The euclidean-geometry tag has no usage guidance.

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**1**answer

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### 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 ...

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votes

**4**answers

597 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

200 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 ...

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vote

**1**answer

224 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 ...

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**0**answers

221 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 ...

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votes

**2**answers

577 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

234 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 ...

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**0**answers

461 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 ...

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**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 ...

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**0**answers

367 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 ...

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votes

**3**answers

403 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 ...

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votes

**1**answer

187 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, ...

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votes

**0**answers

149 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 ...

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**0**answers

216 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 ...

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**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 ...

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votes

**2**answers

104 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 ...

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votes

**3**answers

644 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?

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votes

**1**answer

693 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 ...

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votes

**1**answer

206 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 ...

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**1**answer

222 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 ...

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votes

**0**answers

445 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 ...

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**10**answers

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### 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 ...

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**0**answers

90 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 ...

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votes

**3**answers

911 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 ...

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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 ...

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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 ...

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votes

**1**answer

709 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 ...

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votes

**1**answer

173 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 ...

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**0**answers

201 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 ...

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votes

**4**answers

723 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 ...

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votes

**1**answer

298 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 ...

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**1**answer

727 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?

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**0**answers

264 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 ...

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vote

**2**answers

328 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?

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**0**answers

937 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 ...

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votes

**4**answers

976 views

### Pinball on the infinite plane

Imagine pinball on the infinite plane, with every lattice
point $\mathbb{Z}^2$ a point pin.
The ball has radius $r < \frac{1}{2}$.
It starts just touching the origin pin, and shoots off at angle ...

**0**

votes

**1**answer

402 views

### Action of Isometries on a Line in the Plane

I'm trying to determine the stabilizer of a line in a plane when acted upon by the group of isometries of the plane. Please note that I'm using the notation found in the Wikipedia article on Euclidean ...

**0**

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**0**answers

163 views

### Inertia/Gravity in Distance Geometry

The Cayley-Menger Determinant, D(N), slickly calculates the N-dimensional simplex
volume of any N+1 points. One constraint in our 3D world is that D(4)=0.
Give each point a mass (Mi) and dynamic ...

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votes

**1**answer

562 views

### Polygons uniquely inducing arrangements

A beautiful, relatively recent result is that,
Every simple arrangement $\cal{A}$ of $n$ lines in the plane is induced by a simple $n$-gon $P$.
In a simple arrangement, every pair of lines ...

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votes

**1**answer

560 views

### Ratio of circumscribed/inscribed $(n{-}1)$-gons

As a discrete analog of the MO question,
"Löwner-John Ellipsoid: incribed and circumscribed,"
I've been wondering what might be the maximum ratio
of this quantity?
Let $P$ be a convex polygon of $n$ ...

**5**

votes

**1**answer

361 views

### What can be said about number-theoretic properties of the solid angle measures of polytopal cones in the weight lattice of sl(n)?

The following question might be elementary — it is too far from my area of expertise to tell. It has shown up in my research in an interesting way, which I will not go into here, but I'm happy to ...

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**2**answers

575 views

### Maximum thickness of three linked Euclidean solid tori

Consider three circles of radius 1 in $\mathbb{R}^3$, linked with each other in the same arrangement as three fibers of the Hopf fibration. Now thicken the circles up into non-overlapping standard ...

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**0**answers

959 views

### 3-piece dissection of square to equilateral triangle?

At a workshop it was suggested that it likely remains an open problem
whether or not there is a 3- or 2 -piece
dissection
of a square to an equilateral triangle.
Can anyone confirm that this is ...

**0**

votes

**1**answer

193 views

### Special functions on the unit disk

Let $\mathbb{D} = \{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 < 1 \}$ be the unit disk.
We say a function $f : \mathbb{D} \rightarrow \mathbb{D}$ is a winner if it satisfies the following:
1) it is a ...

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votes

**2**answers

476 views

### Covering a sphere using reflections of an intersection of three lunes

I have been trying to figure this problem out for a while, and while I believe someone must have figured it out hundreds of years ago, I still can't quite get it.
Suppose we have a 3-dimensional ...

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votes

**8**answers

2k views

### Determine if circle is covered by some set of other circles

Suppose you have a set of circles $\mathcal{C} = \{ C_1, \ldots, C_n \}$ each with a fixed radius $r$ but varying centre coordinates. Next, you are given a new circle $C_{n+1}$ with the same radius ...

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**1**answer

951 views

### What is the limit of the “knight” distance on finer and finer chessboards?

Consider the "infinite chessboard" on the plane. Think of it as the lattice $X_1:=\mathbb{Z}^2$, and also finer chessboards $X_n$ corresponding to $\frac{1}{n}\cdot \mathbb{Z}^2$, $n\geq 1$. Given two ...

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**5**answers

978 views

### Circumference of convex shapes

Here is a puzzle I found in "Mitteilungen der DMV" (roughly "Letters of the German Society of Mathematicians") issue 19/2011. It was posed by Alfred Schreiber in "Wie man Hasen fangt" (How to catch ...

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**3**answers

884 views

### (1-Lipschitz) + (length-preserving) = isometry

I am looking for an elementary way to prove the following theorem.
Theorem. Let $\alpha$ and $\beta$ be two simple convex closed curves in $\mathbb R^2$.
Assume
$$\mathop{\rm length} ...

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**3**answers

1k views

### How to solve geometry problems using involutions

Some geometry problems ( like this and this ) have short solutions if we use involutions. What references are there for solving geometry problems using involutions? I am particularly interested in ...