Mathematics

Questions tagged [hyperspace]

For questions related to hyperspace (an $n$-dimensional Euclidian space with $n > 3$).

In mathematics, hyperspace is a space having dimension $n>3$, that is, an $n$-dimensional Euclidian space with $n > 3$.

Sometimes, also referred to as a Euclidian space of unspecified dimension.

23 questions
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When is most of an inner ball outside the hypercube which bounds the outer balls which bound the inner ball?

There is a simple question where a unit square contains four circles of radius $\frac14$ with a fifth inner circle, and the aim is to find that the radius of the inner circle is $r_2=\frac{\sqrt{2}-1}{4} \approx 0.10355$ . Consider the line segment…
Henry
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Topology on the set of all separable Hausdorff topological spaces

It is well-known ( 1 , 2 ) that the cardinality of a separable Hausdorff topological space is at most $2^{2^{\aleph_0}}$ . Therefore, the collection $\mathcal{A}$ of all (homeomorphism classes of) separable Hausdorff topological spaces is a set. Can…
Elia Immanuel Auer
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Prove total edge length of section is constant? A cross-polytope cut by a hyperplane parallel to a face.

The vertices of a cross-polytope can be chosen as the unit vectors pointing along each co-ordinate axis – i.e. all the permutations of $(±1, 0, 0,\dots, 0)$ . The cross-polytope is the convex hull of its vertices. Consider the section of a…
hbghlyj
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Volume of polytope spanned by vertex pairs $(A_i,A_i')$ is invariant under translation of diagonals $A_iA_i'$

2D case Let a convex quadrilateral $Q= \operatorname{conv}\{A_1,A_1',A_2,A_2'\}$ have vertex pairs $$(A_1,A_1'),\quad (A_2,A_2'),$$ and define the “diagonal vectors” $$d_i = \overrightarrow{A_iA_i'} = A_i' - A_i \in \mathbb{R}^2.$$ Form the diagonal…
user1693987
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Construct perspective projection of rotating tesseract by perpendicular lines intersecting ellipse

The contruction was used in two different sources on the web: a Geogebra resource and a video using inRm3D so I think it must be documented and proved somewhere, but I didn't find any. Here is the construction (following the second source): On the…
hbghlyj
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The rotation of a solid object in 3D can be described by a single 3D vector. Is the same true for higher dimensions?

If a solid ball (like the Earth) is rotating in 3D space, you can point a single 3D vector out of the North Pole (according to the right hand rule), with the length of that vector proportional to the speed of rotation, and that single vector…
chausies
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Singular values of uniform random points on hypersphere?

This question is motivated by a self-supervised learning problem in machine learning, but I'll try to strip out as many unnecessary details as possible. In this setting, we have large datasets and we constrain our deep neural network's outputs to…
Rylan Schaeffer
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The Jacobian and Integration of $n$-dimensional polar coordinates

This is my first proof and is most likely going to be crude. Please while reading comment tips about ways to improve my writing. Polar coordinates are transformed through: $$x=r\cos(\theta) \qquad y=r\sin(\theta).$$ Spherical coordinates are…
Brandon Sniady
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Why is $F_n(X)$ closed in the Vietoris topology?

Let $X$ be a Tychonoff space, and let $K(X)$ be the space of nonempty compact subsets of $X$ , equipped with the Vietoris topology. For $n \in \mathbb{N}$ , define $ F_n(X) := \{ A \in K(X) : |A| \leq n \}. $ It is stated (without proof) in Cardinal…
Cascaron Limosnero
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Can the Impossible Cube be isometrically embedded as geodesics in Nil geometry, just as the Penrose triangle is a geodesic triangle?

From the answer https://math.stackexchange.com/a/3743490/ the "impossibility" of the Penrose triangle and the Escher cube in Euclidean space arises because they violate the condition that a closed loop of orthogonal edges must return to the same $z$…
hbghlyj
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Proving $Z_r\cap iZ_r$ a complex hyperspace

Let $X$ be a linear space over $\mathbf{C}$ . If $Z_r$ is a real hyperspace in $X$ , then $Z_r\cap iZ_r$ is a complex hyperspace in $X$ . I know, we have to use the fact: $Ref$ determines $f$ as follows: $$f(x)=Ref(x)-iRef(ix).$$ But I want rigorous…
Pro_blem_finder
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ODE and SDE on Spheres $S^{n-1}$

I'm looking for comprehensive references (books or papers) that cover the theory and applications of Ordinary Differential Equations (ODEs) and Stochastic Differential Equations (SDEs) on $S^{n-1}$ spheres . Additionally, could anyone provide a…
Oliver
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Maximum number of points on the surface of an n-dimensional sphere such that distance between any two points is larger than the radius of the sphere.

1000 alien spaceships meet in a 4-dimensional battlefield. At an agreed time (ignoring relativistic effects on their clocks) every spaceship fires its laser to the spaceship which is closest (assume that all distances between the spaceships are…
Rüdi Jehn
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Integrating a function over the surface of a unit hypersphere

Suppose I have a function $f(\mathbf{x})$ across the surface of the unit hypersphere, where $\mathbf{x}=(x_1,...,x_d)'$ are the hyperspherical coordinates of a point on the surface of the unit hypersphere. I want to integrate this function over the…
Ron Snow
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projections from a plane in $\mathbb{R}^6$ onto three orthogonal planes

Let $\Pi_1=\operatorname{span}\{e_1,e_2\},\Pi_2=\operatorname{span}\{e_3,e_4\},\Pi_3=\operatorname{span}\{e_5,e_6\}$ be orthogonal 2D planes in $\mathbb{R}^6$ . Let $U$ be arbitrary 2D plane in $\mathbb{R}^6$ . Let $P_i:U\to\Pi_i\;(i=1,2,3)$ be the…
hbghlyj
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