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Reshape and Iota — Building Arrays

Two functions you'll use constantly: ⍳ (iota) to generate sequences and ⍴ (rho) to shape them.

Iota: generating sequences

Monadic ⍳ generates integers from 1 to its argument:

      ⍳6
1 2 3 4 5 6
      ⍳1
1

Note

MARPLE's default index origin is 1 (⎕IO←1), so ⍳5 starts at 1. Some APLs default to 0.

Iota is often the starting point for building data:

      2 × ⍳5
2 4 6 8 10
      (⍳5) * 2
1 4 9 16 25

Reshape: ⍴

Dyadic ⍴ reshapes data into a given shape:

      3 4 ⍴ ⍳12
 1  2  3  4
 5  6  7  8
 9 10 11 12

If you provide fewer elements than needed, they recycle:

      3 3 ⍴ 1 0
1 0 1
0 1 0
1 0 1
      2 5 ⍴ 0
0 0 0 0 0
0 0 0 0 0

This is useful for creating arrays filled with a constant, identity-like patterns, or repeating sequences.

Shape: asking about dimensions

Monadic ⍴ returns the shape of an array:

      ⍴ 'Hello'
5
      ⍴ 3 4 ⍴ ⍳12
3 4
      ⍴ 2 3 4 ⍴ ⍳24
2 3 4

The number of elements in the shape tells you the rank:

      ⍴⍴ 3 4 ⍴ ⍳12
2

Ravel: ,

Monadic , (ravel) flattens any array into a vector:

      M ← 2 3 ⍴ ⍳6
      , M
1 2 3 4 5 6

Catenate: ,

Dyadic , joins arrays along the last axis:

      1 2 3 , 4 5 6
1 2 3 4 5 6
      (2 3⍴⍳6) , (2 2⍴10 20 30 40)
 1  2  3 10 20
 4  5  6 30 40

Putting it together

These building blocks combine naturally:

      ⍝ A multiplication table
      (⍳5) ∘.× (⍳5)
 1  2  3  4  5
 2  4  6  8 10
 3  6  9 12 15
 4  8 12 16 20
 5 10 15 20 25

      ⍝ Count of elements
      ×/ ⍴ 3 4 5 ⍴ 0
60

Key points

  • ⍳N generates the integers 1 through N
  • shape ⍴ data reshapes data into the given dimensions (recycling if needed)
  • ⍴ array returns the shape; ⍴⍴ array returns the rank
  • , ravels (monadic) or catenates (dyadic)
  • These functions are the basic toolkit for creating and exploring arrays

Next: Reading APL Right to Left