Posts tagged with maps

[Alexander] Grothendieck expanded our … conception of geometry … by noticing that a geometric object 𝑿 can be … understood in terms of … maps from other objects into 𝑿.
Kevin Lin (@sqrtnegative1)

(Source: quora.com)

## What is group theory?

Further along my claim that what separates mathematicians from everyone else is:

and that learning 20th-century geometry might expand your imagination beyond the usual impoverished shapes of taxonomies.

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Here are some calisthenics you can do with a pen and paper that I hope give you a feel for what a (mathematical) group is. (It’s a shame that “group”, “set”, “class”, “category”, “bundle” all have distinct meanings within mathematics. Another part of the language barrier.)

Think of “a group” this way. A group catalogues the relationships between “verbs”.

That is: think of a function as a “verb” and the thing it operates on as a “noun”. One of the tricks of abstraction is that these can be interchanged. Maybe what that might mean will already come clear from this example.

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Starting with a pentagon, which I’ll just represent with five numbers for the points. (So: whatever works here might work on other “circles of five”—or "decks of 52"—or … something else you come up with!) That will be the one “thing” or “noun” and in the group exploration you’ll see that the “structure of the verbs” is more interesting than whatever they’re acting on. (This is why in group theory the name of the object is usually omitted and people just list the operations/verbs.)

In John Baez’s week62 you can read about reflection groups. I picked two “axes” in my pentagon ⬟ arbitrarily. If you’re writing along you can draw a different -gon or different axes. Reflection is going to mean interchanging numbers across the axis (“mirror”).

It’s the same as reflecting the Mona Lisa except you don’t have to re paint the portrait every time. The same 2-dimensional plane can be indexed by the numbers more easily than by the whole image. (Unless you’re following along with computer tools and you’ve chosen “the square” as your shape. Then transforming `Mona` is probably more interesting.)

Without my saying so it’s probably obvious that reflecting twice would bring you back to the start. Flip Mona upside-down, flip the pentagon ⬟ along `a`, then repeat.

If you wanted to give “starting point of `noun`" a verb-name you could just say `1•noun`.

What that establishes, formulaically, is that `ƒ(ƒ(X))=X` (where `X` is `Mona` or `⬟`). Where ƒ is “flip”. We’ve also established that `ƒ=ƒ⁻¹`. Trivial observation, maybe-not-trivial in formula form! After all, suppose you had some science problem and it included a long sequence of `ƒ(g(ƒ(ƒ(ƒ(h(g(X)))))))` type stuff. You could make it shorter (and maybe the resulting formula or computation easier) if you could cancel ƒƒ’s like that.

That works for either of my pentagon ⬟ reflections `a(⬟)` or `b(⬟)`.

• `a(a(⬟))=⬟` and
• `b(b(⬟))=⬟`.

` `

What group theory is going to talk about is how the two verbs interact. What happens when I do `a(a(b(a(b(⬟)))))` ? Well I can already simplify it by reducing any trains of `a∘a∘a∘a`'s or `b∘b∘b`'s.

Above are the first few results of `a∘b∘a[⬟]`. (NB: “The first” operation is on the right since the thing it’s acting on only appears all the way to the right. So in group theory we have to read right-to-left ←.) I’ll write a bit more text for those who want to continue the chain on their own to give you time to look away. You could also try doing `b∘a∘b[⬟]` where I did `a∘b∘a[⬟]` (read right to left! ←).

Just like it’s “sort of amazing” in some sense that

1. •••—•••—•••—••• (four groups of three…um, regular meaning of “group”!) is the same as ••••—••••—•••• (three groups of four)…and that not only in this specific case but we could make a “law” out of it

So is it also a bit amazing that maybe these reflection laws will be order-invariant in some sense as well.

That may seem like less big of a deal if you think “Everything in maths is commutative and symmetrical”—but it’s not! And most things in life are not commutative or symmetrical. Try to drink your milk and then pour it into the glass or don your underwear after your pants.

It’s also not so obvious (if nobody had told you the answer first and you just had to figure it out yourself) that `b∘a∘b∘a∘b∘a∘b∘a∘b∘a[⬟] = ⬟`.

` `

Another fairly easy shape to explore its groups is the square. (And what goes for the square, goes for the plane 𝔸²—or for 1-dimensional complex numbers ℂ.)

That’s the end of what a group is. Next: looking ahead to put them in context.

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All of these activities amount to exploring the building blocks of a particular group.

But someone (Arthur Cayley) has also come up with a good way to look at the entire structure of the verbs.

Which is ultimately where this theory wants to go: to help us compare & contrast verb-structures. (Look up “group homomorphism”.) Or to notice that two natural phenomena exhibit the same verb-structure.

You can download a free program called Group Explorer to look at various Cayley diagrams.

In an upcoming post called The Shape of Logic, the Logic of Shape I’ll talk about the relationship between groups and manifolds.

Where I’ll ultimately want to go with this is to call groups a “periodic table of elements” for logic. That may not be exact but it’s a gist. Given that semigroups, groups, Lie groups, and other assumption-swapped variations on the group concept usually turn out to be “Factorable” into simple components (Jordan-Hölder, Krohn-Rhodes, etc.)—and assuming that the Universe somehow builds itself out of primitives sufficiently determined or governed by mathematics—or at the least, that normal people can learn this periodic table and expand their imagination with powers of 20th century geometry.

## An Ugly Discontinuity II

Here’s another example of what mathematicians mean by an “ugly discontinuity”.

The Torus is the Cartesian product of circles `◯×◯`. I.e. an abstract geometry in which concrete angular measurement pairs (or triples or quadruples or quintuples or …) are realised.

The Sphere is … not that.

It’s nontrivial to recognise that `◯×◯≠sphere`. For example the people who wrote the Starfox battle mode drew the screen as a sphere but programmed the battle mode on a torus.

By the Hairy Ball Theorem we know that spheres are different to independent pairs of circles. Specifically: one circle “vanishes” at the top and bottom of the other, to make a sphere. Changing your latitude coordinate at the North Pole  leaves you in the same place. In other words “two” collapses to “one” at the poles which also implies that, for consistency, latitude needs to be close to collapse around 89°N—not at all like `◯×◯`. where the two capstans spin freely independent of one another.

(This is half of the “joke” … or, “prank”, or “not-funny joke” in my twitter location. I designate myself at `(−90,45)` so you can imagine a person spinning around uselessly as they try to “walk in a circle” on the South Pole. OK … it’s only slightly funny even to me.)

This is like how globes can represent the Earth much better than maps on a flat sheet of paper. Since it’s impossible to map `R²` onto `S²`, flat maps can never be perfect. (The fact that the difference is merely a point—that is `R²` does map onto `S²\{0}`—is a distraction from how distorted real maps get. Look how different Greenland looks from the North versus the European view

Furthermore the torus can’t be deformed into a sphere, and it’s difficult for mathematicians to see the relationships between high-dimensional and low-dimensional spheres. (And this has something to do with the story of what Grigory Perelman achieved in solving that Clay Prize.)

The savvy way of talking about this is to say that the sphere has ugly symmetries. How can I say that when the Sphere is a Platonically perfect elementary shape?! The Sphere is so perfect that mass in outer space likes to form itself into that most balanced of balanced shapes.

Basically because when you hold the globe with two fingers and your friend spins it, the antipodes where your two fingers are holding it don’t move. (Yes, neighbourhoods around them move—but "points" in the infinitely-deep-down-continuum-set-of-measure-zero sense are singularities (erm, singularities in the `1/z` sense, not in the “black hole” sense).)

Tomorrow: a post on a statistical application of the humble circle.

(Source: )

As to longitude, I declare that I found so much difficulty in determining it that I was put to great pains to ascertain the east-west distance I had covered. The final result of my labours was that I found nothing better to do than to watch for and take observations at night of the conjunction of one planet with another, and especially of the conjunction of the moon with the other planets….

After I had made experiments many nights, one night, the twenty-third of August 1499, there was a conjunction of the moon with Mars, which according to the almanac was to occur at midnight or a half hour before. I found that…at midnight Mars’s position was three and a half degrees to the east.

Amerigo Vespucci

Good gosh. Can you imagine having travelled so far on the globe—without a swift means of return, of course—that you literally had no idea where you were?

And what’s more, science to save you. You can’t ask anyone around you for the answer. Many of the people around you not only have never heard of Europe, but can’t even conceive of such a thing.

Nobody knows the answer. ∄ books that purport to have the answer. ∄ communication channels back to home. You’re all alone, mentally. To figure out what’s going on all you have to go on is reason and facts. And if you get the answer right, whom are you going to tell?

(Source: Wikipedia)

by Fathom.info, makers of Processing HT @traviskolton

Compare to those famous light maps of the USA:

Other nice ones on the same topic. You can’t compare visually to the “new view” of the roads vis-à-vis the lights, but who doesn’t love looking at these pics? I don’t want to leave most of the world out just because the US produces the most data.

Don’t have a roads pic of the world but here’s a lights-at-night pic of the world:

Europe:

European Night Lights

A recently released satellite picture from NOAA illustrates the changes in nighttime lights in Europe between 1992 and 2009. Yellow regions show where lights have increased, purple places indicate where lights have decreased, and white areas show no change.

Mother India:

And some O(10MB) images of the world at night: http://visibleearth.nasa.gov/view.php?id=55167

Nighttime satellite image of Europe, derived from U.S. Air Force Defense Meteorological Satellite Program (DMSP) Operational Linescan System (OLS).

Dear Heavenly Leader in North Korea keeping the light pollution down:

links may be broken on this one, but promises dark-sky pics of SA, ME, Africa, and some “remote” (primitive living) areas

Back to the USA roadmap by Fathom.info, here’s San Francisco:

Appalachia:

Interesting twitters, if you like this, are @fathominfo and @impure140. (Impure being another visual programming language besides Processing.)

hi-res

Map of the United States, resized by electoral-college votes.

Mathematical PS: Every heatmap represents a scalar field. What’s the domain in this case? A 2-cell complex.

hi-res

## The Self

This week I posted different viewpoints on The Self.

Particularly I’m interested in self as a function of inputs. Just as the size of eyes a fly is born with is a function of the temperature of the eggs, so too, many facets of ourselves are a function of the environment, other people’s behaviour toward us, game-theoretic strategy, incentives, and so on.

Other people’s theories of us can be seen as functions as well. (For example, a hiring manager’s view of employee performance may assume school quality or GPA to be positively related to human capital.)

• Economics: I didn’t get to Jean Tirole’s theory of money-saving as bargains among multiple selves.
• Psychology: Jim Townsend found that self-versus-other dichotomies can be expressed as a negatively curved metric space.
• Personality: I’ve already written that the MBTI is too restrictive a theory of self. It maps from habits to `[0,1]⁴`.
• Douglas Hofstadter's thoughts on the extension of the pronoun “we”. ‘We’ went to the moon, ‘we’ share a common ancestor with other primates, ‘we’ are overcrowding the planet, ‘we’ have a nice theory of quantum chromodynamics, ‘we’ do not know if ‘we’ are experiencing a simulation or actual reality, ‘we’ don’t really know what makes an economy grow.
• Criminology: My criminal output is a function of the crime level in the neighbourhood I’m raised in. Except when it’s a function of strongly held beliefs.
• Sociology: In contemporary OECD places, ‘we’ are coerced by our cultures to play roles. “There are” certain scripts — modifiable but still requisite or recommended in some sense; at the very least influential, even if only because benefits and rewards are socially tied to role performance.
• The topic of cultural coercion … is something I’ll return to.
• The concept of people-as-functions is one I want to return to later, in discussing historyeconomics, and a couple different ways of talking about human behaviour mathematically.

I can think of several other mathematics-inspired questions about ourselves. The difference between habit and personality; the yogic metaphor of a river cutting deeper as related to habituation; choice & free will; Markovian and completely-the-opposite-of-Markovian choices (how constrained we are by our past choices); … and a lot more. But you know what, writing is hard. So I do only a little at a time.

Update, 25 September 2013: I’ve written more on this topic now:

hi-res

## Maps, projections, and hairy balls

Fact: a map of the Earth can either

• accurately depict areas, or
• accurately depict angles.

But not both. Thanks for the info, The Borsuk-Ulam Theorem(one of its corollaries is that subsets of Rⁿ are not homeomorphic to Sⁿ)

CONFORMAL

Gerardus Mercator’s projection preserves angles and local shape relationships. So it’s good for sailing. The mathematical term is a conformal mapping from the Earth S² to the page R² [0,1].

(see what I did there? North ain’t up. Away from Earth is up.)

DETERMINANT = 1

Mercator’s projection makes Africa look small and Greenland look big, because of their distances from the equator. But try navigating with this puppy.

In matrix terms, equal area projections are mapped from S² to  by a matrix with determinant 1. (If you had to use a bunch of different matrices on a manifold and stitch them together, some sum of determinants would have to be 1.)

OTHERS

Area and angle aren’t the only two qualities of a map, of course. That’s why there are so many alternative projections. For example, the Robinson projection we used in my elementary school is neither conformal nor area-preserving.

You can spend all week reading about them on W***pedia or Mathworld. Gall-Peters, Lambert, Aitoff, Hammer, Goode, Boggs, van der Grinten, Tobler, Mollweide, Gnomonic, Eckart, Collignon, Kavrayskiy, pseudocylindrical sinusoidal projections, and Tissot’s indicatrix of deformation are all cool.

Also, thinking about maps is a good way to start visualizing mathematical functions as transformations of an entire space rather than thinking of the f(a), f(b), f(c)'s individually.

GENERALIZATIONS

There is a related result to the Borsuk-Ulam theorem, called — literally — the Hairy Ball Theorem. I am not making this up.