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.

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.