "(mathematics)"
• bullet (•) • • U+2022 "black small circle" "often used to mark list items"

I'm mostly interested in the Unicode characters, but Math.SE told me to try here, and LaTeX likewise has:

• \cdot
• \cdotp
• \bullet
• \textbullet

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For example:

• Wikipedia shows multiplication as both "x ⋅ y" ("dot operator") and "x ∙ y" ("bullet operator") in the same paragraph.
• WP List of mathematical symbols uses "middle dot" for multiplication.
• WP Multiplication article uses \cdot for scalar multiplication.
• WP says the matrix dot product should be written using the "bullet operator" character, like "a ∙ b".
• The WP Dot product article uses the LaTeX \cdot character for dot products.
• Wikipedia shows a raised decimal point example "£21·48", which uses the "middle dot" character.
• Wikipedia notates chemistry hydrates like "CuSO4 · 5H2O", which uses the "middle dot" character.
• Combined units can also be written with a dot, like "N·m". Wikipedia uses a "middle dot" for this, not the "dot operator".
• Characters in SI notations has a paragraph on combined units, seeming to say that dot operator is preferred for this, like "N⋅m".
• Wolfram Alpha considers \cdot and · to be equivalent, as well as \bullet and •. Valid?

Of course it's pedantic, they all look the same, and the meaning can always be assumed from context, but I'd like to know, once and for all, which Unicode/HTML and LaTeX characters are semantically correct in each application?

(If there are other examples I left out, feel free to edit the question and add them.)

Answer 1

The following is my personal view on this matter. I have not seen hard rules anywhere.

• Normal mathematics

The \cdot is a multiplication symbol. The following are all equivalent

It is useful in thermo-fluid dynamics formulas where there are a lot of multi character values such as Reynolds number, Prandtl number, etc.

Compound SI units (see siunitx package) is also multiplication

For vector-tensors it is element wise multiplication

• Chemical formula

I don't know much about chemical notation (ask Joseph Wright)

• Decimal marker

The center dot as decimal marker is only found in very old books or non-scientific literature. Because it can be confused with a multiplication sign is is not used in scientific literature any more.

Answer 2

As the Unicode project seeks to determine the semantics of the glyphs rather than their graphical representation, this question is quite interesting.

Although I agree that all of these “bullets” could theoretically look the same (and one wouldn’t need to care about this issue at all), the semantic background can—at least to some degree—be inferred by the official Unicode charts:

Since two of the four symbols, which are subject of the discussion, are placed in Unicode Blocks named “punctuation” (see below), I would advise against using them in mathematical context. In my opinion, the annotation “= midpoint (in typography)” for the 00B7 · MIDDLE DOT symbol speaks for itself and I think one could use it in typesetting poems, for example, to mark stanzas or as a substitute for the regular space.

From my point of view, the 2022 • BULLET should be used as a list symbol exclusively.

As I think it, the 22C5 ⋅ DOT OPERATOR is the mathematical version of 00B7 · MIDDLE DOT and 2219 ∙ BULLET OPERATOR of 2022 • BULLET, respectively, when it comes to appearance. One will probably get the most “regular-looking” result with using the 22C5 ⋅ DOT OPERATOR for regular (scalar) multiplication and the 2219 ∙ BULLET OPERATOR for any other (user-defined) operation on other objects.
(See the image prepared by Danie Els.)

This can be done, from my point of view again, because the Unicode standard does not define this operator to represent any mathematical operation. Interestingly, for the 2218 ∘ RING OPERATOR symbol, the annotation does just that: “= composite function”. So I would say that semantically you are on the safe side, so to speak, as maybe the Unicode Consortium has spoken to mathematicians who told them that the dot can be—and which is being—redefined to meet the mathematician’s need. (This is very interesting when speaking about abstract algebraic structures, such as fields and vector spaces, when one uses only abstract operations.)

Since, your query for “house rules for mathematical typesetting” is not so far off, forasmuch I have experienced that in my math lectures, every professor would define these symbols at the beginning of the semester and there were differences throughout the lectures, of course.

TL;DR:
Use 22C5 ⋅ DOT OPERATOR for multiplication and 2219 ∙ BULLET OPERATOR for own operations. This is only my opinion.

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Excerpts from Unicode specifications

(I think the arrows indicate references to similarly-looking rather than to semantically similar symbols.)

Block “C1 Controls and Latin-1 Supplement”, Section title “Latin-1 punctuation and symbols”

00B7 · MIDDLE DOT
= midpoint (in typography)
= Georgian comma
= Greek middle dot (ano teleia)
→ 0387 · greek ano teleia
→ 16EB ᛫ runic single punctuation
→ 2022 • bullet
→ 2024 . one dot leader
→ 2027 ‧ hyphenation point
→ 2219 ∙ bullet operator
→ 22C5 ⋅ dot operator
→ 2E31 ⸱ word separator middle dot
→ 2E33 ⸳ raised dot
→ 30FB ・ katakana middle dot

Block “General Punctuation”

2022 • BULLET
= black small circle
→ 00B7 · middle dot
→ 2024 . one dot leader
→ 2219 ∙ bullet operator
→ 25D8 ◘ inverse bullet
→ 25E6 ◦ white bullet

Block “Mathematical Operators”

2219 ∙ BULLET OPERATOR
→ 00B7 · middle dot
→ 2022 • bullet
→ 2024 . one dot leader

22C5 ⋅ DOT OPERATOR
→ 00B7 · middle dot

Answer 3

In case anyone also wonder, what's the different between $\cdot$ vs $\cdotp$ besides that the space seems to be equally distributed when using $\cdot$

Here's what I found from Wikipedia on Interpunct. $\cdotp$ is the "centered dot used for interword separation in ancient Latin script."

Hence, as many other peoples have posted above, use $\cdot$ for multiplication, not $\cdotp$

Answer 4

There are no fewer than thirteen vertically-centered dots in Unicode 11.0, as of 2019.

Of these, the following are mapped to LaTeX commands by unicode-math:

· (u+000b7) is \cdotp or \centerdot. I normally use it in text mode because, as a Basic Latin character, text fonts are much more likely to have it. That might make it good for writing chemical formulas and compound units. (I would advise you to avoid using a middle dot as a decimal marker in a LaTeX paper, but if you did, this is what you would use.) If I need to enter a middle dot on my keyboard, I type this one because it needs fewer keystrokes. (And because it’s what my computer had in the ’90s.)

⋅ (u+022c5) is \cdot, which is what I normally use for multiplication.

∙ (u+02219) is the bullet operator, \vysmblkcircle. Wikipedia thinks it’s used for vector dot products and occasionally Boolean and.

The difference between these is usually very subtle, and I wouldn’t worry too much about using the right one unless somebody might search your paper for it.

• (u+02022) is \smblkcircle or \bullet. You would normally use something like itemize to generate a bulleted list in LaTeX.

Also, ̇ (u+00307) is the mathematical accent \dot, a combining dot above.

Most remaining center-dot symbols are punctuation from non-Latin scripts, such as Chinese and Japanese, and you would presumably enter them the same way as other characters from those languages.

Answer 5

Just a complement to all the other answers, and what I came looking for. The middle dot in Hellenistic Greek or Koine Greek context represent a text break, may be interpreted in English as:

• Colon : unicode U+003A
• Semicolon ; unicode U+003B

Middle dot in greek text published by:

• Nestle Aland, use character Middle dot (·) unicode U+00B7
• Crossway, use character Greek Ano Teleia (·) unicode U+0387

It is very common to find printed lexicons using the wrong dot representation, leading the poor student to learn something different. Some confusions I found, instead of using the middle dot, using:

• Raised Dot (⸳) unicode U+2E33
• Dot Above (˙) unicode U+02D9
• Dot Operator (⋅) unicode U+22C5

Answer 6

I had problems with the \cdot command to have the middle dot for chemical formula. I fixed it with ${\cdot}$.