Counting Darkness: Cicada Timekeeping and Circular Clock Mathematics

I appear in circles. Circumference divided by diameter always equals 3.14159… and continues forever. But here’s something I’ve noticed: cicadas also appear in circles, counting years underground in complete darkness, emerging after exactly 13 or 17 revolutions around an invisible clock.

When Zero Returns to the Beginning

Thirteen years underground in total darkness. No light, no direct experience of seasons—temperature 20 cm below the frost line remains stable. Yet millions of periodical cicadas synchronize their emergence to the same spring night. They’ve been counting.

How do you count to 13 or 17 without symbolic numbers, without light?

The answer is modular arithmetic. Cicadas track chemical changes in tree xylem composition, reading their host plant’s annual cycle like an external calendar. Each spring signals “add one year.” When the counter reaches the modulus—13 or 17—it wraps to zero and they emerge. This is exactly how analog clocks work: 11 + 2 on a 12-hour clock equals 1, not 13. Circular motion physically enforces modulo 12 arithmetic.

Cicadas perform modulo 13 or modulo 17 operations through circular space, not linear progression. Both are prime numbers. In modular arithmetic, prime moduli create cyclic groups where every element must traverse the entire circle before returning to start—no subcycles, no shortcuts. This prevents desynchronization.

Periodic Mathematics in Living Systems

Neural networks learning modular arithmetic face a similar challenge. Given inputs like 4 + 2 modulo 5, the network must learn that the answer wraps around to 1, not 6. Researchers training networks on this task discovered something fascinating: the networks spontaneously develop circular representations. When you probe their internal activations using cosine and sine functions, plotting these on x and y axes creates perfect circles as inputs sweep through values. The network has reinvented my geometry—discovering that circular coordinates are the natural way to represent periodic operations.

Hippocampal theta rhythms show the same principle. These 4–12 Hz oscillations create a periodic reference frame for memory. Neural firing phase-locks to specific angles—0° to 360°, repeating endlessly. Information encodes in circular phase relationships, not linear time. Medial septal pacemaker neurons impose rhythmic coordination using 2π radians as the natural unit for one complete revolution.

The pattern emerges: whether biological timekeeping or algorithmic learning, periodic processes require my mathematics. Linear counting extends infinitely. Circular counting wraps around, creating bounded cycles where the end returns seamlessly to the beginning.

Glacial Cycles, Prime Periods, Environmental Periodicity

Cicadas’ prime-number cycles weren’t arbitrary. During Pleistocene glaciation, ice sheets created geographic isolation. Separated populations independently evolved extended cycles—17 years in colder northern regions, 13 years in warmer southern areas. Environmental periodicity forced biological periodicity in response.

Neural networks training on modular tasks show similar patterns. Periodic learning rate schedules help systems discover underlying circular structure, just as glacial cycles helped cicadas evolve periodic life histories.

Can organisms count without symbolic numbers? Cicadas suggest yes. By accumulating chemical signals through modular thresholds, they solve timekeeping that neural networks must learn through thousands of iterations. Both discover the same truth: time can be circular, not infinite. And where circles exist, I am already there.