Graeco-Latin Square - Mutually Orthogonal Latin Squares

Mutually Orthogonal Latin Squares

Mutually orthogonal Latin squares arise in various problems. A set of Latin squares is called mutually orthogonal if every pair of its element Latin squares is orthogonal to each other.

Any two of text, foreground color, background color and typeface form a pair of orthogonal Latin squares:
fjords jawbox phlegm qiviut zincky
zincky fjords jawbox phlegm qiviut
qiviut zincky fjords jawbox phlegm
phlegm qiviut zincky fjords jawbox
jawbox phlegm qiviut zincky fjords

The above table shows 4 mutually orthogonal Latin squares of order 5, representing respectively:

  • the text: fjords, jawbox, phlegm, qiviut, and zincky
  • the foreground color: white, red, lime, blue, and yellow
  • the background color: black, maroon, teal, navy, and silver
  • the typeface: serif (Georgia / Times Roman), sans-serif (Verdana / Helvetica), monospace (Courier New), cursive (Comic Sans), and fantasy (Impact).

Due to the Latin square property, each row and each column has all five texts, all five foregrounds, all five backgrounds, and all five typefaces.

Due to the mutually orthogonal property, there is exactly one instance somewhere in the table for any pair of elements, such as (white foreground, monospace), or (fjords, navy background) etc., and also all possible such pairs of values and dimensions are represented exactly once each.

The above table therefore allows for testing 5 values each of 4 different dimensions in only 25 observations instead of 625 observations. Note that the five 6-letter words (fjords, jawbox, phlegm, qiviut, and zincky) between them cover all 26 letters of the alphabet at least once each. The table therefore allows for examining each letter of the alphabet in five different typefaces, foreground colors, and background colors.

Due to a close relation between orthogonal Latin squares and combinatorial designs, every pair of distinct cells in the 5x5 table will have exactly one of the following properties in common:

  • a common row, or
  • a common column, or
  • a common text, or
  • a common typeface, or
  • a common background color, or
  • a common foreground color.

In each category, every cell has 4 neighbors (4 neighbors in the same row with nothing else in common, 4 in the same column, etc.), giving 6 * 4 = 24 neighbors, which makes it a complete graph with 6 different edge colors.

Read more about this topic:  Graeco-Latin Square

Famous quotes containing the words mutually, latin and/or squares:

    “Let it be an alliance of two large, formidable natures, mutually beheld, mutually feared, before yet they recognize the deep identity which beneath these disparities unites them.”
    —Ralph Waldo Emerson (1803–1882)

    “Americans living in Latin American countries are often more snobbish than the Latins themselves. The typical American has quite a bit of money by Latin American standards, and he rarely sees a countryman who doesn’t. An American businessman who would think nothing of being seen in a sport shirt on the streets of his home town will be shocked and offended at a suggestion that he appear in Rio de Janeiro, for instance, in anything but a coat and tie.”
    —Hunter S. Thompson (b. 1939)

    “An afternoon of nurses and rumours;
    The provinces of his body revolted,
    The squares of his mind were empty,
    Silence invaded the suburbs,”
    —W.H. (Wystan Hugh)