The beekeeper focused his eyes to confirm what he was seeing: each cell within the honeycomb before him was octagonal in shape. As far as he knew, this was impossible. Hexagons were supposedly the most complex shape which could be tessellated, yet somehow, the edges of each octagon were perfectly aligned with those of eight others. He looked at them from multiple angles, wondering if a difference in perspective might cause the anomaly to go away, but the impossibility yet persisted.
Later that night, the beekeeper tried to draw the pattern that he had seen in his hive. He could easily make hexagons tesselate- each edge was shared with another as his pen’s tip slid from place to place- but octagons refused to obey him. He could see the pattern they made in his mind’s eye as a honey-colored phosphene, but no matter how many times he tried, their shapes simply could not be traced in ink. He went to sleep, concluding that what he saw had been some form of illusion, or a temporary madness.
In the morning, the eight-sided cells were still there.
He called another beekeeper, hoping to reestablishing his sense of sanity. “Eight-sided cells?”
“Yeah. That’s what I’m seeing.”
“Well, you’re in luck, then.”
“Oh? And why might that be?”
“Because eight-sided combs should be able to hold more honey than six-sided ones.”
“But, what does it mean?”
“What does what mean?”
“Aren’t you bothered by the fact that eight-sided honeycombs are mathematically impossible?”
“Maybe if you had asked me that ten years ago, but now? Not really,” the other beekeeper began. “Listen, man: I've seen bees do some really weird stuff. When it comes to building hives, they’ll abandon the laws of geometry themselves if they’re getting in the way of productivity. The bees don’t have to play by our rules, because they don’t actually have to think individually. Surround one with mirrors, for instance, and it will become a hive all on its own. That’s the evolutionary advantage of a hive mind: it allows for the simulation of intelligence without any need for the species to obey reason.”
“Wait, what? So, it’s not impossible?”
“Afraid not. You’re just going to have to accept it, you know? Use your middle brain for once.”