Honeycomb

One thing I really do love regarding the discussion of shapes in mathematics is the precision of the phrase “to tile the plane”. The plane. The infinite and singular expanse: That which lacks a dimension I, inter-dimensionally cubed, cannot imagine lacking. As in, there is no way for me to separate length and width. We live on the plane, assume the plane, stack our sentience upon it —consider the crackling yawn of an inflatable Santa. Consider the plane’s assemblage laying in wait for swells of function: Hexagon to honeycomb, form to process.

Shapes on the plane are not “in time”, not on anything, not inhabiting, there is nowhere for them to go. Shapes most fundamentally stay where they’re imagined. They’re also not ideas. They’re topological disks, material without volume. They suggest and communicate without mind or thought. They are thoughts.

Consider the perfectly durable and gapless periodicity of simple polygons. In their infinitely snug duplication they spread without stretching - continuous and uniform. Even those symmetrical forms which leave gaps leave necessary kinshape, they complete their own absence. Circles, squares and triangles may flip, spread, and self-similarly transform in all manner of gratuity without betrayal of the pattern. The Pattern, The Pattern, The Pattern: Nature gives and takes away. The Pattern, stuttering across the horizon, carries nothing but the truth of repetition. ‘tition. ‘tition. ‘tition.

I am on my back again. I’m falling out the bottom and landing on the top, afraid I’ll never be allowed to die. Afraid I’ll never end, find the fling, the final unwinding. Back far enough away from the point of reference and there is no difference and with no difference there is no conclusion and with no conclusion there is no

So. Not all repetition produces symmetry and this brings me great comfort. Difference and discrepancy, those chaos weevils of the geometric guild, infest the law and order. Consider the forbidden quasi-crystalline structures and impossible alloys that, in fact, permit themselves. All hail the anarchist shapes that dimensionalize against the rules of atomic-packing! We live on such an organism. Not Penrose with his kite and dart; not Islamic architects 500 years before him with their Girih on the floors of mosques and madrasas; but the rule of yes itself says yes to that which must not exist: shapes that tile the infinite plane without repeating.

How unfathomable is it to tile the plane in complete and utter difference? Try to fathom it. Try to fathom it five times. Consider that the plane does not cease to spread, and that these equally ceaseless aperiodic bitches do not obey the spreading. They never stutter. They just sprawl. Consider the golden ratio - which is the ratio to each other a set of non-repeating shapes will approach the closer they get to the edge, except there is no edge. How is that possible?

How is it possible that like a game of telephone the pattern can maintain its lock and weave while betraying its own design? The line is there and the shape has changed. The shape is almost there and the line has joined a search party for itself. They spin without finishing a full rotation and in their incompleteness they are somehow even more solid than the repetitiously snug bugs of aforementioned snoozefest.

One advantage of the aperiodic structure transposed to three dimensions: it is less susceptible to acoustic resonance – as valuable as buckyballs for mounting indestructible towers of babel or hypersonic spacecraft. Of all the numeric combinations of closed-shape angulation (an actual infinite number of shapes and combinations), only a few are willing to do this. Only last year did we discover the first one that can do it alone. It took humankind this long to find one. That’s how rare. That’s the significance and power of true aperiodicity. It is secret difference obscured by prolific repetition, and that is the only two-dimensional mirror in which I can see my own reflection.

I can’t see myself in shapes, tarpaulin attached to a fan that I am. The tessellation of order and chaos, though - I know the ever-unfinished spinning very well. And I, too, whisper into my own ear. I do get dizzy. A good amount of the time I can’t predict my own next move. I can say that being unpredictable is bad for Capital and because I am stuffed into this unidirectional situation of length and width with an awareness of my own disadvantage, I find it necessary to be bad for Capital. So I seek to fold my life into a shape that won’t repeat.

So. My aperiodicity is not a form. My devotion to the discovery of difference is a fox making more tracks than necessary. To antagonize the overlords I wish to sit on my own head like a chiral spectre and laugh. I want to wear impossible hats. I have work to do and jobs to quit. I want to love math because it is beautiful and holy; because at its boundaries it illuminates mystery and the mystery is what makes the endless spinning a joyride. Rocket launch me into a black hole at the end. I’m a fractal on my back under terror and I’m done being scared.

So. Here I am taking a long time to write my reasons for loving math and shapes when really what I want to say is just that I love a man with many triangles in his head. My headless triangle who says he is a hexagon but whose honey would suggest otherwise.

I know I am unreasonable. I know I take too long to say things. I could say I am maniacal on my back and all mountain and valley when I lay on my side. I could call him a frozen lake and smooth tundra under silent clouds. I could speak to his softness and loud head; could say his warmth is cold, his perpetual dusk and diffusion of refracted light is still and quiet. Ice and field and temperature and sky and crepuscular glow - all shapeless.

So. To calculate our difference I could call him three sixties minus a screw or a flattened expanse of six one-hundred twenties. I could say I’m a few angles that won’t cooperate, namely thirty-six, seventy-two, one hundred forty-four and two hundred sixteen degrees.

But really I'm a kite on the end of a braided cord that won’t break and a dart sunk into the dirt beside a paper target with no bullseye - which is something only a honeycomb could understand.