APL Wiki logo: Difference between revisions

The APL Wiki logo can be seen as the following numeric matrix, where each number indicates the circle size. This page will explain, step-by-step, an expression<ref>[https://codegolf.stackexchange.com/users/78410/bubbler "Bubbler"], message [https://chat.stackexchange.com/transcript/message/52389201#52389201 "52389201"] in ''The Nineteenth Byte'' chat room. Stack Exchange network, 2019-10-31 23:57</ref> for this matrix — an expression which demonstrates quite a few APL features:

We will follow APL's evaluation from right to left.

The <source lang=apl inline>⍳</source> function takes a number ''N'' and [[index generator|generates indices]] until is has made ''N'' indices. Since we set <source lang=apl inline>⎕IO</source> to 0, we count from 0 until right before ''N'':

Consider a bag with four distinct items. If you stick your hand into the bag and pick two items out, how many different possibilities are there  for which pair you get out? <math>\{(0,1), (0,2), (0,3), (1,2), (1,3), (2,3)\}</math>. APL can tell you this with the <source lang=apl inline>!</source> function:

Notice how APL uses traditional mathematical symbols in a generalised way. The traditional post-fix (after its argument) symbol <math>!</math> is used with a syntax similar to how you'd normally use <math>+</math> or <math>×</math>. In fact, all APL functions can be used infix, like <math>a-b</math> or prefix, like <math>-b</math>.

A really nice feature of APL is its array-orientation. For computations which are defined on single elements, [[wikipedia:map (higher-order function)|map]]ping is implicit:

(What's up with picking zero out of four items? Since all empty hands are equal, there is exactly one such set — the empty set.)

We want to generate the indices using <source lang=apl inline>⍳</source>…

If the arguments of <source lang=apl inline>!</source> were swapped, we didn't need that parenthesis. Enter the [[operator]] (higher-order function) [[swap]] (<source lang=apl inline>⍨</source>) which takes a [[dyadic]] function on its left and creates a new [[derived function]] which is identical to the original, but has swapped arguments:

Notice how we were dealing with integers until now, but then we multiply by a float (non-integer). In APL, you don't need to worry about numeric data type conversions. All numeric types get automatically promoted and demoted as needed. APL will usually use the most compact internal representation.

Also notice that we use a proper multiplication symbol, <math>×</math>, for multiplication. If traditional mathematics has a symbol for a concept APL includes then APL will use that symbol. Another example is <math>÷</math> for division.

In fact, any function will do:

It gets tedious to type the same argument twice. Enter the [[self]]ie operator which shares its symbol with the above-mentioned [[swap]] operator. There's no ambiguity here. ''Swap'' swaps the two arguments, while ''selfie'' uses a single argument twice:

The last step is to round these numbers down. Traditional mathematics writes ''floor'' as <math>⌊x⌋</math> but APL is regular, so no function is denoted by two separated symbols. If the function takes a single argument, then the symbol will be on the left, so we write floor as <source lang=apl inline>⌊x</source>: