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The Cayley tree is a mathematical object related to a Bethe Lattice (see the wikipedia article for an overview here: It’s essentially a branched structure. Each node joining the branches in the “tree” has three branches emanating from it (think of it as a trunk that forks into two branches at the node). Each branch terminates at another node which also branches. A smooth plane has two dimensions and a smooth line has one dimension, but a Cayley tree has an intermediate dimensionality. This can be useful for certain types of calculations: where a fractal surface either better reflects the physics of the system, or where the symmetry of the Cayley tree simplifies the physics and/or math.
In “Variation on a Cayley Tree” some liberties were taken with the strict branching geometry and topology of the mathematical object, in order to map it onto a form that actually resembles a tree. The drawing “Variation on a Cayley tree” is part of a series of drawings and paintings examining tree symbolism in nature, and in the world of scientific ideas.


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Dr. Regina Valluzzi, “The Nerdly Painter” – Art Inspired by Science from a Scientist inspired by Art
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  • F.A. Moore
    F.A. Moorealmost 2 years ago

    I like it, and loved reading about it. In software programming (well, any computer programming), we’re working with trees, and in fact tree notation, in its various forms, incessantly.

    A simple visual example, followed by its notation.

       /  \
      a    d
     / \  /  \
    b  c e  f

    In nested array notation, for example:
    $tree = [ t => [ a => [ b => [], c => [] ], d => [ e => [], f => [] ] ] ];

    (If this doesn’t look right, I plead it’s due to the formatting. ;)

    I’m enjoying your tree series, Regina, and this one has struck a chord of a different kind.

  • Hi, Glad it struck that chord. I took some programming classes, but actually fell out of my chair (spectacularly, I might add). I remember binary trees fondly, and AVL trees are something I’d rather forget. The Cayley tree is sometimes used to model magnetism in funky lattices and occasionally for branched polymers. i’m sure the mathematicians also have their favorite trees.

    – Regina Valluzzi

  • Lynn Gedeon
    Lynn Gedeonover 1 year ago

    Nice pattern!

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