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Higher dimensional space

This part will cover

  • Constructing matrices
  • Constructing 3D arrays
  • The reshape function

The astute reader may have noticed that, although the vector data is much more structured, the dates and times of the measurements have been completely forgotten.

One solution is to use more vectors to organize this data. We represent dates here using floating-point decimal encoded format. This format stores the dates as decimal numbers, where the integer part stores the year, month, and day, and the fractional part stores the hour, minute, and second, yyyymmdd.hhmmss.

For example, 20240806.070130 is year 2024, month 08, day 06, hour 05, minute 01, and second 30.

      TEMPERATURE_PAGE1 ← 21.4 21.8 22.0 21.5 21.3 22.3
      TEMPERATURE_PAGE1_DATE ← 00010101.074200 00010101.084700 00010101.101000 00010101.120100 00010101.143600 00010101.165000
      TEMPERATURE_PAGE2 ← 22.8 21.5 22.1 22.0 21.9 22.4
      TEMPERATURE_PAGE2_DATE ← 00010101.182300 00010101.193000 00010101.211200 00010102.071500 00010102.083000 00010102.094500

and access dates and times using the same index;

      TEMPERATURE_PAGE1[2]
21.8
      TEMPERATURE_PAGE1_DATE[2]
00010101.084700

However, this lack of structure is exactly what introducing vectors was supposed to solve; two closely related pieces of information, the time of a measurement and the value of the measurement, are kept separate when they should logically be part of the same collection of data. Measurement data of this form are usually stored in tables, and it is only natural to try to store them in the same manner in a computer system.

You decide to start over yet again, and decide to store data in a matrix instead

      TEMPERATURE_PAGE1 ← 6 2 ⍴ 21.4 00010101.074200 21.8 00010101.084700 22.0 00010101.101000 21.5 00010101.120100 21.3 00010101.143600 22.3 00010101.165000
      TEMPERATURE_PAGE2 ← 6 2 ⍴ 22.8 00010101.182300 21.5 00010101.193000 22.1 00010102.211200 22.0 00010103.071500 21.9 00010103.083000 22.4 00010103.094500

Matrices are rectangles of data. They can be created by reshaping (⍴) a vector.

Typing the reshape function ⍴

Prefix method: PREFIX r Tab method: r r Tab

Function Valence

The symbol ⍴ actually represents two different functions depending on the manner in which arguments are given.

When applied to a single argument, ⍴X, it acts as the shape function; when two arguments are given one on either side, X⍴Y, it acts as the reshape function.

The former function is the monadic function associated to the symbol ⍴, and the latter is the dyadic function associated with the symbol ⍴.

The reshape function acts by returning an array whose entries are the entries of its right argument, and whose axes are specified by a vector of integers as its left argument, more concretely,

      TEMPERATURE_DATA ← 21.4 00010101.074200 21.8 00010101.084700 22.0 00010101.101000 21.5 00010101.120100 21.3 00010101.143600 22.3 00010101.165000
      6 2 ⍴ TEMPERATURE_DATA
21.4 10101.0742
21.8 10101.0847
22   10101.101
21.5 10101.1201
21.3 10101.1436
22.3 10101.165
      ⍝ The reshaped matrix has 6 rows and 2 columns

turns the vector TEMPERATURE_DATA into a matrix with axes of length six and two, consisting of the entries in TEMPERATURE_DATA.

     PYRAMID_ENTRIES ← 1 1 1 1 1 1 2 2 2 1 1 2 3 2 1 1 2 2 2 1 1 1 1 1 1 1
      5 5 ⍴ PYRAMID_ENTRIES ⍝ 5 rows and 5 columns
1 1 1 1 1
1 2 2 2 1
1 2 3 2 1
1 2 2 2 1
1 1 1 1 1

Strings in APL are vectors of characters, defined using single quotes. The useful ⎕A system constant stores the upper-case english alphabet 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'.

      WORD ← 'STONE'
      WORD
STONE

      ALPHABET ← ⎕A 
      ALPHABET
ABCDEFGHIJKLMNOPQRSTUVWXYZ

If the right argument is too short to fill the array, the reshape (dyadic ⍴) function repeats the right argument's entries.

      5 25 ⍴ ALPHABET 
ABCDEFGHIJKLMNOPQRSTUVWXY
ZABCDEFGHIJKLMNOPQRSTUVWX
YZABCDEFGHIJKLMNOPQRSTUVW
XYZABCDEFGHIJKLMNOPQRSTUV
WXYZABCDEFGHIJKLMNOPQRSTU

      5 4⍴WORD
STON
ESTO
NEST
ONES
TONE

The shape (monadic ⍴) function acts on one array, its right argument, by returning a vector whose entries are the lengths of the axes.

      TEMPERATURE_DATA ← 21.4 00010101.074200 21.8 00010101.084700 22.0 00010101.101000 21.5 00010101.120100 21.3 00010101.143600 22.3 00010101.165000
      TEMPERATURE_PAGE1 ← 6 2 ⍴ TEMPERATURE_DATA
      ⍴TEMPERATURE_PAGE1
6 2
      ⍴100 ⍝ The shape of a scalar is the empty list

      ⍴⎕A ⍝ Number of letters in the alphabet
26

Since elements in matrices are ordered along two axes, an element of a matrix can be specified by two position, the row and column. If only a row position (or column position) is specified, the whole row (or column) is returned.

     TABLE ← 5 5 ⍴ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
     TABLE
 1  2  3  4  5
 6  7  8  9 10
11 12 13 14 15
16 17 18 19 20
21 22 23 24 25

     TABLE[1;1]
1
     TABLE[1;]
1 2 3 4 5
     TABLE[;1]
1 6 11 16 21

     WORD_SQUARE ← 5 5 ⍴ "HEARTEMBERABUSERESINTREND"
     WORD_SQUARE
HEART
EMBER
ABUSE
RESIN
TREND
     WORD_SQUARE[1;]
HEART
     WORD_SQUARE[;1]
HEART
     WORD_SQUARE[5;]
TREND
     WORD_SQUARE[;5]
TREND

      TEMPERATURE_DATA1 ← 21.4 00010101.074200 21.8 00010101.084700 22.0 00010101.101000 21.5 00010101.120100 21.3 00010101.143600 22.3 00010101.165000
      TEMPERATURE_PAGE1 ← 6 2 ⍴ TEMPERATURE_DATA1
      TEMPERATURE_PAGE1
21.4 10101.0742
21.8 10101.0847
22   10101.101
21.5 10101.1201
21.3 10101.1436
22.3 10101.165
      TEMPERATURE_PAGE1[1;1]
21.4
      TEMPERATURE_PAGE1[1;2]
10101.0742
      TEMPERATURE_PAGE1[1;]
21.4 10101.0742
      TEMPERATURE_PAGE1[3;2]
10101.101

     TEMPERATURE_PAGE2 ← 6 2 ⍴ 22.8 00010101.182300 21.5 00010101.193000 22.1 00010102.211200 22.0 00010103.071500 21.9 00010103.083000 22.4 00010103.094500
     TEMPERATURE_PAGE2
22.8 10101.1823
21.5 10101.193 
22.1 10102.2112
22   10103.0715
21.9 10103.083 
22.4 10103.0945
     TEMPERATURE_PAGE2[1;2]
10101.1823
     TEMPERATURE_PAGE2[2;2]
10101.193
     TEMPERATURE_PAGE2[3;2]
10102.2112
     TEMPERATURE_PAGE2[;2]
0101.1823 10101.193 10102.2112 10103.0715 10103.083 10103.0945

Multiple numbers can be specified for both row and column indices.

       ALPHABET ← 5 5⍴⎕A
       ALPHABET[1;]
ABCDE
       ALPHABET[1 2 3;]
ABCDE
FGHIJ
KLMNO
       ALPHABET[1 2 3; 1 2 3]
ABC
FGH
KLM

However again, the data measurements are separated without reason, the problem that introducing matrices was supposed to solve. Going one dimension further, the data can be arranged in a three-dimensional ordered collection of data:

     TEMPERATURE_ARRAY ← 2 6 2 ⍴ 21.4 00010101.074200 21.8 00010101.084700 22.0 00010101.101000 21.5 00010101.120100 21.3 00010101.143600 22.3 00010101.165000 22.8 00010101.182300 21.5 00010101.193000 22.1 00010102.211200 22.0 00010103.071500 21.9 00010103.083000 22.4 00010103.094500
     TEMPERATURE_ARRAY
21.4 10101.0742
21.8 10101.0847
22   10101.101 
21.5 10101.1201
21.3 10101.1436
22.3 10101.165 

22.8 10101.1823
21.5 10101.193 
22.1 10102.2112
22   10103.0715
21.9 10103.083 
22.4 10103.0945

     ⍴TEMPERATURE_ARRAY 
2 6 2

     ⍴⍴TEMPERATURE_ARRAY 
3

     TEMPERATURE_ARRAY[1;;]
21.4 10101.0742
21.8 10101.0847
22   10101.101 
21.5 10101.1201
21.3 10101.1436
22.3 10101.165

     TEMPERATURE_ARRAY[2;;]
22.8 10101.1823
21.5 10101.193 
22.1 10102.2112
22   10103.0715
21.9 10103.083 
22.4 10103.0945

     TEMPERATURE_ARRAY[;;1]
21.4 21.8 22   21.5 21.3 22.3
22.8 21.5 22.1 22   21.9 22.4

Rank

The number of axes of an array is called the rank of the array.

The arrays we’ve constructed so far are of rank 0 (scalars), rank 1 (vectors), rank 2 (matrices), and rank 3. The maximum rank of an array in Dyalog APL is 15.

A useful idiom for getting the rank of an array is the shape of the shape of an array, ⍴⍴X.

Now with your temperature table safely stored in your APL workspace, you can only imagine how many more values you can log and maintain. You excitedly gesture at one of your unimpressed coworkers before you notice you’ve accidentally logged the temperature of the cabin as 226 degrees. Before they have a chance to look at your mistake, you quickly and shamefully change the value.

     TEMPERATURE_ARRAY
21.4 10101.0742
21.8 10101.0847
226  10101.101 
21.5 10101.1201
21.3 10101.1436
22.3 10101.165 

22.8 10101.1823
21.5 10101.193 
22.1 10102.2112
22   10103.0715
21.9 10103.083 
22.4 10103.0945
     TEMPERATURE_ARRAY[1;3;1] ← 22.6
     TEMPERATURE_ARRAY
21.4 10101.0742
21.8 10101.0847
22.6 10101.101 
21.5 10101.1201
21.3 10101.1436
22.3 10101.165 

22.8 10101.1823
21.5 10101.193 
22.1 10102.2112
22   10103.0715
21.9 10103.083 
22.4 10103.0945

That was close!

Changing values in arrays acts in the same manner as it does for the case of changing variables, specify the element(s) to change and assign a new value.

     BOX ← '╔═══╗║TRY║╠═ ═╣║APL║╚═══╝'
     BOX ← 5 5 ⍴ BOX
     BOX
╔═══╗
║TRY║
╠═ ═╣
║APL║
╚═══╝

     BOX[3;3] ← '═'
     BOX
╔═══╗
║TRY║
╠═══╣
║APL║
╚═══╝

     BOX[3;]
╠═══╣
     BOX[3;] ← '║   ║'
     BOX
╔═══╗
║TRY║
║   ║
║APL║
╚═══╝

     BOX[3;] ← '╬'
     BOX
╔═══╗
║TRY║
╬╬╬╬╬
║APL║
╚═══╝