Which is it? (straight line or circle through a portal?)

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Kurushimi
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Which is it? (straight line or circle through a portal?)

Postby Kurushimi » Mon Sep 13, 2010 2:31 am UTC

Image

Which would it be? I'm really not sure.

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Re: Which is it?

Postby Dopefish » Mon Sep 13, 2010 2:38 am UTC

I'm inclined to think that when using fictional technology (portals), you're free to come to whatever conclusion you want, and there are no wrong answers, and as such you'd be correct in saying it's neither, just one of the two, or both, depending on how you decide to treat portal mechanics.

If I were to assign an answer, I'd go with straight line, but that's fairly arbitrary.

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Re: Which is it?

Postby Eastwinn » Mon Sep 13, 2010 2:44 am UTC

What would make it a circle?
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Re: Which is it?

Postby phlip » Mon Sep 13, 2010 2:50 am UTC

If the portals are exactly parallel then it's a straight line... if there even slightly misaligned, then it'll be a circle (the closer they are to parallel, the larger the circle). Basically, it's a generalised circle of radius approximately equal to the distance between the portal divided by the angle between them (in radians).

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Re: Which is it?

Postby Luonnos » Mon Sep 13, 2010 3:34 am UTC

Let us define a line to be a curve with 0 curvature at every point. As we walk along the rope (with not necessarily parallel portals), we find that the rope never bends in reference to our path of travel. This being said, the world does rotate relative to our path, meaning that we have turned the world into a circle, while leaving the rope as a line. :shock:

Also, shouldn't something be holding that rope up?

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Re: Which is it?

Postby phlip » Mon Sep 13, 2010 3:39 am UTC

Luonnos wrote:As we walk along the rope (with not necessarily parallel portals), we find that the rope never bends in reference to our path of travel.

No, if the portals aren't parallel, the rope does bend... if, say, it's normal to the portals at each side, then it must bend between them, but if it's a straight line between the portals, then it must have a bend or corner at the portal transition.

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Re: Which is it?

Postby Syrin » Mon Sep 13, 2010 3:42 am UTC

Call me crazy but this looks like a line with the two endpoints identified. If that isn't a circle I don't know what is.

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Re: Which is it?

Postby phlip » Mon Sep 13, 2010 3:52 am UTC

Well if we want to be technical, then the terms "line" and "circle" refer to shapes in a Euclidian space... and whatever space these portals work in, it's not Euclidian. So the names don't really apply. You could try to extend them, but then it's up to you exactly how you do that (you could focus on its zero curvature and call it a line, or the fact that it's a closed loop of uniform curvature, and call it a circle) but that's up to you.

So yeah, basically: what Dopefish said.

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Re: Which is it?

Postby antonfire » Mon Sep 13, 2010 4:06 am UTC

It's a straight circle.
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Re: Which is it?

Postby Yesila » Mon Sep 13, 2010 5:08 am UTC

antonfire wrote:It's a straight circle.


Maybe it's bi, with hetero-circular leanings.

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Re: Which is it?

Postby Tirian » Mon Sep 13, 2010 4:56 pm UTC

It clearly isn't a line, as the rope is not infinitely long. It also clearly isn't a circle, as there is no point in the space such that every point on the rope is equidistant from that point.

Topology is very capable of answering the question. When you bond the ends of a rope together (assuming that that bond is unbreakable), you form a closed loop. That doesn't change just because the closed loop passes through a portal. That rope cannot be "deformed" into one that doesn't pass through the portal, but this is no more anti-intuitive than a tied rope that passes through a metal ring not being homologically equivalent to a tied rope that doesn't pass through the ring.

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Re: Which is it?

Postby MartianInvader » Mon Sep 13, 2010 5:35 pm UTC

Topologically, it's a circle.

Geometrically, it's a geodesic, which is essentially the generalization of straight line to non-Euclidean spaces. Well, "straight", at least... I don't know if there's any reasonable definition that makes it a "line".

So I might say it's both.
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Re: Which is it?

Postby mnvdk » Mon Sep 13, 2010 8:33 pm UTC

MartianInvader wrote:Topologically, it's a circle.


Why is that so?

(I'm not questioning the truth in this, I'm sincerely interested in a broader explanation :D )
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Re: Which is it?

Postby Tirian » Mon Sep 13, 2010 10:51 pm UTC

mnvdk wrote:
MartianInvader wrote:Topologically, it's a circle.


Why is that so?

(I'm not questioning the truth in this, I'm sincerely interested in a broader explanation :D )


Let's make this a 2-dimensional example just so we can use the language a little more properly. Let's think about the topological Cartesian plane as we normally understand it except that we identify the points (0,0) and (2,0) (as if we had little infinitely small "portals" there). Now for our rope, we'll consider the line segment from (0,0) to (2,0). This is a curve, and as its endpoints are the same (as we have declared it to be) it is a closed curve. It also doesn't intersect itself except at its endpoints, so it is a simple closed curve. MartianInvader and I agree up to this part.

Now, in an ordinary Cartesian plane E, all simple closed curves are homeomorphic to the unit circle, in the sense that there is a continuous function F from the unit interval [0,1] to E such that F(0) is the simple closed curve we're talking about and F(1) is the unit circle. The normal term for simple closed curves of which this is true is Jordan curves (because the Jordan Curve Theorem is very important in lots of fields); it's not entirely intuitive to call them circles since squares and triangles and any squiggle that ends where it starts and doesn't cross itself is also a circle.

However, it is not true in our topology that our simple closed curve is homeomorphic to the unit circle. You can stretch it out and "rotate" your start/end point and do all sorts of interesting things to continuously deform the curve, but you can't make it leave the "portal" and therefore it can never get shaped into the unit circle.

Returning to the 3 dimensional rope problem with all that in mind, let's say that we're dealing with a rope that can be twisted and coiled like a normal rope but can also be stretched and compressed to our heart's content. If we took a length of that rope and tied its ends together, we could shape that rope into a circle lying on the ground (at least enough to the degree that a casual observer would agree that it was a circle) without ever untying the ends. But the rope that is stretched through the portal isn't the same. No matter how much we deform it without untying the ends, it will always be going through the portal. So it happens to be a simple closed space curve that isn't homeomorphic to a circle.

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Re: Which is it?

Postby letterX » Tue Sep 14, 2010 12:04 am UTC

Homeomorphic != homotopic.

The rope between the portals is very much homeomorphic to the circle: as someone mentioned above, it is pretty clearly an interval with its two endpoints identified. When you're building topology from scratch, this is often taken as the definition of a circle, as it's super easy to construct from the real line and quotients.

What you have shown is that the rope between portals is not homotopic to a circle that I draw on the ground (that is, you can't continuously deform one to the other inside the space). But that's just what happens when you live in spaces which are non-simply connected. It's still a circle, though. It just happens to live in a different equivalence class modulo homotopy.

As for it being a line: if it's been pulled taut, then it continues straight in both directions forever (even though forever means looping back on yourself after you go around once). Which is basically the only definition of a line that I can think of outside a strictly Euclidean setting. So I would say that it is definitely a line, which may also happen to be a circle depending on how you're looking at it (In the metric space setting, it is most definitely not a circle).

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Re: Which is it?

Postby nehpest » Tue Sep 14, 2010 12:24 am UTC

OT from the original question, but it seems to me that, though one would likely go through the portal without any ill effects, touching the "edge" of the portal would be highly unpleasant. Weird energy configurations aside...

Image
Image

... this seems like the most probable outcome.
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Re: Which is it?

Postby Dason » Tue Sep 14, 2010 2:36 am UTC

nehpest wrote:OT from the original question, but it seems to me that, though one would likely go through the portal without any ill effects, touching the "edge" of the portal would be highly unpleasant. Weird energy configurations aside...

...

... this seems like the most probable outcome.

You've clearly never played portal.
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Re: Which is it?

Postby mdyrud » Tue Sep 14, 2010 2:37 am UTC

To go further OT, who else has seen the trailer for Portal 2 and is very, very excited?

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Re: Which is it?

Postby Mike_Bson » Tue Sep 14, 2010 2:39 am UTC

mdyrud wrote:To go further OT, who else has seen the trailer for Portal 2 and is very, very excited?

There's a ton of trailers out. I've seen all. Above Reach and New Vegas, Portal 2 is on my top list.

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Re: Which is it?

Postby letterX » Tue Sep 14, 2010 2:39 am UTC

Dason wrote:
nehpest wrote:OT from the original question, but it seems to me that, though one would likely go through the portal without any ill effects, touching the "edge" of the portal would be highly unpleasant. Weird energy configurations aside...

...

... this seems like the most probable outcome.

You've clearly never played portal.

Yes, but in portal, your portals must be nicely attached to walls and other flat surfaces, where trying to walk through the edge results in half of you hitting a wall: no more damaging than running through half a doorway.

However, if a portal were just hanging out in space, with some infinitely thin boundary: that would be less good.

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Re: Which is it?

Postby phlip » Tue Sep 14, 2010 2:43 am UTC

Technically, the same should happen if you're halfway through a portal and try to walk sideways into the boundary - it should act like an infinitely thin blade. And if the portal closes while you're halfway through it, you should get cut in half. Neither of these happens in the game either. Both for gameplay reasons, but still.

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Re: Which is it?

Postby nehpest » Tue Sep 14, 2010 6:03 am UTC

Right, what letterX said. The situation just screams out "Razor Blade from Hell" to me. Then again, a sensible implementation of this scenario would account for such "edge" cases.
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Re: Which is it?

Postby Xanthir » Tue Sep 14, 2010 7:40 am UTC

Quantum loop dynamics in the boundaries produce a macroscopic pressure effect at the dimensional annealing, repelling you from the edges with a force asymptotically increasing as you approach it.
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Re: Which is it?

Postby mnvdk » Tue Sep 14, 2010 1:31 pm UTC

letterX wrote:Homeomorphic != homotopic.

The rope between the portals is very much homeomorphic to the circle: as someone mentioned above, it is pretty clearly an interval with its two endpoints identified. When you're building topology from scratch, this is often taken as the definition of a circle, as it's super easy to construct from the real line and quotients.

What you have shown is that the rope between portals is not homotopic to a circle that I draw on the ground (that is, you can't continuously deform one to the other inside the space). But that's just what happens when you live in spaces which are non-simply connected. It's still a circle, though. It just happens to live in a different equivalence class modulo homotopy.

As for it being a line: if it's been pulled taut, then it continues straight in both directions forever (even though forever means looping back on yourself after you go around once). Which is basically the only definition of a line that I can think of outside a strictly Euclidean setting. So I would say that it is definitely a line, which may also happen to be a circle depending on how you're looking at it (In the metric space setting, it is most definitely not a circle).


Okay, now I haven't really read anything (apart from popular knowledge) about topology as of yet, but I thought that homeomorphic was a much stronger term than homotopic, in that sense that if two spaces are homeomorphic, they are also homotopic.
Clearly, judging from you reply, it is not so.

What is the difference bewteen homotopic and homeomorphic (explained both strictly AND informally/intuitively speaking, if possible...)?
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Re: Which is it?

Postby Tirian » Tue Sep 14, 2010 1:51 pm UTC

letterX wrote:The rope between the portals is very much homeomorphic to the circle: as someone mentioned above, it is pretty clearly an interval with its two endpoints identified. When you're building topology from scratch, this is often taken as the definition of a circle, as it's super easy to construct from the real line and quotients.


I've got three topology books on my shelf, and none of them define "circle" in this way, and neither does Wolfram MathWorld. It strikes me as a very bad idea to do so, since it interferes with the geometric concept of the locus of points in a plane that are a fixed distance from a fixed point, which is a concept that could easily arise naturally in a metrizable topological space and is far more specific that the purely topological notion of a simple closed curve.

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Re: Which is it?

Postby mike-l » Tue Sep 14, 2010 2:21 pm UTC

mnvdk wrote:Okay, now I haven't really read anything (apart from popular knowledge) about topology as of yet, but I thought that homeomorphic was a much stronger term than homotopic, in that sense that if two spaces are homeomorphic, they are also homotopic.
Clearly, judging from you reply, it is not so.

What is the difference bewteen homotopic and homeomorphic (explained both strictly AND informally/intuitively speaking, if possible...)?


Homeomorphic means the two objects have the same topology. Homotopic means the two objects can be continuously deformed into each other in the ambient space. In particular, homeomorphic doesn't care at all about where the objects are living, it's only concerned about what points on the objects are 'close together'. Homotopic on the other hand cares quite a bit about where the objects live. For example, any two circles are homotopic in R^2, but the unit circle is not homotopic to a circle not containing the origin in R^2\0 (R^2 without the origin). Edit: Tirian does that mean that the unit circle is not a circle in the punctured plane? (Not being sarcastic here, I've always gone with a circle is anything homeomorphic to S^1 (topologically a circle that is), but you seem to have a different definition that I can't quite understand the nuances of)

(To be more precise, homotopic is a relation on maps, while homeomorphic is a relation on spaces)
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Re: Which is it?

Postby Tirian » Tue Sep 14, 2010 3:27 pm UTC

mike-l wrote:Homeomorphic means the two objects have the same topology. Homotopic means the two objects can be continuously deformed into each other in the ambient space.


http://mathworld.wolfram.com/Homeomorphic.html

Two objects are homeomorphic if they can be deformed into each other by a continuous, invertible mapping. Such a homeomorphism ignores the space in which surfaces are embedded, so the deformation can be completed in a higher dimensional space than the surface was originally embedded.

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Re: Which is it?

Postby mike-l » Tue Sep 14, 2010 3:54 pm UTC

Tirian wrote:
mike-l wrote:Homeomorphic means the two objects have the same topology. Homotopic means the two objects can be continuously deformed into each other in the ambient space.


http://mathworld.wolfram.com/Homeomorphic.html

Two objects are homeomorphic if they can be deformed into each other by a continuous, invertible mapping. Such a homeomorphism ignores the space in which surfaces are embedded, so the deformation can be completed in a higher dimensional space than the surface was originally embedded.


That's poorly worded. For example, 'O' and 'Q' (the topological spaces looking like those letters) are homotopic but not homeomorphic, but I can deform one into the other (see below). What they seem to be getting at, and this is correct, is that X is homeomorphic to Y if there is a continuous map X->Y that has a continuous inverse. Your definition a few posts back isn't well formed.
Now, in an ordinary Cartesian plane E, all simple closed curves are homeomorphic to the unit circle, in the sense that there is a continuous function F from the unit interval [0,1] to E such that F(0) is the simple closed curve we're talking about and F(1) is the unit circle. The normal term for simple closed curves of which this is true is Jordan curves (because the Jordan Curve Theorem is very important in lots of fields); it's not entirely intuitive to call them circles since squares and triangles and any squiggle that ends where it starts and doesn't cross itself is also a circle.

First of all, if F is a map from [0,1] to E, then F(0) is a point, not a curve. If F is a map from [0,1] to curves in E, then you are definitely talking about homotopy and not homeomorphic-ness. If F is a different map than this, then you'll have to clarify for me what you mean.

To do the example above, we'll make our space 'O', that's easy it's S^1 (the unit circle), and 'Q', for which I'll take my circle S^1, identify a point p where the 'tail' on Q meets the circle, and then [imath]Q = S^1 \cup [0,1]/(p=0)[/imath], that is, a circle and a line with one point on each identified together, and the topology is that a set is open if its intersection with both S^1 and [0,1] is open.

Now, O and Q are not homeomorphic, because I can remove 1 point from Q (namely p) and get 2 disconnected components, but I can't do that with O. (If there was a homeomorphism from O to Q, then if I call t the preimage of p, then I'd have a continuous map from O-t which is connected onto Q-p which is not connected, which is impossible)

But O and Q are homotopic. O sits inside Q in the way we constructed them, and there is a continuous map from Q onto O given by f(q) = q if q is on S^1, and f(q) = p if q is on [0,1]. Now we can construct a homotopy from f to the identity, F:[0,1]xQ -> Q, given by F(t,q)=q if q is on S^1, and F(t,q)=tq if q is on [0,1].
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Re: Which is it?

Postby mnvdk » Fri Sep 17, 2010 11:22 am UTC

Ah, so 'T' would be homotopic to 'I', since we can continuously deform one into the other, but not homeomorphic, since, if you remove the point connecting the vertical to the horizontal line in 'T' you get three components, whereas 'I' can only be divided into two separate components.

Am I right? Or at least intuitively so...
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Re: Which is it?

Postby MartianInvader » Fri Sep 17, 2010 12:12 pm UTC

Strictly speaking, you can't say T is homotopic to I, because 'homotopic' depends on how it's embedded in the space. For example, if we divide the plane in two by removing the x-axis, then put a "T" in the top half and an "I" in the bottom half, then the two aren't homotopic. Being homotopic is a property between two maps into a space, not between two spaces.

However, there is a relatinship called "homotopy equivalence", which is independent of embeddings. Basically, we say two spaces are homotopy equivalent if they have maps to each other such that composing one map with the other (or the other with the one) gives you a map homotopic to the identity.

So T and I are homotopy equivalent, and in fact are both H.E. to a point. A and R are also homotopy equivalent. In fact, all uppercase letters are H.E. to either A,B, or C. Not all lowercase letters are, though... can you figure out which ones aren't?
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Re: Which is it?

Postby snowyowl » Mon Sep 20, 2010 6:53 pm UTC

Xanthir wrote:Quantum loop dynamics in the boundaries produce a macroscopic pressure effect at the dimensional annealing, repelling you from the edges with a force asymptotically increasing as you approach it.

I'm impressed. I got to the end of the sentence before I realised you were making that up.
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Re: Which is it?

Postby Xanthir » Mon Sep 20, 2010 8:29 pm UTC

snowyowl wrote:
Xanthir wrote:Quantum loop dynamics in the boundaries produce a macroscopic pressure effect at the dimensional annealing, repelling you from the edges with a force asymptotically increasing as you approach it.

I'm impressed. I got to the end of the sentence before I realised you were making that up.

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Re: Which is it?

Postby mnvdk » Tue Sep 21, 2010 12:45 pm UTC

MartianInvader wrote:So T and I are homotopy equivalent, and in fact are both H.E. to a point. A and R are also homotopy equivalent. In fact, all uppercase letters are H.E. to either A,B, or C. Not all lowercase letters are, though... can you figure out which ones aren't?


Well, clearly 'i' and 'j' wouldn't be, but all others would...

Am I correct in assuming that 'B' would be in it's own "class" compared to the other letters. That is all the letters in {the alphabet}\{B} would be H.E. to either a point or a circle?
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Re: Which is it? (straight line or circle through a portal?)

Postby HenryS » Wed Sep 22, 2010 2:30 pm UTC

I happen to know someone who works at Valve, and pointed out that Portal is really about applying a physics engine to Euclidean orbifolds.

I think there is a trend towards more complicated and interesting topology and geometry in computer games (Portal, Fez, and upcoming Miegakure to list a few examples). I'm looking forward to the first game to implement hyperbolic geometry. It's possible to use modern graphics cards to do hyperbolic geometry, see http://geometrygames.org/CurvedSpaces/index.html (by Jeff Weeks) for an idea of what is possible.


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