Is it possible to build stuff like a house, or castle, or submarine cave complex, using geometric principles not covered in Euclidean geometry? Seems like the answer would be “no.”
Which would make sense, because actually seeing something impossible like that would be quite mind-bendy.
Euclidean geometry only works on a flat plane. So any building with a curved surface, like a dome, can be said to be non-Euclidean.
You can certainly work the motif in:
The surface of the earth is a non-euclidean plane. However, in small spaces, the deviation from being euclidean is negligible.
Oh. Well that’s boring AF. So R’lyeh could have just been some onion domes or even that castle in The Little Mermaid?
The Wharton Eshrick Museum is a house built by the artist that largely eschews right angles and flat surfaces.
The Wikipedia article on Hyperbolic Geometry has a simple image of a triangle in a “saddle” surface. In 3D, a “cube” will have a warped, melted look like a Dali painting.
You can construct something with a lot of such saddle surfaces instead of semi-spheres. Walking on such a surface will be fun.
If you have a handy little black hole nearby you have some non-Euclidean space to build in.
(Technically, the gravity distortion of the Earth itself will do, but it won’t be obvious to the naked eye.)
I don’t think that’s true. You can have 2D or 3D Euclidean geometry - Euclid wrote about the geometry of solids as well.
Geometry on the surface of a sphere would be non-Euclidean (2D) but Euclidean (3D)
Geometry in the vicinity of a black hole is non-Euclidean in all dimensions we know of
Whether the surface of a sphere is Euclidean or not depends on how you’re describing it. You can describe the sphere embedded in three dimensions, and call its geodesic curves “circles”, and call a pair of antipodes two different points, and so on, in which case it’s Euclidean. But you can also describe the two-dimensional surface itself independent of any embedding, and call its geodesic curves “lines”, and call a pair of antipodes a single point, and so on, in which case you can construct a geometry that follows all of Euclid’s first four axioms, but not the fifth, in which case the sphere is non-Euclidean (in fact, constructions like this are how it was originally proven that the Fifth Axiom is independent of the others).
Euclid had plenty to say about the geometry of three-dimensional solids.
And saddle surfaces are “non-Euclidean” in the sense that it (was) very difficult to do the design calculations to work out if it would break and fall down.
As I recall, the Sydney Harbour bridge design calculations were done by drawing force vectors on paper and measuring how long they were. If you don’t have flat surfaces with Euclidean angles, that doesn’t work. It doesn’t work even if you use a calculator to do the same thing.
The Sydney Harbour Bridge is just a plagiarized Hell Gate Bridge. The bridge engineering philosophy in that era was mostly “we don’t know what we’re doing, so make it insanely strong.”
I guess what I meant is, 2-dimensional shapes on curved surfaces are non-Euclidean.
I could be wrong here, but I’m pretty sure they are Euclidean, unless your example also demands the geometry be forced to conform to the surface. But there’s no reason to do that and Euclidean geometry handles curves as well as any other line. It’s not designed to handle curved space, but that’s different than a curved surface, since no axes need conform to the surface for any particular reason.
A curved surface is a 2-dimensional curved space. Lines and patterns on a curved surface are non-Euclidean.
While I appreciate the pride of a New Yorker, the idea of plagiarism is unnecessarily pejorative. Reusing sound engineering ideas is just sensible, unlike reusing artistic ideas. In any event, the SHB is much bigger than HGB, an observation which means that greater risk was involved than is implied in the idea of plagiarism, because the square-cube principle means that these things don’t scale in linear fashion.
And the devil is always in the detail with these things. Selecting the appropriate ground points to connect, determining the strength of the supporting ground, deciding the actual sizes and gaps between each girder, integrating the design into present or proposed road, rail and tram systems, and so on. You can’t just copy a bridge.
I think most agree that at the end of the day, the SHB, the harbour itself, and the Opera House create an amazing sense of spectacle much greater than reductionist ideas of plagiarism would account for.
Hey, don’t get me wrong. I’ve been to Sydney and I love it. I was especially impressed with the engineering prowess it took to build an entire city there without all of it falling off the ground.
There’s also the Hundertwasser House in Vienna. According to a tour guide, the architect wanted no straight lines anywhere in the building, and would have succeeded except for the elevators. Checking out some pictures, it looks like the windows are flat and straight, but the overall effect is still pretty non-linear.
That’s not the same as non-Euclidean, but might be of interest to the OP.
I’m not sure what OP is asking about. Structures with paradoxical properties, like needing to make five right turns instead of four to get back to a starting point? A playground slide that behaves like Klein’s Bottle? Something like Santa Cruz’s Mystery Spot?
I’m not sure what you’re saying. The 2D surface of a sphere can be represented, with all its geometry intact, in Euclidean 3-space, but Lobachevsky’s 2D hyperbolic plane requires Euclidean 4-space for embedding, IIUC. John F. Nash demonstrated that all suitably-defined geometries can be embedded in a Euclidean space of sufficient dimensionality.
I saw that simulated in VR in a video recently… here’s an example:
