see things differently

Posts tagged with distance

[G]eometry and number[s]…are unified by the concept of a coordinate system, which allows one to convert geometric objects to numeric ones or vice versa. …

[O]ne can view the length ❘AB❘ of a line segment AB not as a number (which requires one to select a unit of length), but more abstractly as the equivalence class of all line segments that are congruent to AB.

With this perspective, ❘AB❘ no longer lies in the standard semigroup ℝ⁺, but in a more abstract semigroup ℒ (the space of line segments quotiented by congruence), with addition now defined geometrically (by concatenation of intervals) rather than numerically.

A unit of length can now be viewed as just one of many different isomorphisms Φ: ℒ → ℝ⁺ between ℒ and ℝ⁺, but one can abandon … units and just work with ℒ directly. Many statements in Euclidean geometry … can be phrased in this manner.

(Indeed, this is basically how the ancient Greeks…viewed geometry, though of course without the assistance of such modern terminology as “semigroup” or “bilinear”.)

Terence Tao

(Source: terrytao.wordpress.com)

Manifolds, Star Fox, and Self-versus-Other

Branes, D-branes, M-theory, K-theory … news articles about theoretical physics often mention “manifolds”. Manifolds are also good tools for theoretical psychology and economics. Thinking about manifolds is guaranteed to make you sexy and interesting.

Fortunately, these fancy surfaces are already familiar to anyone who has played the original Star Fox—Super NES version.

In Star Fox, all of the interactive shapes are built up from polygons. Manifolds are built up the same way! You don’t have to use polygons per se, just stick flats together and you build up any surface you want, in the mathematical limit.

The point of doing it this way, is that you can use all the power of linear algebra and calculus on each of those flats, or “charts”. Then as long as you’re clear on how to transition from chart to chart (from polygon to polygon), you know the whole surface—to precise mathematical detail.

Regarding curvature: the charts don’t need the Euclidean metric. As long as distance is measured in a consistent way, the manifold is all good. So you could use hyperbolic, elliptical, or quasimetric distance. Just a few options.

Manifolds are relevant because according to general relativity, spacetime itself is curved. For example, a black hole or star or planet bends the “rigid rods” that Newton & Descartes supposed make up the fabric of space.

bent spacetime

black hole photo

In fact, the same “curved-space” idea describes racism. Psychological experiments demonstrate that people are able to distinguish fine detail among their own ethnic group, whereas those outside the group are quickly & coarsely categorized as “other”.

This means a hyperbolic or other “negatively curved” metric, where the distance from 0 to 1 is less than the distance from 100 to 101. Imagine longitude & latitude lines tightly packed together around “0”, one’s own perspective — and spread out where the “others” stand. (I forget if this paradigm changes when kids are raised in multiracial environments.)

Experiments verify that people see “other races” like this. I think it applies also to any “othering” or “alienation” — in the postmodern / continental sense of those words.

The manifold concept extends rectilinear reasoning familiar from grade-school math into the more exciting, less restrictive world of the squibbulous, the bubbulous, and the flipflopflegabbulous.

ga zair bison and monkey

calabi-yau manifold

cat detective

B*tchin’ six dimensional 6-cube. The rainbow colours and glass panes really help this visualisation.

Examples of 6-dimensional things

If it’s hard to envision 6 dimensions, consider this: the possible tunings of a guitar constitute a 6-dimensional space. You can tune to EADGBE (standard), DADGAB, drop-D, DADGAD, GCCGCC, BEBEBE, CGCFGE, and many others.

(If you consider notes an octave apart to be equivalent, then we’re talking about a quotient space, each distance being topologically on a loop. But that’s just one system of musical valuation — and like the winding number of a complex number, it’s totally apparent that high octaves do not sound exactly the same as low sounds. And doing a 720° is more impressive than a 360°. If the abstract “loop” is unwound, there is a highest note (“1”) and a lowest note (“0”) that can effectively be played on each string (dimension).)

You can also think about 6-D as being the six columns in a table or array. For example the { RBI, on-base percentage, fielding errors, stolen bases, sacrifice flies, and home-runs } for a number of baseball players.

Or you can think about six security prices moving in parallel, from bell to bell at the NYSE.

Again the lowest price is called “0” and the highest is called “1”. This renaming places the jumping Brownian motions inside a secteract. So instead of six 1-D paths it’s one 6-D path:

Enough examples of 6-dimensional things. Back to the 6-cube itself.

Let’s make one.

The bounds of the secteract (its “corners”? Or should I say its 6-corners.) come from filling in each of six slots with either 0 or 1.

There are 64 ways to do this. (two options for each of six slots = 2^6.) For example (0,0,0,0,0,1) is one, (0,0,0,0,1,0) is another, and (0,1,1,0,1,0) is a third out of the 64.

The R programming language was nice enough to write out all of the vertices for me without my having to type much. Here they are:

> booty=c(0,1) > expand.grid(booty,booty,booty,booty,booty,booty) #rockin everywhere

   Var1 Var2 Var3 Var4 Var5 Var6
1     0    0    0    0    0    0
2     1    0    0    0    0    0
3     0    1    0    0    0    0
4     1    1    0    0    0    0
5     0    0    1    0    0    0
6     1    0    1    0    0    0
7     0    1    1    0    0    0
8     1    1    1    0    0    0
9     0    0    0    1    0    0
10    1    0    0    1    0    0
11    0    1    0    1    0    0
12    1    1    0    1    0    0
13    0    0    1    1    0    0
14    1    0    1    1    0    0
15    0    1    1    1    0    0
16    1    1    1    1    0    0
17    0    0    0    0    1    0
18    1    0    0    0    1    0
19    0    1    0    0    1    0
20    1    1    0    0    1    0
21    0    0    1    0    1    0
22    1    0    1    0    1    0
23    0    1    1    0    1    0
24    1    1    1    0    1    0
25    0    0    0    1    1    0
26    1    0    0    1    1    0
27    0    1    0    1    1    0
28    1    1    0    1    1    0
29    0    0    1    1    1    0
30    1    0    1    1    1    0
31    0    1    1    1    1    0
32    1    1    1    1    1    0
33    0    0    0    0    0    1
34    1    0    0    0    0    1
35    0    1    0    0    0    1
36    1    1    0    0    0    1
37    0    0    1    0    0    1
38    1    0    1    0    0    1
39    0    1    1    0    0    1
40    1    1    1    0    0    1
41    0    0    0    1    0    1
42    1    0    0    1    0    1
43    0    1    0    1    0    1
44    1    1    0    1    0    1
45    0    0    1    1    0    1
46    1    0    1    1    0    1
47    0    1    1    1    0    1
48    1    1    1    1    0    1
49    0    0    0    0    1    1
50    1    0    0    0    1    1
51    0    1    0    0    1    1
52    1    1    0    0    1    1
53    0    0    1    0    1    1
54    1    0    1    0    1    1
55    0    1    1    0    1    1
56    1    1    1    0    1    1
57    0    0    0    1    1    1
58    1    0    0    1    1    1
59    0    1    0    1    1    1
60    1    1    0    1    1    1
61    0    0    1    1    1    1
62    1    0    1    1    1    1
63    0    1    1    1    1    1
64    1    1    1    1    1    1

And there you have it: an electronic realisation of a secteract. Just as real as a Polyworld life-form.

Noncommutative distances between industries

The distance from your house to the grocery must be the same as the distance back, but 20th-century mathematicians speculated about circumstances where this might not be the case.

Very small-scale physics is non-commutative in some ways and so is distance in finance.

But non-commutative logic isn’t really that exotic or abstract.

Imagine you’re hiring. You could hire someone from the private sector, charity sector, or public sector. It’s easier for v managers to cross over into b | c than for c | b managers to cross over into v.

So private is close to public, but not the other way around. Or rather, v is closer to b than b is to v. δ| v, b | < δ| b, v | . (same for δ| v, c |)
Perhaps something similar is true of management consulting, or i-banking? Such is the belief, at least, of recent Ivy grads who don’t know what to do but want to “keep their options open”.

This might be more of a statement about average distance to other industries ∑ᵢ δ| consulting, xᵢ | being low, rather than a comparison between δ| consulting, x | and δ| x, consulting | . Can you cross over from energy consulting to actual energy companies just as easily as the reverse?
Imagine you’re want a marketing consultant. Maybe some “verticals” are more respected than others? So that a firm from vertical 1 could cross over into vertical 2 but not vice versa.
Is it easier for sprinters to cross over into distance running, or vice versa? I think distance runners have a more difficult time getting fast. If it’s easier for one type to cross over, then δ| sprinter, longdist | ≠ δ| longdist, sprinter |.
It’s easier to roll things downhill than uphill. So the energy distance δ | top, bottom | < δ | bottom, top |.
It’s usually cheaper to ship one direction than the other. Protip: if you’re shipping PACA (donated clothes) from the USA to Central America, crate your donation on a Chiquita vessel returning to point of export.

Noncommutative distance, homies. (quasimetric) And I didn’t invoke quantum field theory or Alain Connes. Just business as usual.

“The anchovies were nowhere near the sardines and the tuna. That’s because they were near the pizza toppings.

But it was only a problem because this was a three-dimensional grocery store. If it had been a thirty-dimensional grocery store they could have been near the pizza and the sardines.”

I’ve been putting off a post about phase space for nearly a year now, and watching this talk made me remember that I’ll soon have to do it.

Geoff Hinton talks here about a number of different spaces that are not the physical space that math was initially developed on.

The bag of words model of a document takes each word in a text to be a dimension of the document, with like 80,000 possibilities each or however many words there are in English. (The 80,000 possibilities are the underlying corpus.)
Latent semantic analysis of the bag-of-words type is how Google now does its search rankings. (PageRank only constitutes something like 30% of SERP ranking anymore, because [a] it’s too easy to game and [b] it’s inspecific to what’s being searched on. Domain authority and LSI comprise the rest. <—separate article)
Seeing a pixel-by-pixel representation of a 2-D image as a list vector is problematic because the first pixel in a 200x300 Facebook profile image is next to the second pixel and also next to the 201st pixel. I.e. one needs a 2-array.
Hinton talks about abstract feature space and energies — equivalently evolutionary fitness or economic utility — and ravines and mountains upon this manifold.
The number of dimensions here is like the number of parameters (same as free parameters or degrees of freedom or arbitrary parameters in stats class) and in a neural net each “synapse” or graph edge is a lever you can pull.
The same metaphor — and this is a metaphor in a grand sense which I hope to cover before the year is up — applies to the equalizer on your uncle’s home stereo, i.e. the number of terms in the Fourier decomposition.

Unmeasurable Distances

I wrote earlier about the many different ways to measure distance. One way I didn’t include is unmeasurable distance.

Sometimes A is

tastier,
sexier,
cooler,
more interesting,
or otherwise better endowed

than B … but it’s impossible to quantify by how much. No problem; just say that A≻B but that |A−B| is undefined.

It’s still the case that if A is sexier than B and B is sexier than C, it must follow that A is sexier than C.

Symbolically: A≻B & B≻C ⇒ A≻C.

This concept opens up many parts of human experience to the mathematical imagination.

I will also express my view on moral rates of income tax using orderings ≻.

Oh, and if you’re into this kind of thing: using orders ≻ instead of measurable quantities kind of saved the economic concept of “utility”. Kind of saved it. At least instead of talking about 174.27819 hedons, nowadays you can just say X is lexicographically preferred ≻ to Y. Ordinal utility instead of cardinal utility.

Europe

Data set:
> eurodist Athens Barcelona Brussels Calais Cherbourg Cologne Copenhagen Barcelona 3313 Brussels 2963 1318 Calais 3175 1326 204 Cherbourg 3339 1294 583 460 Cologne 2762 1498 206 409 785 Copenhagen 3276 2218 966 1136 1545 760 Geneva 2610 803 677 747 853 1662 1418 Gibraltar 4485 1172 2256 2224 2047 2436 3196 Hamburg 2977 2018 597 714 1115 460 460 Hook of Holland 3030 1490 172 330 731 269 269 Lisbon 4532 1305 2084 2052 1827 2290 2971 Lyons 2753 645 690 739 789 714 1458 Madrid 3949 636 1558 1550 1347 1764 2498 Marseilles 2865 521 1011 1059 1101 1035 1778 Milan 2282 1014 925 1077 1209 911 1537 Munich 2179 1365 747 977 1160 583 1104 Paris 3000 1033 285 280 340 465 1176 Rome 817 1460 1511 1662 1794 1497 2050 Stockholm 3927 2868 1616 1786 2196 1403 650 Vienna 1991 1802 1175 1381 1588 937 1455 Geneva Gibraltar Hamburg Hook of Holland Lisbon Lyons Madrid Barcelona Brussels Calais Cherbourg Cologne Copenhagen Geneva Gibraltar 1975 Hamburg 1118 2897 Hook of Holland 895 2428 550 Lisbon 1936 676 2671 2280 Lyons 158 1817 1159 863 1178 Madrid 1439 698 2198 1730 668 1281 Marseilles 425 1693 1479 1183 1762 320 1157 Milan 328 2185 1238 1098 2250 328 1724 Munich 591 2565 805 851 2507 724 2010 Paris 513 1971 877 457 1799 471 1273 Rome 995 2631 1751 1683 2700 1048 2097 Stockholm 2068 3886 949 1500 3231 2108 3188 Vienna 1019 2974 1155 1205 2937 1157 2409 Marseilles Milan Munich Paris Rome Stockholm Barcelona Brussels Calais Cherbourg Cologne Copenhagen Geneva Gibraltar Hamburg Hook of Holland Lisbon Lyons Madrid Marseilles Milan 618 Munich 1109 331 Paris 792 856 821 Rome 1011 586 946 1476 Stockholm 2428 2187 1754 1827 2707 Vienna 1363 898 428 1249 1209 2105

Multi-dimensional scaling of the distances:

> cmdscale(eurodist) [,1] [,2] Athens 2290.274680 1798.80293 Barcelona -825.382790 546.81148 Brussels 59.183341 -367.08135 Calais -82.845973 -429.91466 Cherbourg -352.499435 -290.90843 Cologne 293.689633 -405.31194 Copenhagen 681.931545 -1108.64478 Geneva -9.423364 240.40600 Gibraltar -2048.449113 642.45854 Hamburg 561.108970 -773.36929 Hook of Holland 164.921799 -549.36704 Lisbon -1935.040811 49.12514 Lyons -226.423236 187.08779 Madrid -1423.353697 305.87513 Marseilles -299.498710 388.80726 Milan 260.878046 416.67381 Munich 587.675679 81.18224 Paris -156.836257 -211.13911 Rome 709.413282 1109.36665 Stockholm 839.445911 -1836.79055 Vienna 911.230500 205.93020

Plot

require(stats) loc <- cmdscale(eurodist) rx <- range(x <- loc[,1]) ry <- range(y <- -loc[,2]) plot(x, y, type="n", asp=1, xlab="", ylab="") abline(h = pretty(rx, 10), v = pretty(ry, 10), col = "light gray") text(x, y, labels(eurodist), cex=0.8)

hi-res

Topology

Outside of a dog, a book is a man’s best friend. Inside of a dog, it’s very dark.

—Groucho Marx

Directions

Sometimes a precise equation gives you more information than you need, more than you asked for. What if you asked me, “How do I get from Indianapolis to Chicago?” I could answer by giving a precise series of (x,y) coordinates that one must traverse — like a table of all the (Latitude, Longitude) coordinates one passes through on the way. But more likely you just want to know to take 465 until exit X, then 65 northwest until exit Y, and then 90 west until you reach the Chicago part of your directions.

View Larger Map

Similarly if you wanted to get from Indianapolis to Guatemala City, I could just say: take Megabus to Chicago, take the “El” train (blue line) to O’Hare, and fly TACA to La Aurora airport in Guate Guate La Capi.

$Indy --> Chicago --> O'Hare --> La Aurora$

Basically, you just want to know how to connect the parts of your route. You want a topological kind of answer.

Topology Disregards Distance

Topological thinking dispenses with facts like “It’s 180 miles (4 hours) from Indy to Chicago” and just looks at the connections between things. Am I inside or outside the house? Is the storm going to evolve into a hurricane, or will it dissipate? (Dynamical systems question.) Is the economy on a path to sustainable growth, or mired in a self-destructive “spiral”? (Again, dyn sys.) How many ways are there to cross from the Minnesota Public Radio building to the parking lot? (Minneapolis has enclosed skyways to keep you from freezing while you cross the street.)

(an 8_5 knot)

If you disregard distance and think that completely through, eventually you find yourself saying strange things like “A donut is topologically equivalent to a coffee cup” (they both have one hole). Sometimes you want to know how far it is to Chicago, sometimes you want to know how many possible routes you can take to go there. (If you counted all the county roads, switchbacks, and detours, that monstrous entanglement would have a LOT of holes.)

That’s when you use topology.

ANATOMICAL QUANDARY: When you eat food, does it really go “inside” you? After all, it hasn’t traversed a boundary—you have an open path straight through you from mouth to anus (it just temporarily closes the open loop with sphincters and stuff, from time to time). To really be “inside” someone would be like somewhere between the lining of the stomach, and the skin.