Not really blogging

A. Makelov

Monthly Archives: January 2014

Mars = Science fiction

One minute it was Ohio winter, with doors closed, windows locked, the panes blind with frost, icicles fringing every roof, children skiing on slopes, housewives lumbering like great black bears in their furs along the icy streets.

And then a long wave of warmth crossed the small town.

Ray Bradbury, “Rocket summer“

1. The bottom line of this post, in two words

It seems that there is enough textual and contextual evidence to suggest that the “Martian chronicles” can be read, among other things, as the chronicles of an abstract process – the development of science fiction as a genre, progressing from the realm of inventions like humanoid machines and rockets to the point of indistinguishability from magic.

2. Introduction

For all you dedicated Bradbury fans, yes, “Rocket summer“ is the first short story in his “Martian chronicles“. When I was first reading the book, this opening vignette meant nothing to me besides the literal fact that a rocket is being launched, and I quickly forgot about it. But later, with the book finished, and the deadline for my next English paper looming ahead (that’s how inspiration works, apparently), impressions gave rise to ideas and ideas gave rise to a principle at work – behind “Rocket summer“ and behind the rest of the chronicles. Or a piece of rubbish that I managed to convince myself is enough evidence for a principle at work – I can’t really tell the difference; the paper on which this post is based can be found here.

3. Ambiguity in plot

Everyone who’s read the “Chronicles“ knows how confusing and ambiguous the whole thing is. In part, this is because most of the short stories are unrelated, having been published separately by Bradbury before the collection was assembled, and then joined together by short vignettes, such as “Rocket summer“, to the effect that the many Mars-es depicted seem quite different from each other. Moreover, both we, as readers, and the characters we read about, keep encountering the same problem: distinguishing Mars from Earth, past from future, travel through space from travel through time, human from Martian. In “Ylla“, Martians behave just like ordinary people; in “The Earth men“, Martians comically fail to recognize humans as a different species; in “The third expedition“, Mars in Bradbury’s future is indistinguishable from Earth from his past:

Lustig said, “But suppose, by accident, in space, in time, we got lost in the dimensions and landed on an Earth that is thirty or forty years ago.“

In “And the moon be still as bright“, the future is mixed with another past – the colonization of America, and it is extremely unclear whether Spender actually switched with a Martian: the Martian appeared before me and said, `Give me your boots’ […] And the Martian walked down into camp and he’s here now.

4. Ambiguity in genre

The ambiguity in the stories is paralleled by an ambiguity between science fiction and fantasy as genres. While the “Chronicles” are often called science fiction (often enough for Wikipedia to do it), Bradbury insisted that

First of all, I don’t write science fiction. I’ve only done one science fiction book and that’s Fahrenheit 451, based on reality. It was named so to represent the temperature at which paper ignites. Science fiction is a depiction of the real. Fantasy is a depiction of the unreal. So Martian Chronicles is not science fiction, it’s fantasy. It couldn’t happen, you see? That’s the reason it’s going to be around a long time.” because it’s a Greek myth, and myths have staying power.

On the surface, the book employs many of the customary tropes of science fiction, such as space colonization, telepathy, humanoid machines; on the other hand, their rational explicability is either absent or neglected, and the stories are driven by open-ended, timeless philosophical questions set in poetic environments, rather than Campbell-era-style logical puzzles which always have a solution.

5. “Night meeting“

But when it comes to ambiguity, the best example by far is “Night meeting“, where the book seems to be aware of its unifying themes. I’d strongly suggest reading it if you haven’t, for it is a very beautiful piece on its own (the entire collection can be found here). Foreshadowed by the remark “Even time is crazy up here” made by the man at the gasoline station, a meeting takes place between the opposites mentioned above (these being Mars-Earth, past-future, space-time, human-Martian), embodied by Muhe Ca, the Martian, and Tomas, the human:

“You are so certain. How can you prove who is from the Past, who from the Future? What year is it?”

“Two thousand and one!”

“What does that mean to me?”

Tomas considered and shrugged. “Nothing.”[…]

How do you know that those temples are not the temples of your own civilization one hundred centuries from now, tumbled and broken? You do not know.

Most of all, this is saying that looking far enough into the future is indistinguishable from looking into the past – and here comes the principle at work (literally! : ) ).

6. The principle

Any sufficiently advanced technology is indistinguishable from magic.

Arthur Clarke

Clarke’s most famous quote has a fairly unpopular corollary: “Any sufficiently advanced science fiction is indistinguishable from fantasy”. In light of the previous sections, one is inclined to think that the Chronicles’ blend of fantasy and science fiction is working together with elements of the plot to tell a meta-narrative of the science fiction genre, rather than a particular science fiction or fantasy story.

One can further interpret Mars as a metaphor for science fiction itself, and more specifically for its ability to generate wonder. Carl Sagan succinctly captured the attitudes of people from Bradbury’s time towards the red planet:

Mars has become a kind of mythic arena onto which we have projected our Earthly hopes and fears.

Carl Sagan

The concept of a projection is key here, but perhaps in a more intentional context than what Sagan implied. A projection, in both the mathematical and psychological sense, is a transformation whose subject and outcome coincide from some perspectives, and are displaced from others. One of the great merits of science fiction is that it functions largely as such a projection: of current society and technology into future times and alien places, and indeed a projection of our existential hopes and fears. By displacing concepts from their everyday contexts to a genuinely new setting via its novum, it helps us separate the fundamental from the specific – and, consequently, prejudice from rationality. But when the displaced reality is overwhelmingly far from our point of view, the effect wraps around, back in time, to magic, fantasy – and wonder.

Mars, in Bradbury’s Chronicles, is the target of such displacement – the arena where past meets future, science fiction meets fantasy, and so on. In this line of thought, going back to the man at the gasoline station from “Night meeting”, we find the following little jewel:

“How do you like Mars, Pop?”

“Fine. Always something new. I made up my mind when I came here last year I wouldn’t expect nothing, nor ask nothing, nor be surprised at nothing. We’ve got to forget Earth and how things were. We’ve got to look at what we’re in here, and how different it is. I get a hell of a lot of fun out of just the weather here. It’s Martian weather. Hot as hell daytimes, cold as hell nights. I get a big kick out of the different flowers and different rain. I came to Mars to retire and I wanted to retire in a place where everything is different. An old man needs to have things different. Young people don’t want to talk to him, other old people bore hell out of him. So I thought the best thing for me is a place so different that all you got to do is open your eyes and you’re entertained. I got this gas station. If business picks up too much, I’ll move on back to some other old highway that’s not so busy, where I can earn just enough to live on and still have time to feel the different things here.”

“You got the right idea, Pop,” said Tomas, his brown hands idly on the wheel. He was feeling good. He had been working in one of the new colonies for ten days straight and now he had two days off and was on his way to a party.

“I’m not surprised at anything any more,” said the old man. “I’m just looking. I’m just experiencing. If you can’t take Mars for what she is, you might as well go back to Earth. Everything’s crazy up here, the soil, the air, the canals, the natives (I never saw any yet, but I hear they’re around), the clocks. Even my clock acts funny. Even time is crazy up here. Sometimes I feel I’m here all by myself, no one else on the whole damn planet. I’d take bets on it. Sometimes I feel about eight years old, my body squeezed up and everything else tall. Jesus, it’s just the place for an old man. Keeps me alert a nd keeps me happy. You know what Mars is? It’s like a thing I got for Christmas seventy years ago – don’t know if you ever had one – they called them kaleidoscopes, bits of crystal and cloth and beads and pretty junk. You held it up to the sun light and looked in through at it, and it took your breath away. All the patterns! Well, that’s Mars. Enjoy it. Don’t ask it to be nothing else but what it is. Jesus, you know that highway right there, built by the Martians, is over sixteen centuries old and still in good condition? That’s one dollar and fifty cents, thanks and good night.”

7. It all makes sense now

Or at least, some of it makes sense. Remember “Rocket summer”? The rocket, that science-fiction-y thing, is described more like a fire-breathing dragon from some fantasy book; by its technological power, it superposes past and future, showing “last summer’s ancient green lawns”; it also projects summer onto winter, rendering the “bear disguises” of the housewives obsolete. Remember “Usher II”? There literature came to life on Mars.

Finally, remember the ending?

The Martians were there–in the canal–reflected in the water. Timothy and Michael and Robert and Mom and Dad. The Martians stared back up at them for a long, long silent time from the rippling water.

Did they really have to become the Martians, in the end? Was that the only possibility? Maybe “if you can’t take Mars for what she is, you might as well go back to Earth”, in the words of the old man at the gasoline station, so when the radiation on Earth dissipates they will return. Yet Mom and Dad burned the old way of life. And maybe they didn’t have a choice – for Mars is science fiction, and you can’t help but colonize, and become, science fiction one day.

Advertisements

Fundamental domains

This post is about the area of math known as general topology, and I’ll thus assume some basic background (for example, chapter 2 of the textbook by Munkres should be enough). The post was produces with a little modified version of Luca Trevisan’s latex2wp, and the source .tex can be found here. There is a bit of a hope that next time I’ll be able to make the math look better.

What made me write this was the Internet’s apparent lack of a simple but rigorous introduction to fundamental domains and their usefulness in taking quotients by group actions. The goal is, for a space {X} and group {G} acting on {X}, to establish sufficient conditions for the quotient {D/G} of a fundamental domain {D} (to be defined below) by the induced action of {G} to be homeomorphic to {X/G}.

1. A motivating example

Consider the space {\mathbb{R}^2} in the standard topology, and the (discrete) group {\mathbb{Z}^2} acting by translation: {(m,n)\in\mathbb{Z}^2} acts on {(x,y)\in\mathbb{R}^2} by sending it to {(x+m, y+n)}. Suppose you want to get an understanding of the quotient space {\mathbb{R}^2/\mathbb{Z}^2} with respect to this action. Basically, you want to take the real plane and, for each point, glue to it its orbit under {\mathbb{Z}^2}. Working with {\mathbb{R}^2} can make your head hurt, so here’s a trick: consider the unit square {D=[0,1]^2}. It almost contains a single representative of each orbit, except on the boundary. This gives us the much simpler gluing instructions

torus

The resulting space is the torus. The subspace {D} is known as a fundamental domain for the action of {\mathbb{Z}^2} on {\mathbb{R}^2}.

2. Group actions on topological spaces: a crash course

In topology, one often considers quotients of topological spaces by group actions (for example, in the beautiful theory of covering spaces, which you don’t really need to know to get this post). This is a natural extension of group actions on sets which takes into account the continuity of the topological spaces:

Definition 1 A topological group {G} is a space and group {G} such that

  1. The composition map {G\times G\to G} is continuous.
  2. The inverse map {G\to G} is continuous.

Definition 2 A topological group {G} acts on a space {X} if {G} acts on the set {X} and the corresponding action {G\times X\to X} is continuous.

To get some practice with these concepts, do these exercises:

Exercise 1 If {f:X\times Y\to Z} is a continuous map, show that for every {x\in X}, the map {f(x,\cdot):Y\to Z} is continuous.

Exercise 2 If {G} acts on {X}, show that for any {g\in G}, the map {f_g:X\to X} given by {x\to gx} is a homeomorphism.

3. Fundamental domains

One way to define a fundamental domain formally is the following (though we won’t use the full strength of this definition):

Definition 3 A fundamental domain is a closed subset {D\subset X} such that {X} is the union of translates of {D} under the group action:

\displaystyle \begin{array}{rcl} X= \displaystyle\bigcup_{g\in G}gD \end{array}

and such that {\mathop{Int}(gD\cap g'D)=\emptyset} for any two distinct translates.

What we ideally want to get is {D/G\cong X/G}. I know of no result stating that, and maybe there’s a counterexample. The following two propositions detail what I do know in terms of sufficient conditions:

Proposition 4 If :

  1. {X/G} is Hausdorff, and
  2. {D} is compact, or {D/G} is compact,

then {D/G\cong X/G}.

Proof: Observe that if we let {q,p} be the canonical quotient maps from {D,X} to {D/G,X/G} respectively, and {i} be the canonical inclusion {D\to X}, we have that whenever {a,b} are in the same {G}-orbit in {D}, they get mapped to the same element in {X/G}. Consequently, we have the following diagram:diagram

where {f} is continuous and unique by the universal property of the quotient. What does {f} do? Suppose we have an element {a} of a {G} orbit in {D}. Under {p\circ i}, {a} gets mapped to a representative of its {G}-orbit in {X/G}; but {p\circ i = f\circ q}. Hence {f} maps a representative of {a}‘s {G}-orbit in {D/G} to a representative of {a}‘s {G}-orbit in {X/G} (which is exactly the natural map we would expect to get).

Observe that {f} is injective: if {[a],[b]\in D/G} are such that {f([a])=f([b])}, it follows that there are some {a,b\in D} which go to the same element in {X/G}, and thus {a\sim b}, so {[a]=[b]}.

Moreover, {f} is surjective: observe that {p\circ i} is surjective, since {D} contains a representative of each orbit, and thus {f\circ q} is also surjective, which cannot happen if {f} fails to be surjective.

Thus, {f} is a continuous bijection {D/G\to X/G}. Since {D} being compact implies that {D/G} is also compact (being a continuous image under the quotient), and since {X/G} is Hausdorff, {f} is a homeomorphism. \Box

Going back to our example, obviously {[0,1]^2} is compact, and it’s easy to check that {\mathbb{R}^2/\mathbb{Z}^2} is Hausdorff.

Proposition 5 In terms of the notation of the previous proposition, {D/G\cong X/G} if and only if {p\circ i} is a quotient map.

Proof: Observe that {p\circ i} is a surjective continuous map {D\to X/G}, and that {D/G} is the quotient of {D} by the equivalence relation with equivalence classes – the subsets {\{(p\circ i)^{-1}(\{z\}) \ \big| \ z\in X/G\}}. Thus, Corollary 22.3 from Munkres applies to tell us that there is a homeomorphism {D/G\to X/G} \Box

This would hold, for example, if {p} is a closed map, since {i} is a closed map by {D} being closed in {X}, compositions of closed maps are closed, and every closed map is a quotient map.

mathbabe

Exploring and venting about quantitative issues

Rational Altruist

Adventures of a would-be do-gooder.

Gowers's Weblog

Mathematics related discussions

What's new

Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao

Stefan's Blog (archive, for news see blog.krastanov.org)

Most of the blog moved to blog.krastanov.org