hexagonal-tesselation

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Honey bees build complexes of hexagonal wax cells in their nests to contain their larvae and stores of honey. Why do these insects prefer hexagons to, for instance, square cells (which are more straightforward to build)?

There are two possible explanations. One is that hexagon tiles the plane with minimal surface area. This claim (for obvious reasons called the “honeycomb conjecture”) was proved only in 1999 by Thomas Hales, and implies the hexagonal structure uses the least material to create a lattice of cells within a given volume.

Another explanation is that the hexagonal shape simply results from the process of individual bees putting cells together, somewhat analogous to the boundary shapes created in a field of soap bubbles. In support of this theory, it is observed that queen cells, which are constructed singly, are irregular, with no apparent attempt at efficiency.

MIDDLE BRAIN

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.”


Bees in octagonal hives require modifications to their traditional funeral rites .

There are more confusing brain geometries than merely having a middle lobe.

Bees and ants have similar dreams.


North of Reality is an explorable fiction space written by Uel Aramchek. You can receive these pieces overnight via email by signing up here. This is entry #213. Learn how you can receive secret stories via physical mail here.

Injured

“Do you get injured?” She asked.

Chat Noir looked up from the chaise in Marinette’s bedroom to find her looking at him with piercingly blue eyes. He looked down at his hand, fingers tipped with claws, and skin covered in the fine black tessellating hexagons of his suit. He’d wondered about it himself, but Plagg had been uniquely uninformative about the limitations of his strengths, and weaknesses.

“No,” he said, after thinking about it. Marinette tilted her head at him, and he sat up, flexing his fingers, thoughtfully. “Things hurt,” he admitted, “but not as much as you’d think.” He’d been thrown off a building, and could bounce across the road without so much as a bruise, although they hadn’t exactly been comfortable. “The suit’s magic,” he said, softly, “it protects me from a lot.”

Marinette hummed, wordlessly, and turned back to her homework. He’d told her he’d sit patiently and wait for her to finish it, because he wouldn’t be responsible for her being grounded on account of poor grades. He’d planned to let her finish it, without interrupting her, but the question, and subsequent murmur, made that difficult.

“Worried about me, Princess?”

“Yes.” He hadn’t expected that answer, and he looked at her as if seeing her for the first time. She wasn’t looking at him, but he got the impression she was deliberately not looking at him, rather than concentrating hard on her studies.

“Don’t be,” he said, eventually.

“You’re too self sacrificing,” she replied, her voice suddenly quiet. “You take hits for Ladybug all the time. I just,” she trailed off, her head bowing slightly, “wondered if they hurt.”

Chat felt his throat tighten as he looked at Marinette. He tried to reply, but found his jaw worked uselessly until he grit his teeth, his shoulders slumping. “They don’t hurt,” he said, finally. Much.

Marinette nodded, the corner of her mouth twitching faintly in a sad smile. “At least there’s that.”

Reading the neural code for space

The cognitive map for spatial navigation is thought to rely on grid cells. Scientists at LMU and Harvard University have now put forward a mathematical theory that explains key grid-cell features and how these give rise to a neural metric for space

One year ago, the Nobel Prize in Physiology or Medicine went to the discoverers of the mammalian “GPS system” for spatial navigation. Measuring neural activity in cortex, these researchers had found that some cells represent space in a highly surprising manner: As the animal moves through its environment, distinct sets of cells are sequentially activated. Each individual “grid cell” responds to multiple positions in space that form a virtual hexagonal lattice tessellating the environment. This strikingly periodic and beautiful spatial pattern has caught the imagination of experimental and theoretical neuroscientists alike, and has been proposed to constitute the brain’s metric for space.

Theoretical neuroscientists at LMU Munich and the Bernstein Center for Computational Neuroscience in Munich, and at Harvard University, have now put forward a comprehensive mathematical theory that explains many features of grid cell activity that had remained mysterious, and makes specific predictions that can be tested in neurophysiological and behavioral experiments. The framework proposed by Martin Stemmler (LMU), Alexander Mathis (Harvard) and Andreas V.M. Herz (Professor of Computational Neuroscience at LMU) exploits and extends a computational principle well known from sensory systems and the brain’s motor cortex – population-vector decoding: physical quantities such as the angle of a visual stimulus or the direction of a movement can be easily read out from the activity of a population of neurons with different tuning properties if the neurons’ activities are combined in a particular, vector-like manner.

However, the theory proposed by Stemmler and colleagues goes far beyond these previous schemes in that the decoded quantity – the animal’s position – is not a circular variable. “Animals move on two-dimensional surfaces or in three-dimensional spaces, so their position is not a one-dimensional variable either. In addition, population-vector averages across different grid scales need to be combined,” explains Andreas Herz. Yet, the overall conceptual similarity of the new scheme to conventional population-vector decoding suggests that there exist overarching computational principles that operate throughout the brain. “This raises hopes that, in spite of the brain’s complexity, a thorough understanding of the neural basis of many cognitive processes is possible.”

Thanks to the new theory, three basic questions can now be readily answered: Why are grid cells organized into discrete modules, within which all grid lattices share the same spatial scale and orientation, but have different spatial offsets? Why do the spatial scales of the grid lattices form a geometric progression? And why is the observed scale ratio close to 3/2? “If grid cells were not organized into modules with fixed grid scale and orientation, the brain could not use population vectors to represent spatial position,” explains Martin Stemmler. “The geometric progression in the grid scales maximizes the spatial resolution of the neural code, based on just a few bursts of neural activity. Finally, the scale ratio should be close to 3/2 to avoid large-scale errors when information from different grid modules is combined to calculate a single quantity, the animal’s position estimate.”

The theory also shows that grid cells can be used not only to estimate position in world-centered coordinates, but also to tell the direction to some goal – for example, a food source or the animal’s home – in self-centered coordinates, a key requirement for navigation. “Transiently silencing individual grid modules should lead to specific types of navigational errors – a prediction that can be tested in future research”, says Alexander Mathis. Perhaps most importantly, the new work translates the hitherto rather hazy notion of “a neural metric for space” into a precisely defined mathematical framework. The results have already helped to settle a recent controversy in the field – whether the observed distortions of grid patterns falsify the proposed role of grid cells as a metric for space (no, they don’t) – and will, most probably, also guide future experiments.

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