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Loops Across Space
- mydriasis 3y agoLove Greg Egan. Especially loved Schild's Ladder. It was such a cool book, and it's very clear that the author has a serious background to boot, further evidenced by publications like this. Awesome!
- jugg1es 3y agoHow can you have a 3d space with finite volume and no boundary?
- layer8 3y agoThe same way the surface of the earth is a 2D space with finite area and no boundary.
- alchemist1e9 3y agoMöbius strip: https://en.m.wikipedia.org/wiki/M%C3%B6bius_strip https://en.m.wikipedia.org/wiki/M%C3%B6bius_strip
- jugg1es 3y agoWhat does boundary mean in this context then? Because the mobius strip has a boundary on the sides even if it is continuous. I guess boundary means an edge in all directions?
- simonh 3y agoA mobius strip is an example of a space that is finite but unbounded in a single direction. A two dimensional plane that is finite but unbounded in all directions is called a Kline bottle, but you can’t build them for real in 3D space, only distorted approximations. You don’t have to do such geometric contortions though. The surface of a sphere is a two dimensional space that is finite but unbounded.
- ithkuil 3y agoNot sure why you need a Mobius strip when an untwisted rectangle with two opposite edges joined (outside of a cylinder) has the same property. Unlike a sphere that would be a flat 2d space (but unlike a sphere it would be unbounded only in one direction)
- BLKNSLVR 3y agoWhen you reach one of the boundaries you re-enter the space at a different boundary. It was one of the very few things I was able to understand. There's "continuity" between two points that, in the 3D rectangle representation, don't appear to have continuity. This continuity is possible because spacetime may be all warpy like that... You can't exit the space at the boundaries. The door out of the room walks you into the same room from a different door. eg. exit Face C at some point, and re-enter from Face A, because Faces C and A are aligned at that point. I'm lost beyond that point though.
- prof-dr-ir 3y agoin 2d: take a square and glue the two pairs of opposite sides together. If you do this with one pair you get a cylinder, and then gluing the other pair gets you a torus. No boundary is left. In 3d: take a solid cube and glue the three pairs of opposite sides together. Maybe a bit difficult to visualize, but the idea is that if you live in the cube and try to exit it on one side then you re-emerge into the cube from the opposing side.
- dylan604 3y agoWould this allow for Escher type spaces?
- Filligree 3y agoYes, though not with a didicosm. (Unless your standards for Escher-esque spaces are low.)
- kibwen 3y agoEscher's most fantastical spaces would be classified as non-Euclidean, whereas the article here stresses that the spaces it describes are strictly Euclidean.
- falcor84 3y agos/Euclidean/locally Euclidean/
- cubefox 3y agoI think they are rather forms of parallel projection (where farther objects aren't smaller, unlike perspective projection) combined with forced perspective, where things look locally connected from a specific angle: https://en.wikipedia.org/wiki/3D_projection#Limitations_of_parallel_projection https://en.wikipedia.org/wiki/3D_projection#Limitations_of_p...
- perihelions 3y agoIn the same way a sphere, like the Earth's surface, is a 2d-space with finite area and no boundary.
- Sharlin 3y agoIn 2D the most intuitive way is the "Asteroids topology": when you exit from one edge you reappear from the opposite one. This space is flat, finite, and without a boundary. In mathematical terms it’s what you get when you take a rectangle and identify the left edge with the right and the top edge with the bottom one. It is also topologically equivalent to the (surface of) a torus, which is also flat and wraps around the same way. (Note that the embedding of a toroidal surface in three dimensions is curved, but the 2D space itself is flat: the angles of every triangle add up to exactly 180°.) In 3D, just generalize from the 2D case.
- pdonis 3y ago> the 2D space itself is flat It's not that simple. "Flat" is not a topological property, it's a metrical property. The correct statement is not that "the" 2D torus is flat, but that it is possible to put a flat metric on the 2D topological space that is called a "torus". That's what the "Asteroids" space does. But it is also possible to put a curved metric on the same topological space--for example, just use the obvious metric derived from the embedding in 3D Euclidean space that we're all familiar with. The "torus" topological space in itself has no metric, and both the flat "Asteroids" metric and the curved "doughnut" metric are valid metrics on that topological space.
- Sharlin 3y agoYes, good point, I wanted to include the flatness property (which was the third condition mentioned in the article) but simplified a bit too much in the process.
- cubefox 3y ago> It is also topologically equivalent to the (surface of) a torus Interesting, I thought it would be like the surface of a sphere. What's the difference?
- dllthomas 3y agoConnectivity. When you go off the top, you come in on the bottom. When you go North on a globe, you come in elsewhere in the North.
- mike_hock 3y agoTFA tells you how.
- namanyayg 3y agoEgan is one of my favorite sci-fi writers. It takes a while to understand some of the concepts he writes, but imo he's written some of the best works in hard sci-fi that also deal with a lot of philosophy. Great to see him active on his website (though I'll admit I gave up on understanding this around halfway) Editing to add: of his works, Permutation City is one of my favorites. Multiple mind blowing concepts. I'm sure it'll be up the alley of many other HN readers.
- marsten 3y agoInterestingly there are no photos or video of Egan publicly available. All we have over the last 30 years is his/her/its textual output. I'm 50/50 on whether "Greg Egan" is in fact a LLM.
- pavel_lishin 3y agoI've seen LLM output. If Greg Egan is an LLM, you can come to my house, choose any hat within it, and I'll eat it.
- Rediscover 3y agoAbout 20 years ago, there was this individual on Slashdot that was frequently commenting, quite knowledgeably, on many many subjects. I am still wondering if it was a test of an expert engine.
- TheHumanist 3y agoWould you say his writings are over the head of most readers? If not, what are a couple you would start with? Would Permutation City be understandable to a lot of people? You just make him seem like he's writing about very complex topics haha
- namanyayg 3y agoHe definitely does write about complex topics and is one of the harder sci-fi writers I've read. The descriptions of those concepts and the increase in complexity is something that is done over many chapters. But, I've noticed that in some particular parts of the book the "acceleration" of complexity becomes so high that I have to reread things and look up references online to be able to understand and continue. Without giving too many spoilers, Permutation City takes the concept of cellular automata to the next level. It was approachable to me, and perhaps also became my favorite, because I had experience with those in the past. To answer your question, Axiomatic is his collection of short stories and is definitely the most approachable. If you like the style presented there, you will also find it worthwhile to try his novels. There is a certain sense of beauty clear in his longer works, where everything is taken to its pinnacle, and all the ideas fit together like a puzzle in the end.
- andybak 3y agoCan someone please make a simulator of this! I'd love to see a Geometry Wars style shooter or a puzzle game using these spaces.
- pavel_lishin 3y agoCan't grant you that wish, but I can give you a few others! Hyperrogue is a roguelike set in hyperbolic space: https://store.steampowered.com/app/342610/HyperRogue/ https://store.steampowered.com/app/342610/HyperRogue/ And the author has a lot of videos on non-euclidean spaces! https://www.youtube.com/@ZenoRogue https://www.youtube.com/@ZenoRogue There's also Hyperbolica: https://store.steampowered.com/app/1256230/Hyperbolica/ https://store.steampowered.com/app/1256230/Hyperbolica/ And that author also has a lot of videos on its development, non-euclidean spaces, and other games that play with space: https://www.youtube.com/@CodeParade https://www.youtube.com/@CodeParade Not quite in the same family, but maybe a distant cousin - a game where the speed of light is so slow, that merely walking around causes you to experience relativistic effects: http://gamelab.mit.edu/games/a-slower-speed-of-light/ http://gamelab.mit.edu/games/a-slower-speed-of-light/
- RogerL 3y agohttp://www.gregegan.net/DICHRONAUTS/02/Interactive.html http://www.gregegan.net/DICHRONAUTS/02/Interactive.html
- andybak 3y agoThat's fascinating but at first glance it's something different?
- kleer001 3y agoYay, doughnut space! I played with this kinda stuff as a kid back in the QBasic days. Like Asteroids style. Basically: If X>10 then X=0 If Y>10 then Y=0 And then again If X<0 then X=10 If Y<0 then Y=10
- gpderetta 3y agoThe issue is that in 3d the mapping is less straightforward if you want avoid stange effects like being mirrored .
- munificent 3y agoYou actually want: while x > 10 do x -= 10 while x < 0 do x += 10 while y > 10 do y -= 10 while y < 0 do y += 10 What you have will discard some position information every time it wraps. Also, it won't correctly handle changes in position per frame that are larger than 10.
- kleer001 3y agoWell, yes, but I also need to take account of velocity and rotation too. I don't see that particular flow in the manual, and of course everything should be FULLCAPS. https://en.wikibooks.org/wiki/QBasic/Full_Book_View#DO_.._LOOP https://en.wikibooks.org/wiki/QBasic/Full_Book_View#DO_.._LO...
- notfish 3y agoOr just x = x mod 10 y = y mod 10
- munificent 3y agoThat depends on how the programming language implements modulo. Some don't wrap around the way you would need here when the dividend is negative. See: https://en.wikipedia.org/wiki/Modulo#Variants_of_the_definition https://en.wikipedia.org/wiki/Modulo#Variants_of_the_definit...
- 3y ago
- bryan0 3y ago> The first homology group of the didicosm is ℤ4 × ℤ4. What about the other 5 "orientable platycosms"? I assume 3-Torus is another with ℤ × ℤ × ℤ (?)