• The search functionality for the site has been fixed.
• Post and user ratings have been removed from the site; this became a rope for the tug and pull of popularity contests for the lazy. Comments will continue to remain open for the less lazy people seeking to engage in dialogue and debate (admittedly this requires greater functioning cerebral capacity than a Parkinsonian finger twitch).

Thanks as always for everyone’s participation.

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.

1. 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.
2. mmmm, consonance
3. math-ese: ‘mapping’
4. 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’
5. This statement is too general, there are specifications which the reader can discover reading up on the CH…
6. 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
7. “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)

The idea of the cone is to give an easy visualization as to whether two people, who sit at the base of their own cones, are able to exchange information at any given time. While this is technically a depiction of an evolution of a 4-dimensional surface, it might be more helpful if you think of the cone as a circle on the surface of the earth, which radiates outward from you evenly with time – like a rock dropped in a pond. As i sit here in San Francisco, starting at any given instant the radius of my light cone grows with time; in under 2 milliseconds, it has grown to cover Los Angeles; less than 10 milliseconds to encompass Mexico City; under 13 milliseconds to Washington D.C; below 22 milliseconds and there’s Reykjavik; after 30 milliseconds and Paris is part of the union; before 45 milliseconds it’s taken Mumbai…

If i should be lucky enough that a person in one of those locales is looking to exchange information with me, then our light cones need only grow for half the time until they meet — ~in the middle, per se. Should my friend’s cone not intersect my cone until 2 seconds from now, there’s no way i can know anything about my friend for another 2 seconds. Adding another turn to the situation, the information which i’ll hear in 2 seconds is actually ‘stale’ information. In much the same way, but on a harshly smaller scale, that our telescopes which spill forth pastorals while trained to night sky reveal not what is, but rather what was: so too will the information received from my friend not represent what is, but what was.

While we still find ourselves inhabiting just planet Earth, the impact of our separated light cones pales in comparison to the impact of our separated time zones, separated continents, separated hemispheres and slow notification systems; the meaningful events of one person’s day occurring while the other distantly wrestles through fitful sleep serves to sever. It’s only when we’ve gone far off-world that our light cones will define our existence; more than 1.2 seconds to the moon, 180 to more than 1330 seconds for Mars, and on and on.

Now or later, we find in either case when we sit in muted silence looking out on to depthless monochrome skies, midst that lack of information and contact, we begin to infer and make up data where none now exists and we’re left only to imagine the world without you.

1. When discussing what governs our ability to exchange information, i’m blissfully ignoring “spooky action at a distance” without qualm until we have become such masters of our domain that we can employ this in our daily lives

Dear Mr. or Miss 6,000 Year Old Earth Believer,

I’m not here to attack your religious belief system — you believe what you believe, for whatever basis, and that’s that. I’m not even here to point out that there’s supposed to be a separation between church and state in this country.

I am here, however, to say that it is one thing to stand firmly on one’s belief system, but it is a wholly other kettle of fish to deny something and still insist on using the fruits of that thing which you are denying. It would be like my having a fundamental belief that bridge trolls are responsible for creating donuts but then still driving over to Krispy Kreme to shovel their freshly made donuts into my gaping maw.

One of the more frequent tools used in judging the age of organic objects found on our planet is called “radiocarbon dating” which allows the dating of objects back about 60,000 years; the process relies upon judging the amount of a radioactive isotope of the element carbon which remains in the object and which changes over time due to radioactive decay. By your choosing to say that this form of measurement is invalid, you must also say that our understanding of radioactive decay is incorrect. If so, with regards to these other technologies which rely upon the same understanding:

• If part of your power grid is supplied by nuclear generating plants: please stop using our electricity
• The radiology facilities at the hospital: please stay broken-boned and unknowingly tumored
• The bridges and buildings whose metallurgy is verified and made stronger: please stay off, and out, of them
• Nuclear weapons: ¿pretend that all of those tests and attacks never happened?

If you also think that the universe is not billions and billions of years old, then let’s look at that. Our understanding of that age relies upon a number of things too esoteric to provide for clear cutting examples here; there is one verifying, simple, concept though: the way ‘light’ behaves. With that in mind, if you’d like to claim that we have no idea what we’re doing concerning the behaviour of light – that the distances, and therefore the time and age, which we’re looking at is mistaken calculation, then let’s look at what else you should be giving up:

• Airplanes, which rely upon GPS for navigation: please stop boarding them
• The GPS navigation systems in your cars: please go back to printed maps and driving aimlessly lost, too proud to ask for directions
• The hikes through forests and mountains relying upon hand held GPS units: please get lost with your compasses and maps
• Land line telephones, cell phones, the internet, …, basically all modern forms of telecommunication (which are relying both on satellite communication and fiber optics): please stop using them
• Modern stores, UPS/FedEx/DHL/the post office, and any other place using laser barcode readers: please limit your shopping to farmers markets and your shipments to 19th century implements

In conclusion, this isn’t my viewpoint, this is simply the facts being laid out in a manner which demonstrates the fragile house of cards in which you live. I can see how you might not like that, but at the end of the day, and this is the beauty of science, no amount of legislation will change what is a provable fact regardless of what county, state, or country you live in.

Sincerely,
Loki der Quaeler

If you have had interest in the NSK, feel that you’d like to participate in the Congress, and would be able to travel to, and spend time in, Germany in October 2010, please visit the main page for the Congress and then proceed to submit a delegate questionnaire.

One of the unbelievably awful things about the internet is that it mates that ‘bringing together of people’ with the double curse of the average human: (1) the difficulty to discriminate in choice and (2) the propensity to hoard and believe that more is better. What results from this Fly like merging is lived out daily by tens of millions on sites like Facebook. Multiplying the penalty of living this out on such sites is that, unlike some night in 1992 that faded out to muted shades as time went by, massive farms of servers are busy replicating and archiving your mistakes right now, so that the future you, the future friends, the future employers, perhaps the future children, can see it as clearly as though it just happened.

Welcome to the end of valuable friendships: a modern tragic play in four parts.

Looking back, there were those things said in passing over dinner, and the odd second hand tale. It seems like the clues were always there, waiting to be assembled: your dear friend has some good friends who are real douche bags; they haven’t a cupful of wit nor a minute ember of humor to douse with it. You always suspected it but you could never prove it — until now. Day in and day out, your friend’s update feed becomes a longer and longer laundry list of unfunny non-insights and retorts which weren’t even amusing when they first became public domain in the 1980s.
You can no longer deny it: here are the buffoons willingly added, and continuing to be kept, by your friend as their friends… and they’re schmucks: screamingly obvious, self-promoting, flying-a-flag morons. Your mind races:

• How does this person who i’ve come to hold in high esteem have such lousy taste?
• What does it mean for my own self-worth to be a friend of someone who has such lousy taste?

Act II
A hole that can never be filled

20, 50, 100, 200, 500, and on and on; and more and more… Like Romeo is Bleeding, your friend has to fill that hole, but instead of cash in the backyard, they can’t stay away from that Add button. It becomes a tired, hackneyed, ritual which has lost nearly all value and thereby has cheapened that for which it once had value. Nobody has 150 friends. Simply nobody. You can’t say for sure that you’ve even had meaningful conversations with 150 different people in your entire adult life… but there it sits: your friend’s “friend” list. What does it even mean? Even making the statement that someone has 200 friends would have been vapid braggadocio in recent times, and yet here it is as some item of pride on the front page of your-friend-the-prom-queen’s profile. And still everyday, there’s a new one, a new five. Where do they keep finding them? How many items can a collector really pay loving attention to?

---

Act III
Tear Down This Wall! (even if it’s supporting the roof)

Five years ago, the idea of an adult hanging out frequently with both their parents and peers in a social situation would have been solely the hallmark of the tacky white trash.

Five years ago, the idea of a barely known co-worker and a friend-for-the-past-fifteen-years together regularly sharing your conversation and comment would have been impossible — obviously fucking wrong.

Today, these are part of life-de-facto on social network sites like Facebook. Evaporating is the concept of ‘appropriateness’. Bob, from human resources, three jobs ago, is treated to seeing your friend’s children photos – just like you are. Your friend commenting on Brenda’s photo of her baby is given the same profile screen real estate as anything else your friend does, even though you don’t know who in the world Brenda is and couldn’t give a good god damn. On your friend’s profile, you can read your half of the conversation your friend had with their parent when (a) why is it the public’s business, and (b) you didn’t want to know, and (c) seeing only half of something not only makes no sense, but also inspires curiosity in something you didn’t want to know to begin with.

---

Act IV
Everyone is a King or Queen

… so, much like all of the want-to-be-somebodies who flocked daily to Versailles in the 1700s to stay in favour, you too get up every morning to attend your friend’s court. What is it today that might be proclaimed to that court of those awaiting hundreds of friends — that passing remark that you’d like to be a part of because they are, after all, still your friend.
Dirtied and devalued is the notion of privacy: the transience of a special shared moment, lost to archiving on third party servers and replicated; backed up databases; off-site-stored media; there, always stark, never going away; often available for others to read, weeks, months, and years later, and best of-all: out of context. This is a dark rabbit hole from which there’s no return.

---

Epilogue

Perhaps this is just the way humans will evolve, like birds in a pet store, we will become all a large mishmash of people twittering-posting-and-otherwise-babbling, simultaneously, en masse, about nothing of any particular importance. Perhaps that will become what is usual and regular. It’s too early to tell.

One thing seems for certain: it cheapens us all.

---

]]> https://process.org/discept/2009/02/20/the-negative-mutation-of-social-networks/feed/ 2 https://process.org/discept/2009/02/18/life-on-planes/ https://process.org/discept/2009/02/18/life-on-planes/#comments Wed, 18 Feb 2009 18:24:15 +0000 https://process.org/discept/?p=127 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:

1. two oranges
2. a sheet of ordinary letter paper
3. a ruler
4. a sharpee
5. a pen or a pencil (unless you want to use the sharpee on paper)
6. paper towels
7. one object of your choosing, no bigger than the oranges
8. a Pringle²
9. 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:

Whether you’ve decided to participate in parallel with your own materials, or just read along in observation, i’ve placed mile-marker photographs periodically through the article; starting off the photo accompaniment is one of the above inventory list gathered onto my desk. If you do decide to follow along, my one caveat is that you should wash the oranges as much as possible; who knows what sort of funky oranges i bought, but the waxy surface killed my sharpee in short time.⁶

As preparation, around the top of each orange at its belly button i wrote the regular N-E-S-W compass points in their usual fashion. Then, on each orange and treating the belly button like the north pole on the Earth, i drew four lines ‘straight’ from the north pole to the south pole – one for each of those cardinal direction (N-E-S-W) markers. To keep lines as straight as possible, i used the ruler to draw a periodically spaced line; this worked fairly well, but your mileage may vary. At this point, my oranges looked like the photograph on the right.

---

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:

What is this thing that has Santa and sleigh arriving to Sydney houses midst heat waves and beach cavorting? Why are the northern and southern hemispheres forever seasonally separated from each other.

For those with a hint of remembrance from a science class, the proffered answer may be because it’s somehow due to the orbit of the Earth around the Sun being elliptical (like an oval, not a perfect circle). While this is technically true, it is actually pretty close to being a circle. When talking about the shapes of orbits, the term used is ‘orbital eccentricity’. If an orbit is a perfect circle, then its eccentricity is zero; the Earth’s orbital eccentricity around the Sun is 0.016, and the amount of time the Earth spends swinging through the most remote point of its orbit is only about a day and a half longer than it spends swinging through the most near point of its orbit.

The main bringer of seasonal difference is the ‘axial tilt’ of the Earth. If you picture the top of your desk as the plane in which the Earth travels in orbit around the Sun, then pick up your second orange, orientate it such that the north pole is facing the ceiling, rotate it a sixteenth of a turn towards the wall to your left and place it back on the table. This is roughly the 23.4° tilt at which the Earth presently sits⁹. While this isn’t entirely accurate, it’s good enough for this article¹⁰.

Now, place the ‘one object of your choosing’ item on the table such that you can move your orange in an orbit around it — this will be the Sun. This obviously isn’t to scale as the Sun’s width¹¹ is about 110 times that of the Earth’s, but something like that isn’t fitting on your table. Now, making sure to always preserve that 23-ish° tilt, move the orange around your the sun in quarter-orbit movements. Notice how the northern and southern hemispheres vary in closeness to the sun depending on where in the orbit the Earth-orange is, like in my photos below. Assuming a counter-clockwise orbit, from left to right we have Autumn (for the northern hemisphere) / Spring (for the southern hemisphere), then Winter/Summer, then Spring/Autumn, and finally Summer/Winter. It is this changing of who’s-closer-to-the-sun which dictates our winters and summers, and keeps Kiwi kids from understanding why Whoville is buried in snow.

---

1. don’t worry: i’ve got one on social networks which i’ve been writing in quasi-parallel to this
2. … or a hunter-jumper / dressage saddle, if you’re a horse person.
3. Again, don’t stop reading here though.
4. 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
5. Well, nothing except the decent excuse that they didn’t have a mountain of invalidating measurement and observation.
6. Once again, Prockey comes to my rescue…
7. 6.39 x 10^-17 times the area
8. 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
9. … though that appears to vary between 22.1° and 24.5° with a periodicity of 42,000 years
10. 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’.
11. Technically: diameter

]]> https://process.org/discept/2009/02/18/life-on-planes/feed/ 1 https://process.org/discept/2008/12/15/myth-as-asylum-from-questioning/ https://process.org/discept/2008/12/15/myth-as-asylum-from-questioning/#comments Mon, 15 Dec 2008 16:59:38 +0000 https://process.org/discept/?p=90 There is too much religious tolerance in the world today.

This may read as nonsense given the seemingly endless stream of news items in which guy from faith A attempts to kill person from faith B and recent quasi-scrutiny of the belief system birthed by that science fiction author; if so, suspend your disbelief for a moment. Tolerance of religious belief has causally allowed a regular discouragement of scientific inquiry, but what is more damaging than that is that it has allowed illogical ‘explanations’ of real world phenomena to gain social validity. I suggest that by allowing one large portion of people’s lives, of their tenets, to slide by completely unquestioned despite obvious flaws and contradictions, an underlying message of complacency towards the ‘okay-ness’ of irrational and unsupported policy claims is being allowed to permeate portions of society in which it has no place being.

As i was walking around Rome several weeks ago, it was impossible to dodge the overarching influence of Christianity which still shadows life there: from roving small packs of American priests in their 20s on study, to the density of churches, to, of course, the Vatican itself. While i don’t expect the wide spread conversion of churches to Starbucks, it is saddening to note that nearing the end of 2008, with centuries of inquiry and discovery under our collective human belt, with rapid information dissemination at our fingertips, with an increasingly egalitarian view onto our peers, that we can still find such large subscriptions to religious faith.

As is evidenced by published poll data, and anecdotally noted by the turn out at public celebrations of religious events, the vast majority of the human race¹ turn towards religious doctrine for their moral guidance and cosmological explanation. While surveying in the US did appear to give evidence to a shade of a down tick in religious adherence from 1990 through 2000², there was an apparent bounce-back (though not to 1990 rates) reported by 2008.³ Outside the US, the data is not very available (or, at least, very locatable) on a more global scale; though with the world becoming more economically and politically instable⁴ over the past decade, one can imagine an increasing trend in faith during this time.⁵

Confusing this situation even more: during this same recent era, the ability to communicate information across geographically separated communities has become ever easier. This is perplexing since, as communication of information has become increasingly easy, and more widespread, one would think that it must take more effort to actively discount and screen out ideas contrary to one’s personal religious beliefs. Unfortunately, this sort of willful ignorance cannot be easily grouped under the established correlation between poor education and religious belief.⁶ Since the majority of the population succumbs to some flavor of superstitious belief system, what we often see as a reluctance to pose rational questions on the belief systems of others could likely be the fear that their own fragile framework could not withstand a similar scrutiny.

Key to both illusionary magic tricks and religious indoctrination: the participation of two parties is required — someone willing to be fooled, and someone willing to fool. While it is more difficult to see the motivations behind those willing to be duped, it’s not hard at all to see the roots of interest in an established belief system’s discouragement, or forbidding, of rational inquiry into an area which it has already claimed to be able to define through divine insight. No one wants to look like an ass, especially an organization which believes they are able to steer one’s eternal existence; so whether it’s an earth-centric view of the universe, a six day formation of all the universe, or whatever, once the guardians of the belief system have stated X as fact, it is in their interest to discourage examination of X.⁷
The Judeo-Christian belief system carries an extra weapon for this discouragement: in their mythology, the cause of their banishment from an idyllic eden life is the very process of questioning — the search for knowledge, symbolically embodied in that whole serpent-apple routine; to Christians, it is the original, the first, sin.⁸

Once we, as societies, kowtow — giving power to religious ideals by accepting them to be a coherent piece of the fabric of an individual — then it greatly hinders the ability for remotely rational dialogue to enter the equation when a belief system jumps from somewhat loony⁹ to dangerously-and-completely-absent-from-reality loony¹⁰. If we cannot consistently and regularly apply rational scrutiny to matters impacting daily human existence, then there is muted value in those times that we can do it at all.
It is of cold comfort that, for example, sanity finally regained control of the Kansas School Board when the real problem is the large step backwards taken due a fundamental Christian worldview being given such validity within society that a subculture believed it to be something it wasn’t, something it couldn’t be.

Truly, not everything can be presently answered by scientific inquiry and research — and it is completely plausible that it will never entirely be able to be answered — but it is certain that when people are allowed to defer to ritual and willful blindness then inquiry and research is stunted at best, and, at worst, it is prevented outright. Especially in these times in which there is a rash of bad events arising partially from our inability to model a large system¹¹ and partially from our inability to engineer new production solutions¹², it is important that we turn our minds outward with vigilance to ask coherent and decomposing questions, not turn our minds inward to take refuge in constructed fantasy.

As a postscript: it should be noted that, while i am mentioning Christianity in this article, there is no reason this complaint does not apply to all faith based systems (including those ‘alternative’ types — the ‘magick’ spectrum, the wiccan variants, …): they all have, as their basis, a kernel framework of unverifiable supposition over which further ideas, also unverifiable, are added or inferred to form a body of ‘laws’ and a cosmology.¹³

---

1. pie charted for your visual consumption
2. for two frequently referenced sources, there are the ARDA studies and the 2001 ARIS study
3. the Boston Globe had a decent summary
4. including the re-inventing of societies previously dominated by atheism-enforcing governments
5. … given the indications that humans, in a perceived crisis do turn to religion. Anecdotal evidence abounds through Google – here is one of the more recent picks of the litter
6. … those nations which do have less resources with which to educate their population actually do have a more ‘religious’ population (with the United States being the anomalous, and embarrassing, data point in that set) — PewResearchCenter Global Attitudes Project study in 2002.
7. especially when the synthesized ‘fact’ is not based on fact at all
8. think briefly what it means to have a system in which this is the core principle describing the cause for a adherent’s lot in life. ‘you could have had an amazingly paradisiacal life, had only your ancestors not been inquisitive’
9. like “there exists exactly one omnipotent being who watches over everything in the universe”
10. like “there are young children in Nigeria who are actually witches, and whose evil magic is responsible for poor fishing harvests (and not the oil industries dumping regulation-free into the waterways)”
11. in this case an economic system
12. for example a new economically and environmentally sound energy generation source
13. as opposed to something like the field of study loosely termed ‘Physics’, which has a kernel framework of verifiable conjecture from and over which further ideas (including some thought to be unverifiable as portended in certain facets of ‘string theory’) are derived

]]> https://process.org/discept/2008/12/15/myth-as-asylum-from-questioning/feed/ 7 https://process.org/discept/2008/12/08/2009-prediction-extended-neighborhood-watch-nabs-criminals/ https://process.org/discept/2008/12/08/2009-prediction-extended-neighborhood-watch-nabs-criminals/#comments Mon, 08 Dec 2008 17:24:44 +0000 https://process.org/discept/?p=72 Updated: 13.dec.2008; article appended

The world population has a subset comprised of people who, for one reason or another, demonstrate an online desire to constructively expand and, at least in their view, build a better society around them; this phenomenon is repeatedly demonstrated through public knowledge repositories such as those backed by a wiki format. Concurrently, 2008 draws to a close with live web presence being broadcast in increasingly better quality, and with content of every day slice-of-life views.

Using the editors of Wikipedia as an example, we can see that there is a core of people¹ spread around the world who patrol the encyclopedic content watching for forms of vandalism and misinformation. It does not seem outrageous to claim that this behavior satisfies something within individual editors which is akin to the fulfillment of justice and the maintenance of a world view. Similarly, it is unlikely that, for the majority of these individuals, there is something peculiar to encyclopedic bodies which restricts this desire; instead, it is reasonable that this motivating desire could be applied to other arenas were they available.

The other ingredient to this cake, web cams, have had sort of a milk-jug-being-slid-across-the-kitchen-counter existence. Historically, development would occur to a certain resolution quality which was bounded by how much data could be transmitted upstream, during which relative-lull widely available upstream capabilities would surge ahead; rinse and repeat. At present, the quality of web cam broadcasts, thanks to both increasing resolution of cheaper cameras and the decreasing cost for wider upstream bandwidth, has become quite good. Getting an amount of general media exposure recently, there are two good quality example streams from the Tenderloin in San Francisco² at “Adam’s Block”³ — we’re not at the quality of license plate resolution at this distance, but it is generally sharp and with a spry frame rate.

So we have a world population of desktop vigilantes and an increasing population of good quality live web broadcasts of slice-of-life happenings. Further, thanks to the world-wide nature of the internet, observation can occur continually, 24 hours per day, without any one observer needing to perturb their sleep nor social schedule. The last, minimal and non-essential, ingredient would be an even easier way to contact law enforcement local to the geographical area for the camera feed. All of which leads to the prediction that 2009 will see criminal activity being reported by geographically truly remote observers.

---

Update – 13.dec.2008

The SF Chronicle is reporting on recent developments with Adam’s Block.

---

1. the Huggle whitelist lists over 30,000 editors, for example
2. simply the best city in the world
3. for those not familiar with the Tenderloin, it’s one of the more squalid neighborhoods in San Francisco; so, the likelihood of witnessing something dodgy from the stream is decent

]]> https://process.org/discept/2008/12/08/2009-prediction-extended-neighborhood-watch-nabs-criminals/feed/ 2 https://process.org/discept/2008/10/31/an-incongruent-state-of-terrorism/ https://process.org/discept/2008/10/31/an-incongruent-state-of-terrorism/#comments Fri, 31 Oct 2008 19:08:02 +0000 https://process.org/discept/?p=59 As a preface: this article shouldn’t be mistaken as me being wistful for terrorist actions; i am ~infinitely more glad to have the lack of terrorist activity in the geographical places i love than i am pained by yet something else not making sense in the world.

As much as i would wish quietude on my thought processes and to simply enjoy the silence, there is something nagging which i’m unable to shake: there’s something suspicious concerning the state of terrorist attacks in the US and in Europe — an equation which doesn’t appear to add up when written out.

There are certain terrorist groups which advertise themselves as being quite angry at the US and at European countries; voluminous rhetoric in written, aural and visual form is easily encountered on the internet, for example. Judging from their widespread messages, as well as their claimed actual acts: these same groups don’t appear to be of lazy people, nor people without financial means, nor people without access to instruments of destruction.

Meanwhile, in the US and Europe, the public face on anti-terrorism efforts seemingly verges on buffoonery; whether it be employing rules after the fact¹, or instituting policies largely ineffectual in capturing people of forethought², the resulting impression is of a force by which no determined person of moderate intelligence could be deterred.³

In addition to force in numbers at organized country entry points, there exist the tremendously permeable land and sea borders. Historically, these haven’t succeeded as obstacles to traffickers of humans/drugs/cigarettes/alcohol/…; if any given country can’t stop these traffickers at its borders, then there’s very little reason to believe that these same borders aren’t porous to any other type of arriving party.

So, we have angry groups of people already displaying a propensity towards violent acts against certain other peoples; at the same time these other peoples both live in a very travel-open space and display Clouseau-esque defense postures. And yet: there have been basically no terrorist attacks in these countries over the past 7 years.⁴

What’s going on with this apparent imbalance? It could be that the advertised ineptitude is simply a ruse engineered by what is actually a very savvy and wired-in intelligence community, though continued policy decisions in other branches⁵ would have to mean that the ruse is a coordinated effort across several generally notoriously independent government offices.

As a bigger picture, it all continues to make little sense.

---

1. From training the stink eye on shoes only after someone tried to blow their’s up, to banning a very arbitrary amount of fluid only after it was found that people had nefarious plans that employed fluids, and so forth.
2. Such as confiscating media storage devices at borders, when any person serious about getting terrorist data (i don’t even know what this would be) into a country would simply put said data on an internet connected server and then walk physically-empty-handed into the country.
3. For further reading: Bruce Schneier has written and spoken extensively about the logically inconsistent, and unsafe, policies instituted by the TSA in the US.
4. I don’t mean to minimize the impact the London and Madrid attacks have had on the lives of people, but if we were to spin some custom metric of unit ‘(potential-area)(person)/(time)’, it would be damn small.
5. For example, choosing a bizarre moral stance over security as a national policy.

]]> https://process.org/discept/2008/10/31/an-incongruent-state-of-terrorism/feed/ 3