Math – THE PROCESS IS…

Last Train to Transcendental

Loki der Quaeler — Sat, 17 Oct 2009 00:26:41 +0000

Here again we return to infinity (if it can be said possible to return to something which one can’t leave). If you hadn’t heard of it before: there are different types of infinity – different sizes of infinity. This area of mathematics has fascinated and fixated mathematicians for literally millennia, though perhaps one of the most famous and prolific mathematicians to contribute to the area was a man named Georg Cantor. Cantor was very likely bipolar and spent a large chunk of his adult life feeling somewhat insane and persecuted by his peers. This last part wasn’t entirely due to the neurochemical roller coaster as some mathematicians were truly unsettled by his work and lashed out, like that schmuck Poincaré who said that Cantor’s set theory work was a ‘perverse illness from which someday mathematics would be cured.’

To be fair, what Cantor exposed for us does, on first blush, seem to make no sense, at best, and be contradictory, at worst. In this article we’re going to look at just a small sliver and, in that, find something bigger than the universe in which we live.

We’ll be using something you first saw in elementary school: the number line. Here’s one i’m referencing from the Wikimedia Commons:

For the moment, this has more information than we need to bother ourselves with, so just pay attention to the numbers -3, -2, -1, 0, 1, 2, 3. In math-ese, these are examples of what are frequently called the integers – but you may have also heard “whole numbers”. So how many of these integers can we imagine? Well, yes, an ‘infinite’ amount of them.¹ The better questions are things like How big is this infinity? and Can this infinity be compared to the infinity of some other group of infinite stuff?

To improve the former ‘better’ question, we’ll start using a funky word for ‘size’ which is the math-ese word “cardinality”; so we could more formally phrase the former² as What is the cardinality of this infinity? Now, to answer the latter ‘better’ question, and thereby the former, first we need to round up some other groups of infinite stuff. To start to do this, we’ll define a group of numbers that you grew up with knowing as “fractions”; in math-ese, we call these “rationals”. All of the numbers in this group are a number gotten by dividing any integer by any other integer that is not 0.

Cantor came up with an ingenious way to lay out all fractions in a 2 dimensional matrix such that for each integer, there is one listed fraction. Because we can do this “one to one” pairing³ between the integers and the rationals, this means that the integers and the rationals have the same cardinality, or size. This will be where we find the first bump in the weird world of the sizes of infinities, because the astute reader will notice that all of the integers are included in this list of rationals (2/1 = 2, 12/4 = 3, etc.) — and compounding that: more than once (6/3 = 12/6 = 5200/2600 = … = 2) — and yet the infinity which captures the rationals is the same size as the infinity which captures the integers.

While we casually refer to an infinity of this size as “countably infinite”, Cantor decided to start labelling the cardinality of different infinites using the first letter of the Hebrew alphabet, aleph — — and he labelled the cardinality of the infinity which captures the integers, and the rationals, as – spoken as “aleph null”. The grand scheme being that subsequently bigger sizes of infinity would have an increasing subscript — so the next biggest infinity would be labelled as (“aleph one”) and so forth.

Now let’s talk about the “real numbers”; the real numbers can be roughly thought of as any number which can be written as a decimal representation (like 3.500…)⁴. How big is the set of real numbers? Well, if just the integers are infinite, then the real numbers (which contain all of the integers, and rationals, and more) must also be infinite; but how infinite? It turns out that there is no way to do our ‘one-to-one’ trick from the integers to the reals. Cantor (again) presented a robust proof of this which is known as “Cantor’s diagonal argument” – which ultimately means that the real numbers must be of a greater infinity than the integers, et al.
This size of infinity is often referred to as “uncountably infinite”; Cantor, who was unsure whether there was some size of infinity that existed between the infinity of the integers and the infinity of the reals, could not label this infinity and so adopted a convention utilizing the second letter of the Hebrew alphabet (‘beth’) and called this infinity (“beth one”).
Not only is it still unknown whether (a problem which is known as the “Continuum hypothesis”), it appears logically evident that we can never prove it nor disprove it.⁵

If you’ve become somewhat settled with the idea that the rationals, which contain all of the integers, and the integers themselves are groups of the same size infinity, then it shouldn’t be too much of a stunner to learn that the size of infinity of the group of all real numbers is actually the same size of infinity of a group for any continuous range of real numbers. What i mean by ‘continuous range’ is any continuous flow of real numbers between two points; using the number line above as reference, we can consider the group of every real number between 0 and 1 to be a continuous range [between 0 and 1]. Since this equivalence of infinity size is true for all continuous ranges of real numbers, we could then grab a continuous range within that 0-1 range and it, too, would be a group of the same size infinity as all of the real numbers.
This doesn’t translate at all well to what we experience in our physical world and so is often difficult to digest. For example, it would be pure Alice-world were you to operate a power shovel; dig your shovel into the ground and remove a basket full of dirt; rotate and dump that dirt into a pile; dig your shovel into just the center of the pile removing that; find that even though some of the pile is still on the ground, your basket is completely full again.

Ok. So if the real numbers are ‘bigger’ than the rationals (which contain the integers) what else is that stuff hanging out in the real numbers – that stuff which is so numerous as to make the real numbers uncountable? This is the stuff which we call the ‘irrationals’; the irrationals can be thought of as roughly being composed of ‘algebraic’ numbers and ‘transcendental’ numbers. It turns out that our ‘one-to-one’ tool works with the algebraic numbers and not with the transcendental numbers – so here is our real stuff.

Now that we’ve finally arrived at our station, what are transcendentals? Well, we know what some of them are – there are the famous ones shown on the above number line image, like , and – but, truth be told, it can be quite hard to prove that a given number is transcendental and only a small zoo exists. I suppose you could think of them sort of as the ‘dark matter’ of the real numbers.

As an aside, when was finally proven to be irrational in 1882, it sounded the official death knell for a problem more than 2,000 years old called, ‘Squaring the Circle’. The earliest attempts to solve this was first described by the famously named Plutarch 500 years after it was done by Anaxagoras (who was cooling his heels in jail for one thing or another⁶ ). It was a welcome toll as squaring the circle had become such the past-time of math-crackpots that the Paris Academy (Académie des Sciences) had to take an official stand in 1775 and declare that none of its officials would be allowed to review papers claiming to address the solution of this (this, and two other favourites of the time: the trisection of the angle and the duplication of the cube).⁷

We can now finish here with pondering what all of this means when we talk about an “infinite” universe. In what we experience in the physical world and in terms of reality, the concept of an uncountable infinity definitely doesn’t correspond to our idea of physical space as we drill down. From every indication, there is indeed a smallest possible unit of physical space (quantum foam, for example, bumps down to an eensy weensy length of space referred to as the “Planck length“); since there is a smallest definable unit, and if we assume that the universe does expand out “forever”, we can pull out our “one-to-one” tool and see that the group of every smallest definable unit of physical space within the existing universe is countably infinite – its cardinality is .

Rephrased more loose and fast: what this means, in a final twist a weirdness, is that on our above number line the group which contains the continuous range of real numbers between only 0 and 1 has a greater cardinality than the group which is the entire physical space of the universe.

A sloppy quick proof would be to pick the largest possible integer you can think of; now just add 1 to it and you have one even bigger. Repeat endlessly.
mmmm, consonance
math-ese: ‘mapping’
where, technically, there are an infinite number of digits after the decimal point; it could be an infinite number of 0s, in which case 3.00… would represent the same item as the integer ‘3’
This statement is too general, there are specifications which the reader can discover reading up on the CH…
where the ‘one thing or another’ appears to have been that he attempted to explain to people that the Sun was not a deity but rather some physical entity
“L’Académie prend en 1775 la resolution de ne plus examiner aucune solution des Problémes de la duplication du cube, de la trifection de l’angle, & de la quadrature du cercle…” — The last entry on page 2 in Volume 9 of “Mémoires de l’Académie royale des sciences”, published 1786, freely available here (archive.org rocks)

Life on Planes

Loki der Quaeler — Wed, 18 Feb 2009 18:24:15 +0000

I spend some amount of time traveling by air; it’s never enough to satisfy me, but the time-and-space aspects of it always fill me with giddiness. Pondering those aspects gives an excellent opportunity to examine our place in the larger picture of space, for it’s not such a common event that a person can noticeably leave one spatial reality to spend time in another spatial reality. Making the situation more contorted is that we, humans, have attempted to force the application of a framework of calendrical time — an application which doesn’t conform so well to the properties of our planet – of a sphere.

To paint an example: i flew to New Zealand over Thanksgiving. I boarded a plane in Vancouver on a Friday evening around 5.30p; it was wintry – typical for that time of year. I flew for 14 hours and arrived in the morning — but it was now Sunday. Walking around downtown Auckland at 8.00a on what was already shaping up to be a warm summery day (typical for that time of year), i gave my friends T&E a call; they were in California so i knew they’d be awake since it was a few hours ahead: they were answering the phone at 11.00a — but on Saturday. On my return flight, i boarded a plane on Sunday night around 8.00p, flew for 13 hours, and arrived at 12.30p — it was Sunday, early afternoon.

So, how did we get to this state of chronometric cluster fuck? Well, this is just one of the side effects of our living on a sphere.

Instead of one of my usual tilting-at-windmills articles¹, i thought i’d do an article about geometry. Wait – don’t go – you might actually enjoy this. It’s an article about the differences between the geometry we perceive in our every day lives as ground dwellers, and the geometry we’re actually living in on Earth. The three topics i’ll talk about are:

the time zone / calendrical scenario i described above
those flight progress maps they show on video screens during flights
the difference in seasons between hemispheres

Also, this can be an interactive article: should you choose to play along at home, you should scrounge up:

two oranges
a sheet of ordinary letter paper
a ruler
a sharpee
a pen or a pencil (unless you want to use the sharpee on paper)
paper towels
one object of your choosing, no bigger than the oranges
a Pringle²
a decently sharp knife

You’ll also need a responsible adult if you’re a minor (or particularly incompetent with a sharp knife).

A foreword:

This article will be ever so slightly math-y — it’s unavoidable.³ I realize that one man’s orgy of gorgeous shapes and symbols is another man’s snooze-fest, so i’ll do my best to make it candy and consumable, while trying not to dumb it down.

Also, this article has turned into a beastly length, so i’ve made all of the headings collapsible, and collapsed by default (save this one); click on them to expand/collapse the section under each.

Lastly, in the spirit of exclusion, if you’re a person who believes in a flat earth, who has somehow ungnarled your purple polydactyl pointers into using a computer mouse and uncrossed your eyes long enough to read this, you can stop reading here. (Yes, it’s true: there are apparently **still** flat-earth people today⁴ — if you thought that the supporters of a geocentric universe had to come up with some inventive [read: absurd] models to support observational data, they had nothing on the modern flat-earther.⁵)

On with the show:

Time zones and calendrical hoo-hah:

The history of time zones dates back some number of centuries and is well documented on Wikipedia should you be interested; suffice it to say that their usage has been globally adopted by all modern societies on the planet. Part of the problem with time zones is that we’re trying to assign a continually increasing value to a surface that is ‘closed’. By ‘closed’, i mean that the surface does not stretch off infinitely in any given direction and were you to set off in any direction staying on, or parallel to, the surface, you would eventually return back to the spot from which you began (much to consternation of flat earthers and many other people in the 15th century).

As a thought experiment, imagine a benevolent ruler of all the known lands in a time in which there wasn’t clear evidence that we lived on this closed surface. One day he decides that he’s going to travel to the edge of the world, surveying all his land, and bestowing his people with some gifts. He gets in his royal hot air balloon and travels due east along the equator. After every 1000 miles floated, he drops a big bag of gold to his loyal subjects below. With his 25th drop he might start to notice that things seem familiar as he’d be back over his starting location; should he not realize this, however, he will continue his pursuit until death or bankruptcy, never reaching the end of the world.

Common to the need of both the benevolent ruler and the implementers of the time zone system is the ability to say “stop — we’re repeating ourselves.”

Picking one of the oranges, we can see that it is currently divided-by-sharpee into 4 sections, each of which run from north to south. For simplicity, we’ll use these as our time zones (pretend that moving from one time zone to another is a change of six hours and not one hour). To make it visually more easy, draw an equator on the orange and ink a notable black circle at the spot where the equator crosses the pole-to-pole line running out the cardinal W point. Mine looks like the photo to the right. Orientating the orange with the north pole facing your ceiling and the first quadrant to the right of the black circle facing you, pick up the sharpee and write “0” in the first quadrant. Rotate the orange by a quarter turn and write +6 in the next quadrant. Repeat for +12 and +18.
Rotating another quarter turn at this point, we end up back at the quadrant with a ‘0’ in it — and here’s the moment — note that we could put +24 here (and then +30, +36, +42, +48, +54, …) and the clocks for all the little inhabitants on our Planet Orange would be consistent, but not their calendars.

So here’s root of our discontent. In the absurd case, we could let the time zone adding go on ad infinitum but, then, this makes for an even more confusing calendar situation (because then today would be every calendar day from now until the end of time). So what we’ve had to do is to put a stake in the ground, figuratively, and declare an international date line — a line on the planet on the eastern side of which it is one day, and on the western side of which is the next day.

As a brief historical note, the idea of an international date line can be found mentioned as early as the 14th century in a Talmudic commentary, though it seems likely that the idea had more to do with geographic and identity separation than regulating global time. This date line, referred to as the “K’tzai Hamizrach”, was at a longitude 90 degrees east of Jerusalem (which would be basically running just a smidgen east of Taiwan).

Since humans were involved in this whole process, the exact position of the Earth’s date line is steeped in politics and as such looks like a typical case of gerrymandering. This sets up dumb scenarios similar to Iceland insisting on being in the same time zone as the U.K. — like Kiribati extending well east of Samoa, but being exactly one day ahead, and at the same time of day, of Samoa — and like the Line Islands Time (pleasingly abbreviated as ‘LINT’) being outside of the normal -12 to +12 hour offset range at +14.

With all of this in mind, it is an easy thought experiment to design travel in which a given calendar day simply won’t ever exist in the life of a person; had that Air New Zealand flight flown a route that went more to the west initially, it would have crossed the international date line at midnight local time, effectively meaning that no one on that flight would have ever experienced Saturday the 29th of November, 2008: at one moment, the local calendar would have read Friday and the very next second the local calendar would have read Sunday.

Flight Progress Graphics:

If i had a dime for every time i’ve overheard people asking amongst themselves why our plane was flying in that ‘funny curve path’… well i don’t know exactly how much that would be – but i’d have some more money. The reality behind the situation isn’t that the plane is actually flying a funny curved path through space, but rather a problem with map ‘projections’. Without getting too too mathematical, this kind of projection could be thought of as a rule that tells you how to take one point on a surface and place it on another surface.

There’s a couple problems with projections as far as maintaining a good representation of the thing being projected. Two easy-to-talk-about cases would be world maps, which we’ll get to in a second, and something we see all the time: shadows.

With shadows, we’re losing a lot of information in the projection process due to a loss of dimensionality — in other words, we’re taking something which has 3 dimensions (like a chair), and projecting a representation of it (its shadow) into 2 dimensions (like the flat floor). This projection is ‘destructive’, because we can’t reasonably reconstruct all of the information about the original object (the chair) from looking only at its projection (its shadow). Ok, enough about that kind of everyday projection – back to maps…

So at this point, if you’re snarky in a delightfully weird way, maybe you’re asking, “Why do we need to do a projection to take information from one basically 2-dimensional surface (the surface of our planet) and put it on just another 2-dimensional surface (the poster version of the world map)?” The answer to this has to do with ‘curvature’.

A side track into curvature:

Curvature, in this sense, is a mathematical concept related to the bended-ness of an object in space; technically there are a number of kinds of curvature, but we’ll just use the loose word ‘curvature’. The simplest way for us to test the differences in curvature (aside from just staring at the object) is with triangles. You’ll remember from elementary school the lesson that the sum of all of the angles in the corners of a triangle equals 180°.

Let’s belt one out quickly on our piece of paper. Using the ruler, draw a 3″ line parallel to one edge of the paper; now draw a 3″ line from the start or end of that line parallel to the other edge so that you now have an L shape with equals sides — here we have a good old ‘right angle’ – 90°. Now draw a line between the two open ends of the lines to make a triangle. I’ll spare you the trigonometry and you can take my word that each of the remaining two angles are 45°; so adding those two with the right angle equals our promised 180°. This wasn’t nearly rigorous, but trust me: any triangle you draw on that piece of paper will have its corner angles sum to 180°.

Now, pick up the second orange and draw an equator like we did with the first orange. Breaking out your stink eye on the north pole, notice that there is a right angle between any of the lines leaving the cardinal direction indicators. Picking two of the neighbouring cardinal directions, let’s say N and W, draw a right angle symbol, just to remind us. Now, follow the W line from the north pole down to the equator; notice that when they meet, it’s a right angle as well (draw another right angle symbol here). Lastly, follow the N line from the north pole down to the equator; here again, they meet at a right angle (and scribble another right angle symbol here). Holding the orange away from us and looking at this triangle, you’ll note that we now have a triangle in which our corner angles add up to 270°. This is because the orange has positive curvature, whereas the piece of paper was flat (it had zero curvature).

Noting the pattern that positive curvature surfaces have triangle angles that sum to greater than 180° and zero curvature surfaces have triangle angles that sum to exactly 180° then Patty McPattern, the drunk pattern-dog, will correctly guess that surfaces of negative curvature have triangle angles that sum to less than 180°.

Looking up from his Lagavulin, Patty asks, “What’s a negative curvature surface look like?” Well: stare at your Pringle (or dressage saddle) — this ‘saddle surface’ is the classic example of a surface with negative curvature. Ok, you can go ahead and eat the Pringle now, if you want.

Our perception of this curvature depends greatly on the altitude we are above the surface, as well as the amount of the surface we care about measuring. With regards to altitude on the Earth: walking on salt flats, you’d swear there was no curvature at all while if you were lucky enough to fly in the Concorde (damn you, and) the curvature would have been starkly plain to see.

With regards to the amount of surface being measured, if we’re endeavouring to build a sports stadium or a city block on planet Earth, then we needn’t pay heed to the real curvature of the planet. So when measuring the ground for something the size of a football field (or in terms of my orange, which has much much much less surface⁷, the football field would be an area 20 million times smaller than the size of a red blood cell), we act as though we are dealing with something with zero curvature, like our piece of paper. If, instead, we’re endeavouring to pilot a plane from San Francisco to Frankfurt, or lay transoceanic fiber optics, then we need to start thinking about this curvature.

It’s exactly this latter point which entrenches the vast majority of our existence, making it easy to forget that we’re actually living in a number of spatial realities.

Because we have one 2-dimensional surface living on an object with curvature different than the 2-dimensional surface on which we’d like to have our map (like the page of a book, the wall of a room, …), we need to use a projection to get all of the information off our sphere and onto our map. There exist a lot of projections (Globular, Azimuthal Equidistant, Mollweide, Miller Cylindrical, … to name only a portion), but the one that will be most familiar to anyone who’s seen a world map is likely the Mercator projection (named after a 16th century Flemish geographer).

One way to visualize what’s going on with this projection is to grab your orange from the time zone section. In an ideal world, you could cut a single vertical slit from pole-to-pole (for example, along the W line) and scoop out the internal tissue; working more in reality, cut the orange into quarters along the pole-to-pole lines, remove the fruit, dry the skins and place them ordered like they were on the orange next to each other, like in my photo.

By imagining that there are a number of parallel horizontal strips covering these skins, you’ll notice that some horizontal bands have more actual orange region (you can see almost no white area), while other bands have a lot less (there a several gaps of white area). Since this kind of projection maps a horizontal band of the skinned flattened orange to a corresponding horizontal band on our map, the bands which have gaps in between the orange segments must have their information visually spread out to fill in those missing gaps. As a result, the regions near the poles are given disproportionate amounts of real estate on the projected map.

This is the reason that, for example, Greenland and Canada seem so unrealistically large on the usual kind of world map.⁸

Another way to imagine this is by looking at the shortest distance between two points. Fear not, you weren’t lied to by your elementary school math teacher: the shortest distance between two points is indeed a straight line; the fine print on that contract is the sticky wicket. It matters what surface that straight line is living on; for example, when we’re talking about those two points on the surface of the Earth or the orange, that straight line is called a ‘geodesic’ (or less math-y: part of a ‘great circle’).

Pick up the second orange and draw two points – one in the ‘southern’ quadrant of the S-W strip and one in the ‘northern’ quadrant of the S-E strip. Now draw the shortest straight line connecting these two points — a geodesic.

This will take a little mental imagery again (and it’s cheating a bit in terms of legitimate projections): hold the orange in your hand and imagine that the orange is sitting inside of a clear plexiglass cube. Now, imagine that you’ve lifted the strip of skin that is that geodesic directly off the orange (preserving the same shape it was while on the orange) and floated it towards you until it smacked into the plexiglass. What you would see on the plexiglass would be one of those ‘funny curve paths’ people talk about.

The Seasons Change, Change, Change:

don’t worry: i’ve got one on social networks which i’ve been writing in quasi-parallel to this
… or a hunter-jumper / dressage saddle, if you’re a horse person.
Again, don’t stop reading here though.
quite possibly all living together in a trailer park somewhere, and potentially splitting the rent with the chemtrail-conspiracy-theorists and the folks who think that the moon landing was faked
Well, nothing except the decent excuse that they didn’t have a mountain of invalidating measurement and observation.
Once again, Prockey comes to my rescue…
6.39 x 10^-17 times the area
For example, Greenland is almost three times the size of Texas, but the usual kind of world map shows a Greenland that visually would cover nearly the entire continental U.S
… though that appears to vary between 22.1° and 24.5° with a periodicity of 42,000 years
An improvement would be if you could some how have the orange sitting with its center in the table’s surface, since a planet’s orbit follows its center of mass and not its ‘bottom’.
Technically: diameter