Deprecated; please do NOT use this tag, use a more specific tag.

**-4**

votes

**0**answers

36 views

### GEOMETRY PROVE CHORDS CONGRUENT [on hold]

Attached is the picture of the proof. My teacher gave us some hints which were: draw radii, use triangles, and use addition property.

**5**

votes

**1**answer

200 views

### How many convex shapes can be made with the pieces of the Stomachion?

Tangrams are a well-known dissection of the square into seven convex polygons. One fun mathematical question is: how many convex rearrangements of the seven pieces are there?
Answer: there are 12 ...

**2**

votes

**0**answers

46 views

### What is the preimage of the Cassini oval under the stereographic projection?

A Cassini oval in the plane is defined as the locus of points $p$ such that the product of Euclidean distances to the two foci $f_1, f_2$ is equal to some constant. That is:
$$d(p, f_1)\, d(p, f_2) = ...

**64**

votes

**4**answers

5k views

### Is the sphere the only surface all of whose projections are circles? Or: Can we deduce a spherical Earth by observing that its shadows on the Moon are always circular?

Several ancient arguments suggest a curved Earth, such as
the observation that ships disappear mast-last over the
horizon, and
Eratosthenes'
surprisingly accurate calculation of the size of the
Earth
...

**5**

votes

**0**answers

481 views

### Dao's theorem on six circumcenters associated with a cyclic hexagon

This questions from Ngo Quang Duong's paper
In 2013, O. T. Dao published without proof a theorem with title Another seven circles theorem in Cut the Knot, a free site for popular expositionsof many ...

**4**

votes

**6**answers

381 views

### Characterizing Convex Configurations of Quadrupels of Coplanar Points via Linear (In-)equalities between Distance Sums or Differences

Given 4 points $A$, $B$, $C$ and $D$ in general position in the euclidean plane, is it possible to determine from the 6 distances $AB$, $BC$, $CD$, $AD$, $AC$ and, $BD$ alone, whether every point is a ...

**4**

votes

**1**answer

257 views

### Sixteen points circle - A conjecture on Möbius plane

The conjecture refer the reader about the Bundle's theorem configuration. (This conjecture from a note)
Consider the Bundle theorem configuration :
Points $A_1, A_2, A_3, A_4$ lie on a circle,
...

**-1**

votes

**1**answer

48 views

### When is the orbit space of a manifold still a manifold of the same dimension?

$\mathbf{Question}$. Let us assume that $M^n$ is a topological manifold of dimension n, with a group action $\Gamma$, which acts discontinuously and freely on $M^n$. Is the orbit space $M^n / \Gamma$ ...

**1**

vote

**0**answers

142 views

### A chain of six circles associated with six points on a circle (in Mobius plane) [closed]

I found a conjecture: A chain of six circles associated with six points on a circle (in Mobius plane).
This is a generalization of the last my previous question in Three chains of six circles. ...

**-3**

votes

**0**answers

21 views

### Formula for vertices of a Polygon with only 1 vertex at the top and y-axis symmetric [migrated]

I'm trying to find the formula for the vertices of a polygon with n-sides such that there is always only 1 vertex at the top and the polygon is symmetric with respect to the y-axis...
so generally ...

**6**

votes

**2**answers

446 views

### English translation of Lambert's Theorie der Parallellinien?

Does anyone know if there is an available (published or unpublished) English translation of Johann Lambert's Theorie der Parallellinien? I was able to find it online in German by way of the ...

**3**

votes

**0**answers

165 views

### A conjecture on six planes [closed]

When I read Cox's Theorem, and Clifford's Circle Theorem and Miquel six circles theorem, I found the conjecture as folowing. And I checked the conjecture by the Geogebra sofware, the conjecture is ...

**3**

votes

**0**answers

126 views

### Conjecture generalization of Feuerbach theorem and somes another theorems

My question: I am looking for a solution of a conjecture generalization of the Feuerbach theorem in the end of the topic. But I think, I should let you know why I found this conjecture. I thank to ...

**1**

vote

**1**answer

189 views

### Relation of some Euclidean geometry theorems and more conjecture generalizations

In this topic I want to share relation of the Pythagorean theorem, the Stewart theorem and the British Flag theorem, the Apollonius' theorem and the Feuerbach-Luchterhand. Since that I posed two ...

**8**

votes

**4**answers

3k views

### How can I find the average of two 2D curves?

I have a curve interpolation problem.
I have two closed curves that are defined on an X,Y plane. How can I define a 3rd curve that is the average of those two? Programmatically, I have a list of ...

**3**

votes

**0**answers

81 views

### Isometric embedding for manifolds with conical singularities?

Motivation:
In the 2+1 dimensional gravity theory, solutions of Einstein equation are locally with constant curvature except at the locus of sources. In this paper the authors investigate solutions ...

**12**

votes

**6**answers

2k views

### On the number of Archimedean solids

Does anyone know of any good resources for the proof of the number of Archimedean solids (also known as semiregular polyhedra)?
I have seen a couple of algebraic discussions but no true proof. Also, ...

**11**

votes

**0**answers

445 views

### Who conjectured that a transitive projective plane is Desarguesian?

The only known finite projective plane with a transitive automorphism group is the Desarguesian plane $PG(2,q)$ and it seems likely that there are no others, although this is not (quite) proved.
...

**11**

votes

**5**answers

892 views

### Is there a complete classification of constant mean curvature surfaces?

I'm no expert in this field, but I am familiar with the classification of rotationally symmetric surfaces with constant mean curvature by Delaunay. I am aware that once we drop embeddedness and ...

**3**

votes

**1**answer

94 views

### The volume of a region arising from planar linkages

Let $x_0,\dots,x_n$ be a collection of variable points in $\mathbb{R}^2$ and let $c>0$ be a fixed constant. Is there any way I could compute an upper bound of the volume of the region in ...

**13**

votes

**5**answers

1k views

### Which norms have rich isometry groups?

Let $n \ge 2$ be some positive integer. Given a norm $p : \mathbb{R}^n \to \mathbb{R}$, one can inquire about the structure and properties of its isometry group, i.e. the group of all bijections ...

**10**

votes

**1**answer

227 views

### Optimization of points on a plane

Suppose we have $n$ points on a plane. Let $D$ be the sum of the squares of all the pairwise distances between the points. Let $A$ be the area of the convex hull. What is the minimum possible value of ...

**79**

votes

**5**answers

3k views

### Light rays bouncing in twisted tubes

Imagine a smooth curve $c$ sweeping out a unit-radius disk that is
orthogonal to the curve at every point.
Call the result a tube.
I want to restrict the radius of curvature of $c$ to be at most 1.
I ...

**10**

votes

**2**answers

1k views

### Midpoint geodesic polygon / Birkhoff curve shortening

I would like to know under what conditions the process
of creating a midpoint piecewise geodesic polygon converges
on a surface $S \subset \mathbb{R}^3$.
$S$ may be assumed smooth, closed, and ...

**9**

votes

**3**answers

672 views

### Illuminating piecewise-flat manifolds with geodesics

If one issues a geodesic in every direction from a point $p$ on a piecewise-flat 2-manifold, will
it necessarily illuminate the entire surface? I know the answer is 'No,' but I would like to explore
...

**11**

votes

**2**answers

901 views

### Approximating a convex function by a piecewise linear function

Suppose I have a Lipschitz-continuous convex function $f:\mathbb{R}^n\rightarrow \mathbb{R}$. I wish to approximate it on the unit ball by a piecewise-linear function $g:\mathbb{R}^n\rightarrow ...

**3**

votes

**0**answers

60 views

### Equidistribution of Brillouin zones

Answering the question about Limiting shape for Brillouin zones Victor Kleptsyn proved that $N$th Brillouin zone is very close to a circle of radius $c\sqrt N$ (you can find all necessary definitions ...

**12**

votes

**1**answer

330 views

### Limiting shape for Brillouin zones

Is it true that the limiting shape for Brillouin zones (for any lattice) is a circle?
You can find the definition and the step by step construction of Brillouin zones here. This picture is taken from ...

**1**

vote

**0**answers

70 views

### Existence of polytope

Does there exist a polytope in dimensional d consisting of $k>d+1$ faces satisfying that every d faces intersect? I tried 3 dimensional cases, and it seems negative. But is it all negative for any ...

**2**

votes

**0**answers

74 views

### good choice of extension of equivariant map

Let $K$ be a compact Lie group. Let $C_K$ denote the category whose objects are the compact lie groups containing $K$ and whose morphisms are inclusion of the groups. Let $Y$ be a $K-$space such that ...

**9**

votes

**1**answer

545 views

### Ordered geometries from convex subsets of the plane

Motivation
In the Klein disk model of the hyperbolic plane, the points are the interior of the disk, and the lines in $H^2$ correspond to lines intersecting the interior.
Similarly, the Euclidean ...

**-2**

votes

**2**answers

120 views

### Maximal-Orthogonal Convex Hull (or Maximal-Rectilinear Convex Hull) [closed]

Edit : Consider giving a reason for down vote.
In my research, I have come across a this paper from the Computational Geometry field and I am not able to understand the concept of ...

**6**

votes

**1**answer

232 views

### Lattice points on the boundary of an ellipse

How many points of the integer lattice ${\mathbb Z}^2$ can an axis-parallel ellipse of radius $r$ contain on its boundary? (that is, we consider ${\mathbb Z}^2$ as lying in ${\mathbb R}^2$). ...

**4**

votes

**1**answer

163 views

### A conjecture for a curve cuts a curve - variant Cayley-Bacharach's theorem

I propose a conjecture variant of Cayley-Bacharach's theorem.
I'm an electrical engineer, I am not a mathematician. I don't know how to prove this result. Could you give a solution or let me know ...

**1**

vote

**0**answers

40 views

### Thomsen Blaschke condition

I am reading a paper (Paper 1: https://ideas.repec.org/p/cwl/cwldpp/76.html, that cites another paper ( Paper 2) for its proof.
Paper 1, page 1, line 10 says : Consider the topological image G of a ...

**11**

votes

**1**answer

762 views

**30**

votes

**5**answers

878 views

### Can every $\mathbb{Z}^2$ disk be pinball-reached?

Let every point of $\mathbb{Z}^2$ be surrounded by a mirrored disk of radius $r < \frac{1}{2}$,
except leave the origin $(0,0)$ unoccupied by a disk.
Q. Is it the case that every disk can be ...

**8**

votes

**3**answers

830 views

### Solid angles of a tetrahedron

This is a problem I have had for a while. For a triangle, the side opposite the largest angle has the largest length (and similarly for smallest angle). For a tetrahedron, the question is whether the ...

**13**

votes

**3**answers

1k views

### Covering a Cube with a Square

Suppose you are given a single unit square, and you would like to completely cover the surface
of a cube by cutting up the square and pasting it onto the cube's surface.
Q1. What is the largest ...

**7**

votes

**2**answers

741 views

### Which (semi)regular polyhedra are combinations of two others?

The convex combination of convex polytopes is a convex polytope.
An example in $\mathbb{R}^2$ is that a regular octagon
can be obtained as $\frac{1}{2} S + \frac{1}{2} S'$,
where $S$ is a square and ...

**6**

votes

**3**answers

1k views

### Original proof of Pappus' Hexagon Theorem

Does anyone know where I can find an english translation, preferrably online or in a book the library of a small liberal arts college would be likely to have, of the original proof of Pappus' hexagon ...

**5**

votes

**1**answer

313 views

### Geometric applications of Ekeland's variational principle

I'm looking for geometric applications of Ekeland's variational principle in order to see it at work in a context I'm familiar with. Let me recall the principle itself:
Definition. Let $(X,d)$ be a ...

**14**

votes

**2**answers

270 views

### Random rings linked into one component?

Let $S$ be a sphere of unit radius.
Let $C_n$ be a collection of unit-radius circles/rings whose centers
are (uniformly distributed)
random points in $S$, and which are oriented (tilted) randomly ...

**3**

votes

**2**answers

389 views

### The probability distribution for vertex degree in a unit disc graph generated from random points on a plane

Imagine I cover an arbitrarily large plane with randomly placed points at some density $\rho$ s.t. the number of points in any randomly sampled area $A$ (of arbitrary shape and size) is $\approx ...

**3**

votes

**1**answer

427 views

### How to partition a graph into N groups with M elements nearest?

Hi.
This problem probably already here but I could not find the right words to find it.
I have a list with 1700 points (geographic coordinates) and a need to separate into 17 groups with 100 ...

**8**

votes

**2**answers

660 views

### Will a ball fired through a focus of an ellipse eventually tend to a horizontal line?

A couple of years ago I came across this phenomenon which appears to be true although I am having difficulty proving it.
F and F' are foci of a billiard table in the shape of an ellipse. A ball is ...

**3**

votes

**1**answer

385 views

### Holomorphic coordinates on a Kähler manifold

Let $(X,J,\omega)$ be a Kähler manifold. Let $\dim_{\mathbb{R}}(X)=2n$ and we also know that $X$ splits as $X=M\times \mathbb{R}$, where $\dim_{\mathbb{M}}=2n-1$. My question is now: does there exists ...

**4**

votes

**0**answers

287 views

### Averaging lengths and distances

A natural way in which Finsler metrics appear in Riemannian geometry is as averages of Riemannian metrics (e.g., the average of the arc-length elements
$\sqrt{dx^2 + dy^2}$ and $\sqrt{2dx^2 + 3dy^2}$ ...

**3**

votes

**1**answer

253 views

### cover and hide with squares

I am studying two numbers, related to squares, that can characterize a polygon P:
MinCoverNumber = the minimum number of axis-aligned squares required to exactly cover P (the covering squares may ...

**23**

votes

**2**answers

2k views

### Functions whose gradient-descent paths are geodesics

Let $f(x,y)$ define a surface $S$
in $\mathbb{R}^3$ with a unique local minimum at $b \in S$.
Suppose gradient descent from any start point $a \in S$
follows a geodesic on $S$ from $a$ to $b$.
(Q1.)
...