Tiny ideas to encourage girls to explore mathematics and physics - 6: Example: "Geometry in a Different Place"

Mauri Favaron
Posted November 30, 2009 from Italy

In secondary school we got the first exposition to geometry. Now, on the other side of the desk, we look at those little "us, before" and have (I confess) to admit how abstract and self-standing the objects of geometry look.

A problem might be that geometry, the way it is typically presented, seems so unrelated to a child's life. It seems something abstract you have to learn just because it is.

But: is it possible to place geometry in a wider context? To understand it has been made by someone, to address some very practical need, and might be reformulated, would this need change.

Personally, I find the idea that other geometries can be devised in ecological contexts different to mine fascinating. And I'm not dealing with the "non-Euclidean geometries": departures may be much deeper, even in the way of conceiving the space itself, and its most basic elements.

This my contribution is just what the title suggests: a tiny idea. It isn't rigorous (and I have not the skill of making it). Nor I pretend it is "definitive". If anyone of you may help me placing this framework in logical terms, I'd be sincerely glad (and, I promise, will do my part of work, even under direction).

On the other side: in my (little) experience this way of presenting geometry as something constructed is attractive. And, besides indicating other strange possibilities, may help giving geometry a meanings.

So, imagine. Imagine being in a garden, with plants, flowers and grass. Your children, or pupils, are sitting in a circle around you, waiting for a new story. There is (let's imagine that, too, it costs so nothing!) a thin, gentle breeze. And (this is a miracle where I stay, but let's imagine) silence (but for some little bird).

We are. We are, now. Let's begin with the story. It deals with Agw (a place somewhere (or nowhere) in the world), and the way its mathematicians intend with the word "geometry".

Here we are...

You must remember! ;-)

We have just seen. The “space” is that large huge hollow thing with three dimensions (length, height, depth) you can fill with many things: points, lines, triangles… And also villages, pools, sea, cattle, vegetable, people, and anything you can imagine to occupy its portion of the Space.

Well, I know you remember this very well, I was just kidding.

But, have you wondered why we began exploring "space" and geometry? Just to give you some other homeworks?

No. Sure, no. Geometry, you'll discover too soon, is useful.

But today I want to tell you a different story.

You know, geometry is not something existing from the origin of intelligent life on this planet. It has been invented, some thousands years ago, by someone who needed it. We don't know who this (or these) genius was: at the time writing did not exist, so we don't know her or his name. But sure was some real person with some very practical problem to solve.

Which need? So much time has passed... But this does not prevent us from doing some intelligent guessing, isn't it?

Let's imagine: all may have began because people needed to measure in some way the surface of their fields. Where? Of course, where fields existed in the same way we are used to them - maybe in the Middle East, some 10000 years ago...

You know: if you have to plough a field, you (and, maybe, your cattle) will have to work hard. But of course, the larger the surface, the huger the workload. But also, the yield will be greater. As the farmer, it would be really nice to have some way to predict how long you will have to work, and how much yield will you harvest (hoping nothing goes wrong).

How do you do? Using area! Both the workload and the yield will be roughly proportional to area: double it, and you get twice your grain.

But to compute the area, you have first to invent lines, and their intersections, points. You have to build a geometry which is useful to you, the farmer. And, of course, the king or queen who will tax you depending on how wealthy you are supposed to be (once again, based on how much land you grow.

So now: you are studying one of the many possible existing geometries: that of land surveyors and farmers who settled the Middle East in pre-historical time, and then the ancient Greece. In that specific place and time people went literally mad with abstractions, and cleaned up what I imagine was a much simpler and intuitive, but less precise, body of knowledge. The result is what we now know as “geometry”.

Think a moment. In “space” there is an infinity of points. But: may you really distinguish any of them? Right, by their position. But apart of position, what do they have of different?

My feeling (a deep one, for sure) is: nothing! Would it not be because the place they are, you would have no way to distinguish these points. They are “infinitely small”, pure position, so have no color, weight, flavor, texture, or whatever property you may think of.

And: would you move from one position to the other, will the space itself prevent you from doing? No, of course. Given two points in positions, say, A and B, you may imagine to fly from the first to the second along infinitely many strange trajectories. If you feel really lazy, you also have the opportunity to fly from A to B along the straight segment connecting them - and this is precisely the shortest path from A to B.

That is: in principle, you can navigate space without limitation. The only limit is your fantasy.

Is the “real” space something so? This place is a part of space: can you really go wherever you want?

Of course no, it is filled of so many things. Walls, plants, roads, people (moving and not), and an immense variety of characters and things.

So many, that we need a map just to figure out where to go.

Sure, a space of points all so almost-identical to one another (that is, identical, except for their position), which you can freely navigate, and in which you can trace lines and figures at your will is very useful to a land surveyor or a queen.

But: is it the “only” kind of space (and geometry) you may devise?

The geometry of Agw

I imagine you know what Agw is.

Some people say this place (Agw is a place) does not exist. You may feel free saying so, if you like (in which case you are in good company).

But, I assure you: places as Agw exist and are, indeed, very common.

First of all, you have to know Agw is in a rain forest. Trees there are not like the ones you can see in a garden. They are immense, so tall they may cover a skyscraper with their shadow. Behind these trees there are others, a bit less tall, and other more. Until you arrive at ground (of course, if you are able to see where ground is).

Agw is a very nice place to live in, provided you know how to do.

But, we have to admit, in Agw there is no field. Even the schools are somewhere within the trees, among twigs and tangles.

As people has no field, no Agwean (this is the name of the inhabitants of Agw) did ever felt a need to invent geometry as we know it. And in fact, all children in Agw learn a geometry in which there is no “wide space” as we are used to, but rather a huge set of “localities”.

In Agw geometry, any two localities are clearly different and well defined, as you would say of the Tour Eiffel the Mosque of Suleymaniye, the Simon Bolivar Park at Bogota, the St. Peter Cathedral in Rome, the Uhuru Highway at Nairobi, or any other place of “our” world. When you are there you are there and can’t mistake it.

Of any two places you also can say whether they are “connected” or not. Place A is connected to B, if you know there is a path joining A with B. I know this seems a bit of nonsense but, we all have to admit, these “paths” may be very complicated and tortuous.

An interesting thing in the space as known by all Agwean is, no one can guarantee any two localities chosen at random are actually connected. And in fact, many are not. How different from our space, where any two points can be connected with a straight line!

Of course, if you are a little child you will know very few localities. But as you grow, you discover many many more - and this process continues the whole life. Although smaller, we are not that different from the Elephants: as you know, the queen of each flock remembers so many places she has seen in her long life, to be able to go where water is. As the big elephant, we add more and more localities to our own list, and also more connections as we find new paths.

A very important concept in Agwean geometry is that of “minimum length connection” (Agwean mathematicians are no less prone to use difficult terms as ours). By this, they intend that, if you choose two localities at will, say them (again!) A and B, maybe there is not a direct path joining them. But there may be one or more other points C1, C2, …, Cn so that A is connected to C1, C1 to C2, …, Cn to B. In general (as the forest is very large) many such connections can be found (someone of infinite length). Of these, there will be one, or few, with the minimum possible number of intermediate localities: this is the “minimum length connection”.

But of course, as no one in Agw is concerned with giving distances a number, they don’t know what the actual length of the minimum length connection is (they really don’t bother).

A curious phenomenon, observed both among the Agwean and us, is that as people grow, their sense of “minimum length way” worsen, although for different reasons. For us, because as we become older we also become a bit lazy. For them, because as you learn more and more localities, invariably the number of intermediate points joining A with B increases (even though, if we want to be completely sincere, the actual distance between A and B remains exactly the same - unless of course A is on the ground and B on a tree, in which case distance too increases, as the tree grows!)

Agweans know, and use extensively, triangles, quadrangles, and any kind of polygon or geometrical solid. Only, they define them their way, as “connections between vertices”, all vertices being their localities of course.

Now, homework: may you imagine a geometry useful to dolphins? (Hint: imagine being a dolphin, who can swim, dive many tens of meters, jump in air, and ride waves; which kind of “geometry” would be of some use to you?)

For us adults

As I imagine it, the Agwean “geometry” is the result of a “refinement” process on an undirected graph (which needs not be immersed in three dimensional space, although this view may help visualizing it).

More precisely, I imagine a sort of “learning process” in which sets of vertices V1, V2, …, Vk, … are obtained by progressively “adding” new vertices. Same with edge sets E1, E2, …, Ek, …

In case of a real person this process will sooner or later end, with life and/or will to explore. But why limiting ourselves? We can imagine an endless refinement! If you imagine the graph as immersed in a conventional N-dimensional Euclidean space, you may also do something like "passing to the limit" - and see what happens ;-)

“How” refinement may take place? In principle, no restriction is necessary. But in practice, there may be some preference rules, allowing us to know in more depth places we can access and we are interested in because of any reason. In this case, it is not hard to imagine some kin of optimization rules, which may “differ” from “culture” (ecological setting and niche) and “culture”. Each might give rise to subtly different geometries.

The way vertices and edges “self-add” is visible in many natural learning processes, for example the formation of erosion networks, or the shaping of bones to optimize response to load. Learning about our own’s territory is, I imagine, something similar in its dynamics.

My core assumption is, “space” is a category highly conditioned by culture. I assume any Agwean would have no difficulty to understand the logical structure of our commonplace Euclidean space, as we can with Agwean space. But, I imagine they would find it less natural, less aesthetically attracting. Exactly as we (mostly) do.

An interesting (and sad) thing happened during the Western invasion of lands far away, during the colonial age. The Europeans entered cultivated fields without even realizing they were inside a farm - just, this “farm” was made of trees and tortuous and almost invisible (but optimal in that context) trails. Did colonialist mindset carry with it a geometric superiority presumption, in addition to many other things?

Most important, and possibly valuable, the concept that geometry, as we know it, is the distilled abstraction of one way to conceive “space”, of the many existing. Placing geometry in a context, showing its initial use as a very practical tool much before architecture and other arts co-opted it shifting its attention to proportions and other abstract categories, may make it much less afar, and more concrete.

On a more practical side: does the "Agwean space", in the limit, provide a model for our space? Or, better, for the way we encode and categorize it in our brains? Its now commonplace that some people prefer a more directly "Euclidean" mental representation, made of direction lines, distances and skillful navigation. While others prefer a more empirical, "Agwean" I may say, representation of landmarks and paths connecting them.

Imagine letting abstractions stay aside for a moment and limiting attention to the part of space we actually can touch and sense (mostly, a tiny part of the Earth surface and subsurface, including our artificial caves, "houses"). In this case, and in our ecological/cultural context the two strategies are practically equivalent (in the sense, both allow their users to navigate their territories avoiding dangerous places and finding shelter and food; none of the two would be subject to natural selection because "unfit"). But: in a different ecological context? Agweans, we can imagine, may do quite a little with a Euclidean mindset...

Last, and to conclude. Agweans "exist", or at least "existed". They were not a unique population. Many cultures developed very advanced technologies allowing them to settle and live in rain forests, which in ecological terms are quite poor environments despite the apparent immensity of biomass. This technology, seen from a Western standpoint, may seem "primitive" or inept. It isn't. Sure, it is not based on iron forging or other "Westerner-visible" technologies. But, it involves a very detailed knowledge of territory (much more detailed we are used to), and practical deep understanding of "biology". Both biology and territory knowledge may have been encoded in forms so diverse from what we can now imagine, that understanding them fully might be not possible within a time span.

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