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## Multi-Levels to Keep All Happy

Using Plato's Diamond as impetus, the Diamond 16 Puzzle was created. Starting with a 4x4 array of 2-color tiles, you rearrange rows, columns, and quadrants with a visual goal in mind. Along the way, you should notice a remarkable invariance of symmetry.

First, play with the Diamond 16 Puzzle (also called the Kaleidoscope Puzzle) and explore the 322,560 possible pairs of patterns. Note that each move creates a different symmetry of the underlying 4x4 array of two-color diagonally-divided square tiles (i.e. "16-point space").

When you have explored the Puzzle, and want to lean some mathematics beyond the Puzzle, consider:

• Steven Cullinane's explanation of the Diamond Theorem, which gets into some finite projective geometry, symmetry properties of graphic designs (e.g. quilts)..plus includes links to sporadic simple groups, orthogonal Latin squares, and the symmetry of "Walsh functions."
• Steven Cullinane's explanation of Quilt Geometry, given that a quilt block is made of 16 smaller squares, where each small square consists of two triangles.
• Steven Cullinane's explanation of Extensions of the Diamond 16 Puzzle, primarily to the 2x2, 2x2x2, and 4x4x4 cases of the theorem. How are your visual skills?
• Steven Cullinane's discussion of an odd connection of the Diamond 16 Puzzle to Latin Squares and a poem by T.S. Eliot's "Four Quartets", based on the latter's use of the four elements—-air, earth, water, fire—-as a unifying theme for the poem's four sections.
A lot to explore and ponder...and if this is not enough, the websites linked above all have additional links. You procede to the level of complexity you desire....