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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 for this matrix — an expression which demonstrates quite a few APL features:

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      ⎕IO←0
      ⌊∘.+⍨.5×4!⍨⍳5
1 2 3 2 1
2 4 5 4 2
3 5 6 5 3
2 4 5 4 2
1 2 3 2 1

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

Counting from 0 or from 1

A computer console: ⎕

Whether to count from 0 or from 1 is an old disagreement among programmers. Many APLs let you choose whichever convention you want, but they tend to use 1 by default. To switch convention, we set the variable ⎕IO:

      ⎕IO←0

By the way, IO stands for Index Origin.

We can already now observe a couple of APL's characteristics:

• The name ⎕IO begins with the special Quad character (a stylised console) which symbolises the computer system itself. APL has no reserved words. Rather, all built-in constants, variables, functions and operators have the prefix ⎕ indicating that they are part of the system. Because of this, we call them quad names.

• Assignment is not done with = like in many other programming languages, but rather with ← which also indicates the direction of the assignment: Whatever is on the right gets put into the name on the left.

Generating indices

The ⍳ function takes a number N and generates indices until is has made N indices. Since we set ⎕IO to 0, we count from 0 until right before N:

      ⍳5
0 1 2 3 4

How many subsets?

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? $\{(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)\}$. APL can tell you this with the ! function:

      2!4
6

Notice how APL uses traditional mathematical symbols in a generalised way. The traditional post-fix (after its argument) symbol $!$ is used with a syntax similar to how you'd normally use $+$ or Failed to parse (syntax error): {\displaystyle ×} . In fact, all APL functions can be used infix, like $a-b$ or prefix, like $-b$.

Anyway, how many sets of four could you pick? Obviously, only one; all the items:

      4!4
1

A really nice feature of APL is its array-orientation. For computations which are defined on single elements, mapping is implicit:

      0 1 2 3 4!4
1 4 6 4 1

(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.)

Order of evaluation

We want to generate the indices using ⍳…

      ⍳5!4


That didn't work! This is because APL dispenses with traditional mathematics' confusing and inconsistent precedence order, replacing it with a simple right-to-left rule:

      (⍳5)!4
1 4 6 4 1

Swapping arguments

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

      4!⍨⍳5
1 4 6 4 1