Laminate

Laminate (,[0.5]) is a case of the Catenate function with axis that joins its argument arrays along a new length-2 axis. It applies whenever the function axis is non-integral, but is typically written with a half-integer axis. Possible choices of axis range from ,[⎕IO-0.5], which places the new axis before any argument axis, to ,[r+⎕IO-0.5], where r is the shared argument rank, which places it after all argument axes. Laminate appears as a stand-alone function in some languages, with the name Laminate in A+ (~) and J (,:), and Couple in BQN (≍) and Uiua (⊟). These languages implement the ,[⎕IO-0.5] case only, as the Rank operator can be used to add the axis in a later position, in accordance with leading axis theory.

Examples

A typical use of Laminate is to form a matrix from two vectors, using them as rows (examples use ⎕IO←1).

      'abcd' ,[0.5] 'efgh'
abcd
efgh

However, Laminate can be used more generally to pair up cells of the arguments, for example, by laminating along the last axis to pair elements of two shape 3 3 arrays.

      (3 3⍴⍳9) ,[2.5] 3 3⍴'abcdefghi'
1 a
2 b
3 c

4 d
5 e
6 f

7 g
8 h
9 i

As with Catenate, a scalar argument is subject to scalar extension: its rank is brought up to match that of the other argument by repeating its value.

      'x' ,[0.5] 'efgh'
xxxx
efgh

Extension support

APL's scalar extension is retained in A+ and J, but removed in BQN, which instead requires argument shapes to match exactly. Since J extends scalar extension to leading axis agreement for arithmetic, Laminate is one of a fairly small number of cases where scalars are treated specially. J extends Laminate further to handle any combination of argument shapes by padding a lower-rank argument with leading 1s in the shape and then padding any short axes with fills. This matches the padding behavior used by Rank and other operators to merge result cells into a single array.

Language	Syntax	Conformability
APL	`,[0.5]`	Possibly scalar or singleton extension
A+	`~`	Scalar extension
J	`,:`	Scalar extension; pads to conform rank and axes
BQN	`≍`	Strict
Uiua	`⊟`	Strict

Additionally, in J, laminating two scalars gives a result of shape 2 1 rather than 2.

Documentation

In APL: see Catenate
J NuVoc, Dictionary
BQN
Uiua

APL built-ins [edit]
Primitives (Timeline)	Functions
Scalar
		Monadic	Conjugate ∙ Negate ∙ Signum ∙ Reciprocal ∙ Magnitude ∙ Exponential ∙ Natural Logarithm ∙ Floor ∙ Ceiling ∙ Factorial ∙ Not ∙ Pi Times ∙ Roll ∙ Type ∙ Imaginary ∙ Square Root ∙ Round
		Dyadic	Add ∙ Subtract ∙ Times ∙ Divide ∙ Residue ∙ Power ∙ Logarithm ∙ Minimum ∙ Maximum ∙ Binomial ∙ Comparison functions ∙ Boolean functions (And, Or, Nand, Nor) ∙ GCD ∙ LCM ∙ Circular ∙ Complex ∙ Root
Non-Scalar
		Structural	Shape ∙ Reshape ∙ Tally ∙ Depth ∙ Ravel ∙ Enlist ∙ Table ∙ Catenate ∙ Reverse ∙ Rotate ∙ Transpose ∙ Raze ∙ Mix ∙ Split ∙ Enclose ∙ Nest ∙ Cut (K) ∙ Pair ∙ Link ∙ Partitioned Enclose ∙ Partition
		Selection	First ∙ Pick ∙ Take ∙ Drop ∙ Unique ∙ Identity ∙ Stop ∙ Select ∙ Replicate ∙ Expand ∙ Set functions (Intersection ∙ Union ∙ Without) ∙ Bracket indexing ∙ Index ∙ Cartesian Product ∙ Sort
		Selector	Index generator ∙ Grade ∙ Index Of ∙ Interval Index ∙ Indices ∙ Deal ∙ Prefix and suffix vectors
		Computational	Match ∙ Not Match ∙ Membership ∙ Find ∙ Nub Sieve ∙ Encode ∙ Decode ∙ Matrix Inverse ∙ Matrix Divide ∙ Format ∙ Execute ∙ Materialise ∙ Range
Operators	Monadic	Each ∙ Commute ∙ Constant ∙ Replicate ∙ Expand ∙ Reduce ∙ Windowed Reduce ∙ Scan ∙ Outer Product ∙ Key ∙ I-Beam ∙ Spawn ∙ Function axis ∙ Identity (Null, Ident)
Operators	Dyadic	Bind ∙ Compositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner Product ∙ Determinant ∙ Power ∙ At ∙ Under ∙ Rank ∙ Depth ∙ Variant ∙ Stencil ∙ Cut ∙ Direct definition (operator) ∙ Identity (Lev, Dex)
Quad names	Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector

Laminate

Examples

Extension support

Documentation

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