At the very least an Org document should begin with a title and a date. For example, the first two lines of this file are:

#+TITLE: Example Org File
#+DATE: 2020-01-31

Me non paenitet nullum festiviorem excogitasse ad hoc.

Consider the payoff matrix for Roshambo. If we want player one's options to be represented by the first column and aligned right while player two's options are represented by the first row and center-aligned, we can get a decent looking payoff matrix with the following markup in Org:

|          | Rock | Paper | Sissors |
|      <r> | <c>  |  <c>  |   <c>   |
|     Rock |  0   |  -1   |    1    |
|    Paper |  1   |   0   |   -1    |
| Scissors |  -1  |   1   |    0    |

Which produces the following table:

As can bee is seen above, columns can be aligned right with <r>, aligned center with <c>, and aligned left with <l>.

A table with a caption is easy to produce. Consider the magic square of order 3. We can compute the magic constant for magic square of order n using the formula \(M = n (n^2 + 1) / 2 \). This gives us a magic constant of 15 for the order 3 magic square. We can construct the magic square in Org with a caption like so:

#+NAME: tbl:ms-o3
#+CAPTION: Magic Square of Order 3
#+ATTR_HTML: :width 25%
| <c> | <c> | <c> |
|  2  |  7  |  6  |
|  9  |  5  |  1  |
|  4  |  3  |  8  |

The next code block is used to check if the styling can display 80 character lines without overflow. The block is made up of lines contains the first 80 characters of a base-85 encoding called Z85.

0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ.-:+=^!/*?&<>()[]{
0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ.-:+=^!/*?&<>()[]{
0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ.-:+=^!/*?&<>()[]{

Now we must check how a code block that overflows looks.

+ (nullable NSURL *)URLForResource:(nullable NSString *)name withExtension:(nullable NSString *)ext subdirectory:(nullable NSString *)subpath inBundleWithURL:(NSURL *)bundleURL API_AVAILABLE(macos(10.6), ios(4.0), watchos(2.0), tvos(9.0));

Text characters set below or above the line of type for a font/size combination are necessary for technical writing. Subscripts and superscripts are achieved in the same way as in TeX or LaTeX.

The radius of the sun is R_sun = 6.96 × 10⁸ m. On the other hand, the radius of Alpha Centauri is R_{Alpha Centauri} = 1.28 × R_sun.

There are many special symbols and non-Latin alphabets that can be inserted in an Org document. The syntax is like TeX or LaTeX too.

Given a circle Γ of diameter d, the length of its circumference is πd. We can type α as \alpha, β as \beta, and χ as \chi. If you need a symbol in a word or formula it can be terminated with a pair of curly braces, so the previous πd is achieved by typing \pi{}d.

Some other symbols include © as \copy, † as \dag, and × as seen in the subscript/superscript section above as \times. There's ± as \pm or \plusmn. There's ½ as \frac12 and there's ¾ as \frac34. There's x → ∞, typed as x \to \infin. Don't forget the en dash – or the em dash —, respectively -- and ---.

Get a list of special symbols with org-entities-help or you can add your own in the variable org-entities-user.

Inline math such as \(a^2\)is inserted when \(\LaTeX\)is delimited by \( … \). Whereas display math is inserted when \(\LaTeX\)is delimited by \[ … \]. Consider \(n \ge 1, F_0 = 0, F_1 = 1\), then we can express a recurrence relation in dispaly math as

\[ F_{n+1} = F_n + F_{n-1} \]

Given a proposition \( P \)and a proposition \( Q \)we can define some logical modifiers, combinatory terms, and relations using truth tables. We will denote true with a T and false with an F.

The following is a summation with index k from 1 to n - 1.

\[ \sum_{k=1}^{n-1} a_k = a_1 + a_2 + a_3 + … + a_{n-1} \]

The following is a summation with index j from 0 to n - 1.

\[ \sum_{j=0}^{n-1} x^j = 1 + x + x^2 + x^3 + … + x^{n-1} \]

There are several external links in this document. Org can link to many different types of resources. The two most common types of resources are web links "https" (or "http") and "file" links.

A clickable URL to the On-Line Encyclopedia of Integer Sequences follows https://oeis.org. The next example is a descriptive link to the NIST Digital Library of Mathematical Functions.

With an internal link we can refer to a previous section like the section on Math. Or we can link to some named element such as the wallpaper group figure. Or we can link to a section 2 without a description. Section links are of the form [[*section name][optional description]], or to use a section's custom ID to form the link [[#custom-id][optional description]], and to link to a named entity use the form [[entity-name][optional description]], e.g. for the wallpaper group figure the link is formed with [[fig:wg][wallpaper group figure]].

Phasellus laoreet lorem vel dolor tempus vehicula. Donec sed odio operae, eu vulputate felis rhoncus. Fictum, deserunt mollit anim laborum astutumque! A communi observantia non est recedendum. At nos hinc posthac, sitientis piros Afros.

Give a man a fish and feed him for a day. Don't teach a man to fish and feed yourself. He's a grown man. And fishing's not that hard. — Ron Swanson

Sed haec quis possit intrepidus aestimare tellus. Lorem ipsum dolor sit amet, consectetur adipisici elit, sed eiusmod tempor incidunt ut labore et dolore magna aliqua. Fabio vel iudice vincam, sunt in culpa qui officia. Morbi odio eros, volutpat ut pharetra vitae, lobortis sed nibh.

It would be very nice if I could illustrate the various techniques with small demonstration programs and could conclude with "… and when faced with a program a thousand times as large, you compose it in the same way." This common educational device, however, would be self-defeating as one of my central themes will be that any two things that differ in some respect by a factor of already a hundred or more, are utterly incomparable.

History has shown that this truth is very hard to believe. Apparently we are too much trained to disregard differences in scale, to treat them as "gradual differences that are not essential." We tell ourselves that what we can do once, we can also do twice and by induction we fool ourselves into believing that we can do it as many times as needed, but this is just not true! A factor of a thousand is already far beyond our powers of imagination!

[…]

It is in the same mood that I should like to draw the reader's attention to the fact that "clarity" has pronounced quantitative aspects, a fact many mathematicians, curiously enough, seem to be unaware of. A theorem stating the validity of a conclusion when ten pages full of conditions are satisfied is hardly a convenient tool, as all conditions have to be verified whenever the theorem is appealed to. In Euclidean geometry, Pythagoras' Theorem holds for any three points A, B and C such that through A and C a straight line can be drawn orthogonal to a straight line through B and C. How many mathematicians appreciate that the theorem remains applicable when some or all of the points A, B and C coincide? Yet this seems largely responsible for the convenience with which Pythagoras' Theorem can be used. — Edsger W. Dijkstra

Cum sociis natoque penatibus et magnis dis parturient. Morbi odio eros, volutpat ut pharetra vitae, lobortis sed nibh. Sed haec quis possit intrepidus aestimare tellus.

Tyger Tyger, burning bright,
In the forests of the night;
What immortal hand or eye,
Could frame thy fearful symmetry?
    — William Blake

Paullum deliquit, ponderibus modulisque suis ratio utitur. Quisque placerat facilisis egestas cillum dolore. Nihil hic munitissimus habendi senatus locus, nihil horum?

Once upon a midnight dreary, while I pondered, weak and weary,
Over many a quaint and curious volume of forgotten lore—
    While I nodded, nearly napping, suddenly there came a tapping,
As of some one gently rapping, rapping at my chamber door.
“’Tis some visitor,” I muttered, “tapping at my chamber door—
            Only this and nothing more.”
    — Edgar Allan Poe

Inmensae subtilitatis, obscuris et malesuada fames. Salutantibus vitae elit libero, a pharetra augue. Pellentesque habitant morbi tristique senectus et netus. Fictum, deserunt mollit anim laborum astutumque! Cum sociis natoque penatibus et magnis dis parturient.

Vivamus sagittis lacus vel augue laoreet rutrum faucibus. Quae vero auctorem tractata ab fiducia dicuntur. Cras mattis iudicium purus sit amet fermentum. Morbi fringilla convallis sapien, id pulvinar odio volutpat. Sed haec quis possit intrepidus aestimare tellus.

Magna pars studiorum, prodita quaerimus. Cum sociis natoque penatibus et magnis dis parturient. Cum ceteris in veneratione tui montes, nascetur mus. Pellentesque habitant morbi tristique senectus et netus. Quo usque tandem abutere, Catilina, patientia nostra?

Some example from a text file.

Org footnotes are just links that start with fn:. The footnotes may be numbered, named, anonymous and can be inline or defined in a footnotes section. A numbered footnote looks like this:

The Org homepage[fn:1] now looks a lot better than it used to.
...
[fn:1] The link is: https://orgmode.org

Named and anonymous footnotes look like these:

[fn:name]

A named footnote reference, where name is a unique label word, or, for simplicity of automatic creation, a number.

[fn:: This is the inline definition of this footnote]

An anonymous footnote where the definition is given directly at the reference point.

[fn:name: a definition]

An inline definition of a footnote, which also specifies a name for the note. Since Org allows multiple references to the same note, you can then use [fn:name] to create additional references.