As a seven-year-old, when travelling by road between my grandparents’ homes, I would eagerly wait for the next signboard to flash by, jotting down the name of the village. Later, I took immense satisfaction in poring over that list.
I’ve had an inexplicable connection with maps for quite some time!
In fact, in my grandparents’ backyard, I dug an intricate network of underground tunnels and passageways, dotted with the clichéd mud mountain. I loved building bridges over exposed patches of otherwise underground passageways. Though the subterranean path sensed the complex world around it, it mostly followed its own course.
But only as much as the surroundings would allow.
Maps would be rather boring if all they contained were paths that never intersected – not just with themselves, but also with the surroundings: There is not much one can siphon from an army of parallel lines marching down the page…
When we observe the behavior of each individual element in a meeting of several elements of similar type, yet distinct, we understand the element’s true nature. For instance, a particular road, twisting alone at first, may later stick close to another road for the rest of its journey, hinting that the original road was not ‘important’ to begin with.
Years later, when I explored vector spaces and groups, I was delighted to discover that it was the functions between them that revealed the most about these structures!
***
At the time, I was staying in Ottenbach, Switzerland. There, I continued studying Google Maps, but with a renewed sense of purpose: planning cycling routes.
One evening, I panned onto Aussichtspunkt mit Sitzbank (bench overlooking a vista), the end of my walking path…
…and there I found it: first cutting its way through Ottenbach, the road then sloped downward toward a river. Finally, it snaked up the far side of the valley, bypassing a town shaped more by the land than its neighbors — a town named ‘Buttwil’.
I originally learned about homomorphisms in mathematics, but I now see them everywhere. At one level, homomorphisms construct equivalences between objects that seem different when viewed superficially but share strikingly similar internal structures – so much so that the outer labels are irrelevant in the larger context.
I shut the laptop, tingling with anticipation: I knew where I was going. I just needed to trust that embedding.
In essence, a map is a large homomorphism that allows one to explore our three-dimensional world through a two-dimensional one.
***
When I set off the next morning, I found myself in that queer ‘in-between’ region just past the river, as the flat valley began. The vast landscape warped my vision: I was clearly moving, but as I glanced between the churches of Ottenbach and Aristau, I might have been stationary. That produced an oddly light feeling in my stomach, egging me to go on.
Though the steep climb and freezing weather weren’t easy, the prospect of sitting on a bench, playfully recognizing the structures around Ottenbach, piecing the bits of the scenery together was too delicious.
En route, I had to improvise – not knowing the exact way, only the general direction to Buttwil – once squeezing my cycle through a series of poles, apparently there to keep cars off.
Eventually, I reached the summit. Sitting down, I tried to take the landscape in, the ‘in-between’ region ringing softly in my ears. Each time I saw something from Buttwil, I went back to my memory of that object as viewed from Ottenbach…effectively constructing an impromptu homomorphism.
I took extreme comfort as I imagined how this side of the valley might have looked had I been at the Aussichtspunkt mit Sitzbank, content that the question had no answer, only overlapping speculations in my mind. From here, however, I could see many more Aussichtspunkt mit Sitzbanks – each as obvious as mine. I got on my bike…
Numbers Matter 