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Tag Archives: geometry

I still say raytracer in my head, but...I will modernise my internal vocabulary eventually...

Raycasting Rendering Examples

Have finally gotten around to compiling the Raycaster Rendering Demos that Terry and I cobbled together (mostly at the airport) at tigjam UK 2008 from an engine he whipped up some months ago. There are 12; here’s a montage of 10:

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Oh, everyone go check out the second issue of the carnival of mathematics!

Serial: Chapter 3: C – Columbus

Isaac, dearest,

I was leaving through the recently published memoirs of Christopher Columbus (curiously written in the third person; surely such an odd style cannot have been introduced by the translator, whatever other factual errors he makes), last year, and I felt you might be amused with the following story; at least insofar as it contrasts with your recent proof of the finiteness of the the geometrical space in this universe.

Ah, no point faffing about; I will just quote the passage from near the end of the book you now

Columbus sat in his study, overlooking Cádiz, with a Bible on his desk. He stared across to the horizon, and thought.

“How terrible it must be,t o be a highly situated savage, to see the world about you to the horizon and think there nothing more, to feel one’s self trapped on so small a disk of earth”.

Though, then he fancied that maybe a normal man might be able to live feeling so encaged, but not an explorer such as he.

“God has set forth, he had been told, a bigger cage for us in which to live, many orders of magnitude larger than the disk to our horizon, yet finite nonetheless. More than enough to go on exploring for generations”, he reassured himself.

And yet, he felt he could not accept this: How could God imbue the greatest of his men with the spirit of exploration, if there might be some age, maybe in a few centuries, when there will be no to explore, to discover, to find, on this planet. He would have verify this himself, to see the edge of the world with his own eyes, or alternatively spend the rest of his life travelling farther and farther out, boundless.

With this monumental task in mind, he petitioned the king for enough gold to sail for five years continuously west and, after five years petitioning, he was granted this sum.

One year of eager preparations passed before he was ready to set sail to see if the Atlantic actually had an edge. Six months at sea and land ahoy! What joys he experienced, what vindication: some islands, then a whole new expanse of land – the Vatican said that there was nothing west of Europe: they were wrong, he now knew!

This meant for him one of two things: We live on a much, much larger world than the Vatican claim, or alternatively, and his personal theological views led him to regard this as being much more likely: the world is infinite in extent, that one can travel forever in any direction without reaching an edge – this would be the only possible world he could see god having created – otherwise what a cruel fate would await the noble pursuit of exploration!

Okay, so here there’s a big chapter about America, which I’m not going to quote for you in full, but essentially what happens is that I decide that I must move south along the coast, because, even if it does go on forever in both directions, the only alternative is to give up and go back and there’s no way I’m gonna do that. So we travel down around, and west for a few more months, until hitting land again. Bang. And they anchor. And, lo, what’s this? A VILLAGE. Sweet. Then:

Shortly after entering the village, he found that he himself did find the people look quite familiar, some words of their language, their customs, from his travels east of India; had these people, from the other side of Europe, travelled here first, made these vast journeys? The very notion seemed impossible; the Indians and Chinese hated sailing, he knew. And yet, as his crew went further inland, he began to find more and more similarities until there was no doubt left: they were in China.

This world repeated itself then, he realized – it was not Rome’s disk, nor was it an explorer’s world – “What a cruel trick to play”, he thought “To think, if you had a powerful enough telescope you might catch sight of yourself, looking away, that the world was, despite all of it’s illusions, really finite in extent; that one might still sail west forever, but never come across anything new, that the world was finite – there were no barriers to travel, but, nonetheless, there was only a finite amount to discover in it – how cruel a discovery for an explorer to have make!

Of course, *everyone* knows that Galileo believed the earth to be shaped like an aubergine. So this casts everything, really, in to the most terrible of doubts, don’t you think? But then again, it is a charming story, neh?

Anyway, I hope to see you in the new year.




Hmmm...I wonder if such a short post is going to mess up formatting.

Serial: Chapter 2:C/A – Blackness

I was alone, and in darkness. It stretched out forever, and yet it was not a vertigo I felt at the feeling of this expanse, but rather found myself suffocating at the immediacy and intimacy of the blackness.

Finsler, whatever.

Serial: Chapter 2: A – Riemannian

In a dream again, I found myself in a labyrinth. This was not a labyrinth of walls, though, it was a labyrinth of space and, instead of feeling claustrophobic confinement, I felt the oppression of a space too rich, too rich and dense, to meaningfully comprehend.

This was a desert, with occasional rocks, trees, allowing me some orientation with respect to my world and, yet, I couldn’t shake off a feeling of general disorientation.

I set my sight on a particular tree, which was standing next to a sharply jutting piece of sandstone several paces away. I made my way towards it, or rather, tried to – I found after one or two paces these to be on the periphery of my vision, though a little closer – I turned to face them again, and stepped forward: walking towards this tree was like keeping one’s balance on a bicycle – it took constant corrections.

I picked up a small stone, some black, rough stone, from the sand, and tossed it forwards a short distance; I found that I was getting the hang of this desert, and was able, with some small exertion of effort, to walk to it and pick it up again, though I could only move but slowly.

I picked up two stones, and made an effort to toss them both t the same point a few metres away. Indeed, they looked to land rather closely together. And, yet, as I walked towards one, I found the other getting further and further away, until, by the time I had reached my destination, the other one was almost as far from me as I was now from the point at which I had thrown it.

The geometry of this place was…thick; it seemed that even the slightest error in orientation, on a walk of but a few paces, could result in you being further away than when you started.

I then knew, as one does in dreams, that my destination – the place where I needed to go, was many miles away, that I could spend a lifetime of lifetimes trying to find it and, even if someone could point me in its rough direction from where I was, I would have little chance of ever finding it.

Isotropy groups eat your heart out!

Serial: Chapter 2: C – Cube

I fell asleep to find myself standing on an interior wall of a cube. There was no gravity it seemed, though I was, it seemed, bound to walk on it’s surface.

This was not too small a chamber; it stretched almost a hundred metres in each direction, and the isolation or artifice of this situation did not in and of itself worry me. But, after a few minutes of exploration had passed, I noticed that this cube was not as spacious as it had seemed to me at first glance.

I stood at one corner and asked myself “Would it matter me if I was to stand at any other corner?” With this thought I could feel the space I had perceived contract about me. I thought “At this corner, is there any difference between facing one other corner, or any other one?” And with this, and all it’s variations and implications, I felt my freedom disappear, to be replaced with a noose, tightening about my neck.

A thousand doors, all locked, invite thoughts on tantalizing possibilities shut off to one's self.

Serial: Chapter 1: C/A – High Dim.

The following night I had another dream: I awoke inside the dream, to find myself breathing deeply, calmly, in some spacious, geometrical environment. I looked about, trying to take in in my surroundings, only a fear began to grab at me: the space was uninhabited by anything else, it just went on forever in all directions, and yet there was more: as it dawned upon me that I was in a space of dimensions many more than three, that space was going to infinity not merely in three directions, but twenty – this frightened me, I found myself unable to move, praying for some relief to the immense extent of this world.

My prayers were suddenly answered: I found myself trapped in a cramped prison cell; trapped between forty different walls on forty different sides.

lavish resplendent; now...where did I leave that stag-bust?

Serial: Chapter 1: C – Quotient

It took me some time to get to sleep that night. When I did, my fears blossomed into a nightmare:

I was, alone, in a dining room, with a heavy mahogany table and the walls were covered with a rich velvety wallpaper. Silver service laid on lace tablecloth, adorned about with plates of food.

In contrast to the evening’s fears of expanses, I now began to feel uncomfortably contained in my body, in this oppressive room; my apprehensions grew with every breath, and then I noticed some changes that restricted my enclosure upon me in a way I could not have expected:

There were no straightforward physical barriers developing, nor did the walls begin to close in on me; rather I began to perceive things differently – or rather, the reality of the situation changed. I had been looking at the contents of my plate, glancing fearfully away from it up to the ceiling every so often, lest it come crashing down upon me. But then the changes began: suddenly, the lampshade on the ceiling and the plate on the table were not separate things, but rather one thing; still the lampshade and the place, but now thoughts of separating them were impossible.

I pulled my eyes away from this transfiguration, allowing my sight to escape to the door, only the door was now not only the door, but simultaneously the lower shelf of the bookcase on the opposite end of the room, and then the bookcase as a whole became the bookcase and the table; this room was folding in on itself, but in no straightforward way; I tried then to look down, and found that the floor had become the floor and the ceiling, and then, all at once, up was combined with down.

I suddenly awoke, with the thought “What must be becoming of me?”

What if this entire scene, not just a horizontal slice of it, was the horizon of some other scene?

Serial: Chapter 1: A – Horizon

I had found myself walking down a pier about eveningtime; the sun was setting.

I focused on the horizon; how abstract an entity this is; it is not a physical “thing”, but depending on your position you can, thanks to the obliging sphericity of the planet, draw a line (well, a circle) on the globe, or a map an point at it and say “that’s the horizon I see”, which is really quite nice, when you think about it; at least our horizon at any time is a definite line of land, as opposed to the abstract line of land infinitely far off, as it would be if we lived on a plane. But, the notion of horizon is inseparable from this unreal prospect, at least in our thoughts.

How much depth is contained in this horizon, how fantastic it is. Then I thought, as one might, “what might the horizon look like in a higher dimensional world? – What might it be like to look out in to the distance to find yourself confronted with a horizon of not one dimension, a line, but two, a plane? I closed my eyes, and tried visualise this but my body instead froze, my heart was set racing – the overwhelming extent of this sight suddenly struck me, and I found myself unable to move for several minutes, unable to think about anything else, my mind fixated, unable to open my eyes lest I find myself faced with this terrible distant entity; a whole world might indicate the limits of my field of vision, a whole world that existed only as a conceptual nicety to me.