Posts tagged with latitude

## 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)

## Europe

Data set:
`> eurodist                 Athens Barcelona Brussels Calais Cherbourg Cologne CopenhagenBarcelona         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       1418Gibraltar         4485      1172     2256   2224      2047    2436       3196Hamburg           2977      2018      597    714      1115     460        460Hook of Holland   3030      1490      172    330       731     269        269Lisbon            4532      1305     2084   2052      1827    2290       2971Lyons             2753       645      690    739       789     714       1458Madrid            3949       636     1558   1550      1347    1764       2498Marseilles        2865       521     1011   1059      1101    1035       1778Milan             2282      1014      925   1077      1209     911       1537Munich            2179      1365      747    977      1160     583       1104Paris             3000      1033      285    280       340     465       1176Rome               817      1460     1511   1662      1794    1497       2050Stockholm         3927      2868     1616   1786      2196    1403        650Vienna            1991      1802     1175   1381      1588     937       1455                Geneva Gibraltar Hamburg Hook of Holland Lisbon Lyons MadridBarcelona                                                                   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   1157Milan              328      2185    1238            1098   2250   328   1724Munich             591      2565     805             851   2507   724   2010Paris              513      1971     877             457   1799   471   1273Rome               995      2631    1751            1683   2700  1048   2097Stockholm         2068      3886     949            1500   3231  2108   3188Vienna            1019      2974    1155            1205   2937  1157   2409                Marseilles Milan Munich Paris Rome StockholmBarcelona                                                   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  1798.80293Barcelona        -825.382790   546.81148Brussels           59.183341  -367.08135Calais            -82.845973  -429.91466Cherbourg        -352.499435  -290.90843Cologne           293.689633  -405.31194Copenhagen        681.931545 -1108.64478Geneva             -9.423364   240.40600Gibraltar       -2048.449113   642.45854Hamburg           561.108970  -773.36929Hook of Holland   164.921799  -549.36704Lisbon          -1935.040811    49.12514Lyons            -226.423236   187.08779Madrid          -1423.353697   305.87513Marseilles       -299.498710   388.80726Milan             260.878046   416.67381Munich            587.675679    81.18224Paris            -156.836257  -211.13911Rome              709.413282  1109.36665Stockholm         839.445911 -1836.79055Vienna            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