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13
votes
2answers
766 views

Right triangle with edge lengths equal to regular unit polygon edge lengths

This question came up naturally recently from a blog post of John Baez. There is an observation of Euclid that edges of a pentagon, hexagon, and decagon inscribed in a unit circle form the edges of a ...
11
votes
1answer
423 views

Heronian triangle with two sides that are prime

Can any prime number form a Heronian triangle with a second prime as another side? I cannot find a second prime to form a Heronian triangle with either 23 or 167. I have checked up to the 10^7th prime ...
3
votes
2answers
191 views

Equipartition of the circle [closed]

Browsing an old technical studies pupil's school book, I have found the description of a method to place at equal distance $N$ points on the circumference of a circle. I am looking for a proof of this ...
57
votes
10answers
11k views

Is Euclid dead? [closed]

Apparently Euclid died about 2,300 years ago (actually 2,288 to be more precise), but the title of the question refers to the rallying cry of Dieudonné, "A bas Euclide! Mort aux triangles!" ...
24
votes
1answer
642 views

The Eyeball Theorem generalized

I have not seen the 2D Eyeball Theorem—that tangents from the centers of two circles, each encompassing the other, intersect each circle in the same segment length—generalized to higher ...
7
votes
2answers
432 views

Inequality involving the side lengths of a quadrilateral

If $a$, $b$, $c$ and $d$ are the four sides of a quadrilateral, the problem is to show that $ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$. I've verified it to be true for quite a large number of ...
6
votes
1answer
693 views

Shrink polygon to a specific area by offsetting

I have a 2D polygon that I want to shrink by a specific offset (A) to match a certain area ratio (R) of the original polygon. Is there a formula or algorithm for such a problem? I am interested in a ...
15
votes
1answer
466 views

Is the perimeter of an ellipse with integer axes irrational?

Let $Q$ be an ellipse with integer-length axes $a$ and $b$: $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \;.$$ The perimeter of $Q$ is given by the complete elliptic integral of the 2nd kind, $E(\;)$: $4 ...
2
votes
2answers
215 views

Largest inscribed rectangle inside a convex polygon

It has been proved by Radziszewski in this paper K. Radziszewski. Sur une probleme extremal relatif aux gures inscrites et circonscrites aux gures convexes. Ann. Univ. Mariae Curie-Sklodowska, Sect. ...
2
votes
3answers
357 views

Repeating an operation infinitely makes any convex $n$-gon a regular $n$-gon?

For any convex $n$-gon $P_{0,1}P_{0,2}\cdots P_{0,n}$, let us consider the following operation : Operation : Let $k=0,1,\cdots$. Take $n$ points $P_{k+1,i}\ (i=1,2,\cdots,n)$ outside of $n$-gon ...
3
votes
1answer
131 views

About the 'minimum triangle' which includes a convex bounded closed set

Question : Is the following true? "Letting $K$ be a convex bounded closed set on a plane, then there exists a triangle $M$, which includes $K$, such that $|M|\le 2|K|$. Here, $|M|,|K|$ is the area of ...
1
vote
2answers
115 views

Finding the probability that $n$ points which are randomly selected from $d-1$ dimensional spherical surface are 'semi-spherical'

Let $S^{d-1}=\{(x_1,\cdots,x_{d})\in {\mathbb R}^{d}|{x_1}^2+\cdots+{x_d}^2=1\}$, and let us call the intersection of any $d-1$ dimensional subset which passes through the origin and $S^{d-1}$ 'a ...
22
votes
3answers
757 views

Tetrahedron insphere iteration

I know that iterating the following incircle construction approaches an equilateral triangle in the limit:       Starting with any triangle $T$, one forms $T'$ by connecting ...
3
votes
1answer
295 views

Generalising Euler's formula to ellipses and three dimensions

Let $D$ be the closed unit disk, $T$ a triangle and $E$ an ellipse with $E\subset T \subset D$. Without loss of generality say that $E$ is centred at Cartesian coordinates $(c, 0)$ with $0\leq c \leq ...
4
votes
1answer
174 views

Soddy-type relation for Steiner chains

For Steiner $n$-chains of circles of radii $r_1,\dots,r_n$ tangent to an inner circle of radius $r_-$ and an outer circle of radius $r_+$, is there a Soddy-type relation between the $n+2$ quantities ...
28
votes
3answers
1k views

About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals

Question : Letting $S{^\prime}$ be the area of the inner pentagon made by the five diagonals of a convex pentagon whose area is $S$, then find the max of $\frac{S^\prime}{S}$. ...
1
vote
0answers
98 views

Equidissection of square [duplicate]

Monsky's Theorem states: One cannot dissect a square into an uneven number of triangles with equal area. I was wondering how close one could get to a equidissection, i.e. For $n$ an uneven ...
0
votes
0answers
97 views

About the area of the region where the paper is twofold when you double a piece of paper in the shape of a triangle

Suppose that you have a piece of paper in the shape of a triangle $ABC$ whose area is $S_0$ and that the area of the region where the paper is twofold when you double the paper in two along a line is ...
40
votes
1answer
3k views

Probability that a stick randomly broken in five places can form a tetrahedron

Edit (June 2015): Addressing this problem is a brief project report from the Illinois Geometry Lab (University of Illinois at Urbana-Champaign), dated May 2015, that appears here along with a ...
9
votes
1answer
346 views

Can Tarski decide constructibility in elementary geometry?

Can the decision routine for Tarski's Elementary geometry be extended to decide when an existence claim in that theory can be instantiated by a compass and straightedge construction? The answer does ...
5
votes
3answers
1k views

Finding an invisible circle by drawing another line

A friend of mine taught me the following question. He said he found it on a book a few years ago. Though I've tried to solve it, I'm facing difficulty. Question: You know on a plane there is an ...
4
votes
4answers
466 views

Conditions for when an off-centre ellipsoid fits inside the unit ball

An ellipsoid $E$ has centre $\vec{c}=(c_1,c_2,c_3)$ and semiaxes $t_1$, $t_2$ and $t_3$ aligned with the $x$, $y$ and $z$ axes. What are the necessary and sufficient conditions on $\vec{c}$, $t_1$, ...
6
votes
1answer
140 views

2-layer tilings with a center-of-gravity constraint

I've encountered a tiling problem with a physical constraint that might place it outside the literature on tiling. "Tiling" is a bit of a misnomer; it is a special type of cover. All the tiles are ...
1
vote
1answer
132 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 ...
5
votes
1answer
212 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
1answer
272 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 ...
3
votes
0answers
138 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
3answers
756 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
4answers
904 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
1answer
112 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
1answer
307 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
0answers
44 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
1answer
402 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
1answer
326 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
1answer
163 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
0answers
100 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
2answers
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
1answer
105 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
4answers
582 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
1answer
196 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
1answer
197 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
0answers
202 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
2answers
506 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
1answer
224 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 ...
12
votes
0answers
439 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 ...
12
votes
5answers
993 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 ...
10
votes
0answers
361 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
3answers
392 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
1answer
182 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
0answers
137 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 ...