**1**

vote

**1**answer

136 views

### Embedding of Two Objects Into Higher Dimensions With Their Sum

Given two vector sets, $\vec x_i$ and $\vec y_i$ (for $i$=1,2,...N, but the dimensionality of each vector can be more than N), let their sum set be $\vec z_i = \vec x_i + \vec y_i$. It's easy to ...

**6**

votes

**1**answer

228 views

### Symmetric black-hole curves

Is there a curve $C$ that connects $(0,1)$ to $(a,0)$ for some $a>0$, and, when reflected
to $C'$ in the $x$-axis, the shape $S=C \cup C'$ has the property that each horizontal
light ray entering ...

**14**

votes

**1**answer

303 views

### Smallest regular simplex containing the unit cube in $R^n$

What is the length $e_n$ of the edge of the smallest $n$-dimensional regular simplex $S_n$ containing the $n$-dimensional unit cube $Q_n$?
In particular, is there $n$ such that ...

**4**

votes

**0**answers

164 views

### Tetrahedra passing through a hole

Assume a plane $P\subset\mathbb R^3$ has a hole $H$, and that the hole is topologically a compact disc. Being so, $P\setminus H$ does not separate the space. A regular tetrahedron $\sigma^3$ (of ...

**8**

votes

**3**answers

833 views

### On maximal regular polyhedra inscribed in a regular polyhedron

Let T, C, O, D, or I be regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron, respectively. Suppose that the outer polyhedron have edge-length 1.
For example, it's easy to prove that ...

**17**

votes

**4**answers

965 views

### Metrics for lines in $\mathbb{R}^3$?

I seek a metric $d(\cdot,\cdot)$ between pairs of (infinite) lines in $\mathbb{R}^3$.
Let $s$ be the minimum distance between a pair of lines $L_1$ and $L_2$.
Ideally, I would like these properties:
...

**1**

vote

**1**answer

117 views

### Angles and projective metric

Unless I am very wrong, the following seems to be true:
If the angle between two vectors in $\mathbb{R}^{n}_{++}$ is small, then the
value of the Hilbert projective metric between them is also ...

**9**

votes

**1**answer

344 views

### Which values can attain the minimum solid angle in a simplex

Given a simplex $S$ with a vertex $v$ by the solid angle at this vertex I mean the value $\hbox{vol}(B \cap S)/\hbox{vol}(B)$ where $B$ is a small enough ball centered at $v$ (for example, in the ...

**1**

vote

**0**answers

45 views

### Components of Intersection of Ellipsoids

Let $\Sigma$ be an $n-1$ dimensional ellipsoid in $\textbf{R}^{n}$ and $S$ the unit sphere.
I would like to understand the connected components $C$ of the intersection of $\Sigma$ and $S$.
In my ...

**6**

votes

**1**answer

459 views

### Does an origin-centered ellipse in the plane intersect each $L^p$-circle at most 8 times?

The question is in the title: Let $E$ be an origin-centered ellipse in ${\mathbb R}^2$ and let $S$ be an "$L^p$-circle": $S = \{(x,y) : |x|^p + |y|^p = \text{const}\}$, where $1 \leq p \leq \infty$. ...

**4**

votes

**1**answer

343 views

### Biggest ball included in an intersection of balls

I would like to prove that for any family of balls $\{B(c_i,r_i)\}_i \subset \mathbb{R}^d$ such that $\{c_1, \dots, c_n\} \subset \bigcap_i B(c_i,r_i) $ and $\forall i, r_i \geq 1$, there exists a ...

**1**

vote

**1**answer

172 views

### Straight Line Passing Through a Convex Region

Is there any test to tell me whether a straight line in a 3D euclidean space passes through a bounded closed convex region? To focus on a more specialised version of the problem, you can assume that ...

**1**

vote

**0**answers

105 views

### Boundary surfaces in a 3d Voronoi tessellation with obstacles

Let $x_1,\dots,x_n$ be a set of points in $\mathbb{R}^3$ and let $\mathcal{O}_1 ,\dots, \mathcal{O}_m$ denote a set of polyhedral obstacles. What is the name for the surfaces that describe the ...

**5**

votes

**2**answers

1k views

### What is the best *general triangle*?

During courses on geometry it is sometimes necessary to draw a triangle on the blackboard that can easily be recognized as a general triangle. It must not be rectangular and must not have two or more ...

**3**

votes

**1**answer

110 views

### Symmetry group for the frame bundle of a G-space

Let $Q$ be a smooth manifold, and let $G$ be a Lie group which acts smoothly on $Q$ on the left.
Question 1: does the group $G$ act naturally on the tangent bundle $TQ \to Q$?
My motivation here ...

**9**

votes

**4**answers

614 views

### Applications of n-dimensional crystallographic groups

I would like to know what are the applications of the theory of $n$-dimensional crystallographic groups (aka space groups)
1) in mathematics
2) outside of mathematics,
besides the applications to ...

**2**

votes

**1**answer

202 views

### Orthogonality between vectors whose components increase

This is a cross-posting of a MSE question (which did not receive any feedback there so far).
Say that a vector $x=(x_1,x_2, \ldots ,x_n)\in {\mathbb R}^n$ is nondecreasing if $x_1 \leq x_2\leq \ldots ...

**1**

vote

**1**answer

251 views

### A little question on certain parallel-lines-preserving maps

Let $\alpha:\mathbb{R}^n\to\mathbb{R}^n$, $n\geq 2$, be a $\mathbb{Q}$-linear bijection with the following properties:
1) $\alpha$ sends straight affine $\mathbb{R}$-lines to straight affine ...

**2**

votes

**0**answers

246 views

### Find minimum-area ellipse enclosing a set of ellipses, all centered at the origin

Given a set of N > 2 (two-dimensional and coplanar) ellipses, all centered at the origin, how do I find the ellipse with the minimum area which encloses all of them?
Background:
Thanks to Will Jagy ...

**5**

votes

**2**answers

723 views

### Find minimum-area ellipse which encloses two ellipses

I need an efficient algorithm to find the ellipse with the smallest possible area which encloses two given ellipses. The given ellipses are constrained to have coincident centers at the origin but can ...

**4**

votes

**1**answer

254 views

### How well do random projections preserve the distance between a point and a linear subspace?

Let $x_1,\ldots,x_k \in \mathbb{R}^d$ be $k$ unit vectors in $d$ dimensional Euclidean space, and let $S = \mathrm{span}(x_1,\ldots,x_k)$ be a linear subspace defined by these points. Let $P \in ...

**15**

votes

**0**answers

493 views

### Does every connected set that is not a line segment cross some dyadic square?

A dyadic square is a subset of $R^2$ of the form $x + 2^{-n} [0,1]^2$ with $x \in 2^{-m} Z^2$, for integers $m,n \geq 0$. We say that a set $A$ crosses a square $S$ if there exists a connected subset ...

**17**

votes

**5**answers

1k views

### Definition of area

I am looking for an attractive, but rigorous definition of area;
say in Euclidean plane. Probably there is no short definition. It is OK to make it even longer, but can it be built from useful parts ...

**6**

votes

**0**answers

153 views

### Can a closed disc in the plane be partitioned into three disjoint sets which are pair-wise isometric?

Any progress on the following: Can a closed disc in the plane be partitioned into three disjoint sets which are pair-wise isometric, i.e. each set is an image of the others under an isometry?

**10**

votes

**0**answers

378 views

### What is the field generated by an Archimedean Spiral?

Let $F \subset \mathbb R$ be the set of lengths of line segments that one can construct, starting from the points $(0,0)$ and $(1,0)$, using a straightedge, compass, and an Archimedean spiral - the ...

**9**

votes

**3**answers

414 views

### Mutually tangent ellipsoids in 3 space

I recently heard a claim that for any n, it is possible to arrange n ellipsoids in 3 space such that each pair of ellipsoids is kissing. Is this true, and if so, how?
Edit: By kissing, I mean that I ...

**0**

votes

**1**answer

188 views

### Length of intersection of intervals

Can anyone prove this statement? It seems true, but I'm finding it tricky to give a concise proof.
Fix $\alpha\in[0,1]$. Let $\mu$ be Lebesgue measure. Define $B(c,r)\equiv[c-r,c+r]$, where $[\cdot, ...

**2**

votes

**0**answers

151 views

### Series for envelope of triangle area bisectors

The lines which bisect the area of a triangle form an envelope as shown in this picture
It is not difficult to show that the ratio of the area of the red deltoid to the area of the triangle is ...

**2**

votes

**0**answers

221 views

### Axiomatization of the incidence geometry of the Euclidean plane

There are several well-known axiomatizations of Euclidean plane geometry, the language of which is usually considered to include at least the relations of
incidence (point-line, point-segment, or ...

**20**

votes

**6**answers

2k views

### Does every ellipse inside a tetrahedron inside a ball fit in a triangle inside the ball?

In three-dimensional euclidean space, consider the closed unit ball $B$. Let $T$ be a tetrahedron, and $E$ an ellipse, with $E \subset T \subset B$. Does there necessarily exist a triangle $T'$ with ...

**2**

votes

**2**answers

107 views

### Worst-case nearest-neighbor distances between regions

Suppose that $S_1,\dots,S_n$ is a collection of disjoint shapes in the plane, and let $\mathcal{X}$ denote the set of all $n$-tuples of points $\lbrace x_1,\dots,x_n\rbrace$ such that $x_i\in S_i$ for ...

**8**

votes

**3**answers

694 views

### In which geometries do triangles have an Euler line?

In Euclidean geometry, the centroid, orthocenter and circumcenter of a triangle lie on a line.
In which other geometries does this hold?

**2**

votes

**1**answer

696 views

### Why 2 as an exponent in the euclidean vector space?

Let us develop the question:
Let us focus on finite real vector space, equiped with a norm. A priori, one does not make the hypothesis that the norm is derived from a scalar product.
Which ...

**3**

votes

**1**answer

212 views

### Inter-Kissing Number for Non-Spheres

In 3D, the maximum number of spheres which can inter-touch is 5 (mathoverflow.net/questions/106120). This maximum reduces to 4 for unit spheres.
Is there a different shape (e.g., an egg, or a ...

**5**

votes

**1**answer

238 views

### Inter-Kissing Number for Spheres of Different Sizes

What is the maximum number of spheres that can be placed in 3D such that all inter-touch?
One can of course place four unit spheres tetrahedrally and then add a smaller sphere in the
middle, so this ...

**4**

votes

**0**answers

460 views

### sine and Archimedes' derivation of the area of the circle

The elementary "opposite over hypotenuse" definition of the sine function defines the sine of an angle, not a real number. As discussed in the article "A Circular Argument" [Fred Richman, The College ...

**8**

votes

**10**answers

7k views

### College (Euclidean) geometry textbook recommendations

I will be teaching a mid-level undergraduate course in Euclidean geometry this fall. Has anyone taught such a course, who can recommend a good textbook?
My students will mostly be future high school ...

**0**

votes

**0**answers

93 views

### Error Metric which incorporates both mean & standard deviation of data in euclidean space

For simplicities sake (the actually problem is more complex)...Let say I have a set of n 3d points, whose position move over time. For all pairs, I have calculated the mean and standard deviation of ...

**4**

votes

**3**answers

992 views

### Reconstructing an Euclidean point cloud from their pairwise distances

I have a collection of points $P_1, ..., P_N$ in some Euclidean space $\mathbb R^m$ and the coordinates $x_1, x_2, ..., x_N$ respectively associated with them, where $x_i$ is the usual Cartesian tuple ...

**5**

votes

**3**answers

2k views

### Minimum distance between two arbitrary circles in space?

What is the minimum distance between two arbitrary circles in space?
I am working out the problem with Maxima, but I am surprised by how complicated this rapidly turns out to unfold for such a ...

**15**

votes

**3**answers

1k views

### Can Morley's theorem be generalized?

Morley's theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle.
In a talk some years ago, David Rusin made the ...

**6**

votes

**1**answer

725 views

### Quadrature of the Lune

What is a good reference for the following result which I believe is proved by Tchebotarev.
There are exactly 5 types of Lunes that are squarable. (Hippocrates produced three and then two more were ...

**3**

votes

**1**answer

177 views

### Flattening a corner in a convex $d$-polytope (into $d-1$ dimensions, without overlap)?

I'm interested in the following question, which seems to be assumed all over the place (at least for 3 dimensions) in convex geometry, and which I cannot find a proof of.
Suppose we have a corner ...

**3**

votes

**0**answers

203 views

### Finite subgroups of the unimodular group

This is related to this MO question (and others as well).
Hoping that this will not turn out to be too broad, I would like to know about the 'state of the art' of:
1) The problem of classifying ...

**6**

votes

**4**answers

760 views

### Isomorphic but non-conjugate subgroups of $GL(n,\mathbb{Z})$ ?

I've been asked some questions by a friend interested in crystallography, and the following questions (I'm not an expert) came spontaneous to me:
1) Are there two finite subgroups ...

**6**

votes

**1**answer

299 views

### Reorienting a ladder among $\mathbb{Z}^2$ poles

Imagine an object, which I'll call a ladder $\cal{L}$, a "racetrack" shape
composed of a rectangle of length $L$ capped at either end by
semicircles of radius $r$; so it is $L+2r$ tip-to-tip.
View ...

**5**

votes

**1**answer

757 views

### theorems equivalent to the parallel postulate

Is there a good survey article listing all the theorems of Euclidean geometry that are equivalent to the parallel postulate?

**3**

votes

**0**answers

277 views

### Minimum solid angle and aspect ratio of an $n$-simplex

In computational geometry and other fields, it is of interest to have degeneracy measures for shapes of simplices, which quantitatively seperate the regular simplex from degenerate simplices.
In two ...

**1**

vote

**2**answers

336 views

### Triangles with Congruent Corresponding Sides that Cannot fold into a Tetrahedron

I've been trying to find, without much success, 4 triangles whose corresponding sides are congruent that cannot be folded into a tetrahedron.
Anyone has any clue how to approach this problem?

**6**

votes

**0**answers

990 views

### Interpolating points with minimum curvature constraint

I have $n$ points $p_i$ strictly interior to a rectangle $R$,
and I would like to connect them with a curve $C$ whose curvature is as low as possible.
Let $\kappa_\max(C)$ be the sharpest (largest ...