# Tagged Questions

**3**

votes

**1**answer

296 views

### Least cardinality of a set of points in the plane

What is the least possible cardinality $K$, of a set S of points in the plane, such that there exists a point P in the plane and an open ball B centered at P, such that for all points X in B, not all ...

**3**

votes

**2**answers

436 views

### Shortest “painting” of the sphere

Let $S$ be the sphere in $\mathbb{R}^3$ and $C:[0,1]\to S$ a continuously differentiable curve on $S$. Let $T:[0,1]\to\mathbb{R}^3$ denote the tangent vector of $C$. Let $P(t)$ be the plane containing ...

**11**

votes

**2**answers

2k views

### Which platonic solids can form a topological torus?

8 cubes can be joined face-to-face to form a closed ring with a hole in it, with each cube sharing a face with only two others. The same can be done with 8 dodecahedrons.
Is the same possible with the ...

**1**

vote

**1**answer

222 views

### Isocontours of depth and magnitude of gradient

We are interested in characterizing a 2D surface $z(x,y)$, where $(x,y)$ is the regular 2D Cartesian grid. Let $\nabla z = (z_x, z_y)$ denote the gradient. The surface is a "general" one, that is, ...

**2**

votes

**1**answer

329 views

### weak metric space

In the definition of a metric space, replace the triangle inequality by the weaker inequality
d (x, z) ≤ C max {d (x, y), d (y, z)},
where C is a positive constant (depending on the "metric", ...

**12**

votes

**1**answer

685 views

### Fixed point theorems and equiangular lines

I've been thinking about the equiangular lines (or SIC-POVM) conjecture, and my conclusion is that the best means of attack would be through some kind of fixed point theorem -- I'm thinking ...

**-3**

votes

**1**answer

2k views

### How to transform a plane into a sphere? [SOLVED] [closed]

Given a 2-dimensional array of MxN heights, how to transform it to a sphere? Every element of this array is just a 3D point (x,y,z) where z represents some height. One has to transform this array into ...

**-6**

votes

**4**answers

561 views

### What is the max number of points in R^3, interconnected by generic curves?

The largest complete graph that embeds in 2 dimensions is $K_4$, while the largest complete graph that embeds in 3 dimensions is $K_{\infty}$, right? However, I don't know any constructive proof of ...

**4**

votes

**4**answers

895 views

### How to partition R^3 into pairwise non-parallel lines?

Problem. How to partition R^3 into pairwise non-parallel lines?
A possible solution is to stack infinitely many ``concentric'' hyperboloids, by increasing radius and decreasing slope. And don't ...

**14**

votes

**5**answers

947 views

### Is there a topological description of combinatorial Euler characteristic?

There are a collection of definitions of "combinatorial Euler characteristic", which is different from the "homotopy Euler characteristic". I will describe a few of them and give some references, and ...

**11**

votes

**4**answers

899 views

### How to compute the (co)homology of orbit spaces (when the action is not free)?

Suppose a compact Lie group G acts on a compact manifold Q in a not necessarily free manner. Is there any general method to gain information about the quotient Q/G (a stratified space)? For example, I ...