The euclidean-geometry tag has no wiki summary.

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### 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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### 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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### 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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### 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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**1**answer

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### 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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**1**answer

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### 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

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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**1**answer

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### 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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**1**answer

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### 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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### 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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### 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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**1**answer

297 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

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### 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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### 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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### 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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870 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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### 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 ...

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**1**answer

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### 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 ...

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### 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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**1**answer

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### 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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**1**answer

548 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 $n$ ...

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**1**answer

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### 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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### 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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### 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 ...

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**1**answer

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### 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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### 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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### 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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### 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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### 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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### (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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### 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 ...

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### Why do all incidence theorems follow from Pappus' theorem?

In Hilbert and Cohn-Vossen's ``Geometry and the Imagination,"
they state in the last paragraph of Chapter 20 that "Any
theorems concerned solely with incidence relations in the
[Euclidean ...

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### The Ramanujan Problems.

I originally thought of asking this question at the Mathematics Stackexchange, but then I decided that I'd have a better chance of a good discussion here.
In the Wikipedia page on Ramanujan, there is ...

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### Quadrilateral from 4 random points

Given 4 random points in 2D, how do I compute the area of the quadrilateral formed by the points?
I'm aware of formulae giving the area when I know the sides a,b,c,d and the diagonals p & q.
But ...

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### An elementary problem in Euclidean geometry [closed]

This problem was first put to me by Luke Pebody (who did not know the answer at the time) and after some work I am yet to find a proof or counterexample. I would be grateful of any insights.
Call a ...

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### Efficient visibility blockers in Polya's orchard problem

Polya's orchard problem asks for which radius $\rho$ of trees at each lattice point within a distance $R$ of the origin block all lines of sight to the exterior of the orchard.
...

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### Geometric meaning of trigonometric relations

According to a paper by Zhiqin Lu in the Mathematical Gazette (the British publication, not the Boston-area newsletter, if that still exists (or even if it doesn't)) in 2007(?), if $u+v+w=\pi$ and ...

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### Helix translates as geodesics

I believe one can fill $\mathbb{R}^3$ with
horizontal translates of the helix
$(\cos t, \sin t, t) \;,\; t \in \mathbb{R}$,
so that every point of $\mathbb{R}^3$
lies in exactly one helix.
I am ...

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### Generalization of plane geometric trees?

View a plane tree drawn in $\mathbb{R}^2$ as a joining of geometric (straight) segments at endpoints such that (a) they avoid intersecting one another (except where they share a vertex), and (b) they ...

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**1**answer

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### Subspace of $\mathbb{R}^n$ spanned by the image of convex $(n-1)$-polyhedra under the face-counting map

Fix $n \in \mathbb{N}$. A convex polyhedron $C$ in $\mathbb{R}^n$ is the convex hull of finitely many points with nonempty interior. For $H$ a supporting hyperplane, ie $C$ is contained in one of the ...

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### Angle of a regular simplex

I find the following question embarrassing, but I have not been able to either resolve it, or to find a reference.
What is the vertex angle of a regular $n$-simplex?
Background: For a vertex $v$ ...

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### Open problems in Euclidean geometry?

Which are some (research level) open problems in Euclidean geometry ?
(Edit: I ask just out of curiosity, to understand how -and if- nowadays this is not a "dead" field yet)
I should clarify a ...

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### Can we alter the axioms of Euclidean space to have $\mathbb{Q}^3$ as a unique model?

I posted this question at math.stackexchange.com but didn't get an answer.
Motivation
Physicists are in search for a model of discrete space(-time) for a long time. So I wondered why not start with ...

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### Characterizations of Euclidean space

I posted this question at math.stackexchange.com but didn't get an answer. Is it a dumb question, eventually?
There are three ways of characterizing the abstract Euclidean space $E^n$ that are quite ...