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The *geometry* in both Antichamber and Manifold Garden is Euclidean, but the *topology* is weird (Antichamber is also weird for some other reasons). Also "Chyr observed that the game may appear to have non-Euclidean geometry similar to the game Antichamber.[5]" appears that [5] is the source of that quote, but it is actually some article about ...
A topological space X is called locally Euclidean if there is a non-negative integer n such that every point in X has a neighborhood which is homeomorphic to real n-space R n. [2] A topological manifold is a locally Euclidean Hausdorff space. It is common to place additional requirements on topological manifolds.
Like the Möbius strip, the Klein bottle is a two-dimensional manifold which is not orientable. Unlike the Möbius strip, it is a closed manifold, meaning it is a compact manifold without boundary. While the Möbius strip can be embedded in three-dimensional Euclidean space R 3, the Klein bottle cannot.
A manifold can be constructed by giving a collection of coordinate charts, that is, a covering by open sets with homeomorphisms to a Euclidean space, and patching functions [clarification needed]: homeomorphisms from one region of Euclidean space to another region if they correspond to the same part of the manifold in two different coordinate ...
In mathematics, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fiber bundles.
In geometry the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R 3 under the operation of composition. [1] By definition, a rotation about the origin is a linear transformation that preserves length of vectors (it is an isometry) and preserves orientation (i.e. handedness) of space.
A relatively 'easy' result is to prove that any two embeddings of a 1-manifold into are isotopic (see Knot theory#Higher dimensions). This is proved using general position, which also allows to show that any two embeddings of an n-manifold into + are isotopic. This result is an isotopy version of the weak Whitney embedding theorem.
The 3-sphere is naturally a smooth manifold, in fact, a closed embedded submanifold of R 4. The Euclidean metric on R 4 induces a metric on the 3-sphere giving it the structure of a Riemannian manifold. As with all spheres, the 3-sphere has constant positive sectional curvature equal to 1 / r 2 where r is the radius.