Future Views Magazine
University Fees and Debt
Who is the main beneficiary of university education, the student, the employer or the nation?
Units of measurement and the end of space.
Six Point Pollution Solution
Part One: "Flowers In The City"
brighten our cities with many tiny garden?
Six Point Pollution Solution
Part Two: "The Transformation of Waste"
Instead of landfill, waste could be a valuable resource.
Hidden blueprints for the new cities of the sea.
Arabian Mights & Maybes
Plans for the future carefully copied by Brenda.
Arabian Might May Raise The Sea
Desalinated reverse rivers powered by nature
The Unpublished Manuscript
A hint as to what may lie in the future.
A curious twist to the laws of perception.
The Slop House
an alternative to the traditional public house
You Are A Winner
A willing victim is the best victim.
Maybe our prisons are not as wonderful as all that.
The Bright New Pants Manifesto
The hilarious solution to invevitable change
A raging argument is discovered
What is infinity? Arri, Gerri, Rick & Euan discuss some curious ideas.
"Look at this," Arri exclaimed.
Euan and I crossed the room to join him.
We were inside the Royal Observatory on Blackford Hill in Edinburgh, Euan and I had just been given a guided tour of the telescopes.
Arri stepped back, there was a binocular viewer on the bench.
"Take a look through that," said Arri.
Euan, ever the extrovert, got there first. I waited, listening to Euan enthusing over whatever it was he was looking at.
Finally it was my turn. I think I had to endure all of forty five seconds patience before adjusting my vision gazing at what first seemed to be a white- on- black ink- blot. As the focus sharpened I realised I was looking at a photograph of the Milky Way, thousands upon thousands of stars - more than that, even.
A white box at the centre invited me to zoom in to the selected area. The 'Zoom' control took me in to the next view, the stars within the central area - again there were thousands upon thousands of stars. At the centre was another selection square. I zoomed in again, and as before found myself looking at numerous tiny white dots of varying shapes and sizes. There was one more central selection and again I was looking at many stars.
By this point I had the clear impression that if all the stars were equidistant at say, a hundred million light years from earth, our night sky would be a shimmering white ceiling without any gaps. That is how many stars we were looking at.
I stepped back, suitably impressed.
"Amazing, isn't it?" said Arri.
Euan and I agreed.
We had arrived the evening before and were about to embark on filming the heavens in all their spectacular glory. First, however, we had a day of familiarisation. Never having been inside an observatory before I found the visit fascinating.
"What is the limit of the universe and could it ever be seen by astronomers?" Euan asked.
We made our way to the visitor's dining room where lunch was waiting.
"That's what astronomers have been grappling with for years," replied Arri.
"Interesting thought," I remarked, "I suppose it would be like looking at infinity."
"Electron microscopes can look at electrons," Arri remarked.
"Pah! " Euan dismissed him, "not at all, they can only deduce what electrons are like by observing reactions."
"Can't they see atoms?" I asked, and blushed as I realised this had to be impossible.
"Atoms are too similar in size to photons," Gerri, ahead of us, explained, turning, "there's no way photons can resolve anything as small as an atom."
We stood behind Gerri with our trays in the dinner queue.
"And there's no end to the universe, as far as we can tell," quipped Arri, "therefore it can never be possible to see the end of the stars."
"By definition," said Euan, "looking at infinity is impossible."
"How do you mean, 'by definition'?" I asked.
"Well," Euan shrugged, "infinity lies beyond the end of measurement."
Gerri turned again, "the term 'Infinity' refers to a place or a value that is relative to where 'I' am - whoever and wherever that 'I' may happen to be."
Arri's eyes lit up, "Infinity is always the exact same distance away from me as it is from you, regardless of what separate planets or galaxies we may inhabit."
Gerri was being served, roast beef, yorkshire pudding, small potatoes, peas, carrots and gravy. It looked pretty good, my mouth watered.
We were silent as we each in turn collected our dinner and made our way to a table.
The view was fantastic, we looked out across hundreds of miles of plains, mountains and the sea to a distant horizon.
Arri laughed, his mouth was full. We looked at him but had to wait before he could speak.
"That means," he said at last, "wherever I am and whoever I might be, the 'Centre of the Universe' is always beneath my feet! "
We looked at one another, he was right. We laughed too, it seemed an odd concept.
Gerri frowned, "it also means that there is no such thing as the 'Centre of the Universe," he said.
"True," I remarked, "if the universe is limitless - "
"Infinite," Euan interrupted.
"Infinite," I continued, "then it has no boundary and can have no centre."
The others nodded.
We ate in silence.
"How do we know?" Arri asked.
"Know what?" Asked Euan.
"That the universe is not and cannot be a finite object with a solid periphery and a single, geographic mid- point?" Arri concluded.
Gerri shrugged, "if it were, there would have to be something else lying beyond that sphere, blob, square, spiral or whatever, and that something else would automatically be part of what we call, 'The Universe' because that is what 'universe' means." Gerri looked from one to the other, I pondered his suggestion.
"Everything that is encompassed by Infinity," Arri added.
"How is it that mathematicians can talk about infinity as though it were a place?" Euan asked.
"How do you mean?" I asked.
"Well," Euan continued, "mathematicians have speculated on what happens to radiant lines at infinity and have come up with diagrams of dots spread out along the line representing infinity, arguing if a single line were to stretch to infinity by the time it got there its thickness would have diverged into a series of separated points."
"That's odd," said Arri, "is there anything to suggest a line should change its width on arriving at infinity? I don't see why it shouldn't still be the same width along its entire length"
Euan laughed, "if I travel to infinity in my imagination, the moment I arrive it will once again be infinitely far away." He looked at us with a somewhat smug look, I laughed.
"We were taught at school," said Gerri, "that parallel lines meet at infinity," he paused to be sure he had our attention.
"However, since 'infinity' lies in every direction, surely, then, all lines must meet at infinity?"
"True," I remarked.
Euan's eyes widened, I sensed he was having one of his, 'Eureka' moments.
"Infinity is a circle that lies beyond the horizon," he said, "however, because it lies in every direction, it is, rather, a sphere that lies beyond the periphery of our universe." He held up his finger, "note, not our solar system but the entire universe. However, because 'sphere' suggests tangibility, it is more like a sphere made of dust, a dust which does not exist. Infinity, therefore, is a sphere that does not exist, and yet, it is there and has both a name and a mathematical value." Euan looked pleased with his definition. Gerri, on the other hand, looked out- gunned.
"Aha," I remarked, "infinity is a sphere with an infinite diameter and an infinite circumference which means its circumference and diameter are both the same, which is impossible."
"Except at Infinity because that's where all points in the universe meet," said Gerri.
"If it's where everything meets," said Euan, "it's not a unit but a place, a point,"
"It's both," Gerri and I said in unison.
"I'll get the desserts," said Arri, rising. "Trifle?"
Euan and I nodded.
"Cheesecake," said Gerri.
Gerri cleared his throat and leaned forward. "One argument says that if a line were drawn from the centre of the universe - the point beneath my feet in the here and now - and stretched all the way to infinity, by the time it got there it would have diverged to such an extent that it would make contact with every single point on Infinity, no gaps. The line's thickness would itself have become infinite."
Euan and I gave him our full attention, he continued.
"The other argument is to say there would be no divergence, the line would be the same width at the furthermost end as at the beginning and centre. Since there is no end, infinity never being reached, it could be said that the line arrives back where it started," he paused to see if we were following his argument, which we were (I think); "except that it does not," Gerri was now the one having a 'Eureka' moment, "it vanishes." He hit the table with his finger. "In that vanishing, it meets every other line. Infinity therefore has the mathematical characteristics of a point, a point through which every point of the universe meets - and that point," he waved his finger, "is right here, in the centre, under our very feet."
"Or noses," I quipped.
"You've spent too long looking at your naval," Euan laughed.
Gerri scowled, "I never cut you down when you made your point," he complained.
Arri was back, and grinning from ear to ear. I wondered if he had had an incident with a waitress.
"There's one thing none of us has thought of," he said as he handed round the desserts, "the idea of 'infinity' might be wrong, a cop- out, a name for something which does not exist." He laughed as he took his seat evidently finding his conclusion very funny.
I found it funny too but Euan and Gerri looked like they had had the wind taken from their sails.
"Think about it," he continued, "if I travel as far as the biggest number of light- years away from here, 'Infinity' will still be the biggest number of light- years away from wherever I arrive. 'Infinity' is just a term invented to compensate for the tedium of inventing new names for numbers."
"Ad-infen-itum," I quipped.
I leaned forward, it was my turn for 'Eureka!'. "Anyway," I said, "If infinity lies beyond all numbers, what is the last number before infinity?"
I felt pleased with myself.
"Ah," Gerri was evidently an expert on this, "the biggest number is a Googolplex."
"A what?" Arri's eyebrows shot up.
"A 'googol' is ten with a hundred noughts, adding 'plex' means it's raised to the power of itself," Gerri explained.
He took a breath and continued, "Ten is always the start of a new sequence, which ends in nine." He looked at us each in turn, we were nodding. Gerri continued, "the biggest number, therefore, is, 'googol- nines' , the googol equivalent to nineteen - "
"Nine hundred and ninety nine, or the googol equivalent to nine- billion- nines," I said.
"I stand corrected - googol- nines point nine repeater- plex," Gerri stabbed the air triumphantly.
"Googol- nines point nine repeated raised to the power of itself?" Arri asked.
"That's it," Euan was nodding.
"Infinity lies just beyond the last number."
"A long string of nines," said Euan, "followed by a decimal point followed by a long string of nines." He frowned, "but point- nine repeater is infinite! " Euan looked at us. I pursed my lip, raised my eyebrows and nodded.
"Not necessarily," said Gerri, "it could be argued that the nines stop repeating once they have reached the end of all our numbers."
"Googol- nine point- nine repeater- plex," I said.
"Exactly." Gerri nodded.
"Exactly what - or rather, what, exactly?" Arri said, struggling to follow.
"That's the number of times nine is repeated after the decimal point," Gerri explained.
"That's a lot of nines," said Arri thoughtfully.
"Yet still just short of the next whole number," explained Gerri.
"Which does not exist because the googol is the largest of all units," Euan said.
"Infinity is next," I said, "it lies just beyond the end of all those nines."
"At the end of all numbers, or units, there is a gap," said Euan.
"An infinitely large gap?" Arri asked.
"Or infinitely small," said Euan.
"How's this," he continued, "nought point googolnines- point- nine- repeater- plex noughts - one."
"Meaning?" I asked.
"That's how many noughts there are before the 'one' at the end," Euan explained, "that is the size of the gap between the very last number and infinity." He looked at Gerri.
"Why not nine? Nought- point- one is much further from two than nought- point- nine," said Gerri.
"You mean," I said, "infinity lies just beyond all the noughts after the decimal point ending in nine?"
Arri looked from me to Gerri to Euan.
"Why not all the nines? Nines all the way?" he asked.
Euan shook his head, "it has to be all the noughts ending in one because it has to be as close to nought as possible, not as close to the next number as possible. Nought- point Googolplex- nines: number of noughts, then one." He squiggled on the table with his finger.
"That's the number of noughts?" Arri asked.
Euan nodded, "the number of noughts before the last 'one' - "
"Which has to be there or all you have is nought, you know, nothing," I said.
"- has to be the biggest number of noughts on the other side of the decimal point followed by a solitary number one," Euan finished.
Gerri had been concentrating, "ah yes," he muttered, envisioning it in his head.
"Sounds infinitely small," Arri quipped.
"It is," I assured him, "almost."
"And then we have infinity?" said Arri.
"Yes," said Euan.
"Good, we can go there," I laughed.
"No," Euan exclaimed, "because the moment you arrive, infinity is again the same distance away."
I thought a moment, "what if I use millimetres as the unit, then measure the same distance in miles?" I grinned, "it would become a much more manageable distance."
"Why not convert nanometres to light- years?" Arri suggested.
"That's daft." Euan scowled, "we're missing the point."
We groaned at his inadvertent pun.
"Okay then what is the point?" Gerri was growing impatient, we had strayed from the serious side of the debate.
"You can't measure the gap between numbers and infinity," said Arri, "because 'infinity' is always infinitely far away, therefore, the gap between Infinity and the last number is itself, infinite."
"No wonder mathematicians went mad," Euan remarked sardonically.
"The point is," I said, evoking another groan, "we are dealing with two entirely different kinds of unit - "
"Absolute units versus relative," Arri snapped his fingers.
"Infinity is a unit defined relative to Origin," said Gerri with a tone akin to revelation.
"Like 'Horizon' I suppose," Euan observed.
"Great," said Gerri, "now we're getting somewhere. Instead of arguing about how many units there are to infinity, we ought to be asking ourselves how many units are there which are defined as 'Relative to Origin' rather than relative to any fixed point." Gerri looked at us earnestly.
"We've found two so far," I said.
"There could be more," said Arri.
"Horizon, infinity, up, along?" Euan counted them off on his fingers.
"No, no, no," Gerri could not conceal his annoyance, "those aren't units," he threw in a disparaging remark.
Arri frowned, "I bet there are more but I can't think of any right now."
I shrugged, "maybe there aren't, maybe that's it."
"That's it indeed," said Lotte, making us jump, "time to go- o! " Eta added in a sing- song tone. It seems they had both been standing listening to our last exchange. They sniggered as they gestured us to follow them. We had to attend a talk about photographing nebulae and creating those awesome pictures that make outer space look like clouds of coloured gas.
Arri was not yet prepared to drop the subject, "infinity is a unit measured relative to origin and is therefore also a constant," he said as we left the table. "Constants are the building blocks of the mathematician's universe."
Lotte turned to him, "that sounds awfully, ahem, profound, especially coming from you," she exclaimed with not a little incredulity.
We grinned as we left the room.