Saturday, November 22, 2014


Gamepuzzles: Paradigms of a Rational World

As of 2014, I've been designing and making puzzles and games for 35 years. For so seemingly trivial an activity to become the meaning and purpose of my earthly existence, some intellectual underpinning was required. Like other forms of art, my designs exhibit aesthetics, conceptual integration and, hah, psycho-epistemological relevance. How do they do that?

My designs are based on mathematical phenomena, geometric tilings or, as they say in the ed biz, "tessellations". Now a tessellation is something that fills up space with permutations and combinations of shapes. Here are a few samplings:

M. C. Escher’s designs come to mind. And permutations and combinations echo the very structure of the Universe, from the way the simplest building blocks, like protons and electrons and neutrons, combine to form every element that exists. The periodic chart of elements organizes them into groups by the number of each bit in the core and its rings, that mimic the way planets orbit suns, and suns cluster in galaxies, and galaxies orbit in galactic clusters, and whatever makes up the next layers of hierarchy in an infinite Universe.

Tall order to replicate that grandeur in what some would consider merely a toy. Let's see an example you can hold in your hand.

"Polyominoes" are made of squares joined on their edges. They start from the singularity and can go on forever, combining and growing. Here are the 21 shapes from 1 to 5 squares in size (Tetris players will recognize the 4’s):
At level 6 there are 35 unique shapes (hexominoes). At level 7 we find 108 heptominoes. Level 8 gives us 369 octominoes.

That's as far as we go in actually producing working models. After only 8 iterations its volume exceeds human comfort, and in a few more steps the number of variations exceeds the number of atoms in the known Universe. Here is an amazing construction by Karl Wilk, astronomer by training, that took him months to solve. He calls it Cyclops, and it contains all the sizes from 1 through 8 forming concentric rings within a symmetrical ring of the 1285 enneominoes. Even the holes are symmetrically distributed:

Further research by mathematicians and computer programmers turns up 4655 dekominoes, 17,073 "11-ominoes" and 63,600 "12-ominoes". We can find reference to counts up to 28, giving 153,511,100,594,603 distinct shapes. See an extensive list here:

That's over 153 trillion if you're counting in U.S. sequences, or 153 billion in Europe, where they stick "milliard" after million, and "billiard" after billion. By any name it's a huge number. By comparison, our planet contains a little more than 7 billion humans, each of whom contains trillions of cells.

Now why would people spend so much effort to enumerate oddball shapes made of squares? Or other polygons? Because it is the nature of scientific curiosity and the human need to know what exists and how it works, the epistemological pursuit of an understanding of reality. It is our finest power.

I'm using the polyomino family as an example of the evolution and propagation of entities, both theoretical and actual, that illustrate how the Cosmos works. In effect they are clusters of pixels or "cellular automata". Human knowledge is likewise accumulated one bit, one byte, one meme, one layer at a time, integrated into the previously known structure.

That polyominoes even at the earliest levels have thousands of ways to connect and form coherent constructions, such as rectangles, squares, symmetrical shapes and even 3-D figures, is a model of how elements combine to form molecules, compounds, and all the stuff of the real world around us that we humans need to know about for our survival. And at some point in the continuum, we learn how ideas themselves form and fuse.

What is especially satisfying in working with such puzzle sets and producing different solutions to variously posited goals is that such activity feeds the mind's need to experience efficacy. To tackle a problem and resolve it in intelligent and creative ways is thoroughly satisfying. It serves our self-esteem. It exercises those capacities of mind that a rational consciousness needs for reaching its fullest potential.

As an artist and occasional poet, I see the working of these kinds of puzzles as a microcosm of problem-solving in the larger world. To coordinate, integrate, get disparate parts to work together in constructive ways without depriving any component of its individual attributes parallels the collaboration of individual humans for mutual benefit, not destroying, depriving or mutilating anyone for someone else’s benefit. In a rational world, each individual has the rights and freedom to find his or her best niche.
And in life as in puzzles, there is more than one answer. Having many options, not being shackled by totalitarian demands, is one of the hallmarks of individual freedom. And individual freedom is the fountainhead of creation. Celebrate uniqueness. Do no harm. Make glorious combinations…….

Visit for more of my playable art. They are unique in all the world, and the pleasure lasts. If you decide to order, please add a note of where you heard about me and you'll receive an extra gift.
Please feel free to comment on these ideas. I am working on a book about human consciousness, and your thoughts, civilly expressed, are most welcome. -- Kate

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