# Questions tagged [hyperspace]

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

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• 2,464
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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 ...
• 2,464
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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) ...
• 1,025
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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 ...
• 2,200
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### Book about explanation of 4th dimension using analogy with 2d object interacting with 3d

I'm searching for a book about the explanation of 4th and higher dimensions using an analogy with 2d space creatures (flatland) with 3d dimension (as humans). I need to to add and update Polish ...
• 215
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### How can I prove $M+t$ is a hyperplane if $M$ is a maximal subspace

Let $M$ be a non-empty proper subset of a vector space $X$ over $\mathbb R$ and $t$ belongs to $X$, then $M$ is a maximal subspace if and only if $t+M$ is a hyperplane and $t$ belongs to $t+M$.
1 vote
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### Nuclear norm of matrix comprised of points sampled uniformly at random from the hypersphere?

In a subfield of machine learning called "self-supervised learning", many methods constrain network representations to lie on the hypersphere. I want to understand how such representations ...
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### Relationship between trace norm (a.k.a. Schatten-1 norm) of a matrix and the vector norm of the matrix's row average?

I'm trying to understand whether a connection exists between two seemingly different optimization problems in machine learning. Setup: Suppose I have $N$ points $x_1, ..., x_N \in \mathbb{R}^D$, where ...
138 views

### 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 ...
1 vote
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### Maximum number of points on the surface of a 4D 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 ...
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One hypersphere at $x^2 + y^2 + z^2 + w^2 = 1$ intersects another hypersphere at $(x - 1)^2 + y^2 + z^2 + w^2 = 1$. (EDIT to address comments) The intersection results in a (3-d) sphere with radius $\... • 424 0 votes 0 answers 121 views ### How many different 3D parallelepiped are in 4D hyper-parallelepiped? Let's say I have three 3D linearly independent vectors, these vectors form a parallelepiped. This parallelepiped has 6 faces, but only 3 of them are "unique" (the other three can be obtained ... • 11 0 votes 0 answers 230 views ### What is the determinant of the Jacobian of four-dimensional hyperspherical coordinates? I am trying to do a four-dimensional change of variables problem, and I am working with hyperspherical coordinates. For$(x,y,z,w)=\varphi(\rho, \varphi, \theta, \lambda)$, I have$x= \rho \sin\varphi ...
If $W$ is a $k$-dimensional subspace of an $n$-dimensional vector space $V$, then $W$ is the intersection of $(n- k)$ hyperspaces in $V$. Rephrasing problem to my taste: If $W$ is a $k$-dimensional ...