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Questions tagged [geometry]

For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, and angles.

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615 views

Why do all the Platonic Solids exist?

In three dimensions it is quite easy to prove that there exist at most five Platonic Solids. Each has to have at least three polygons meeting at each vertex, and the angles of these polygons have to ...
20
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0answers
459 views

Matching red to blue dots

I have two red points, $r_1$ and $r_2$, and two blue points, $b_1$ and $b_2$. They are all placed randomly and uniformly in $[0,1]^2$. Each dot points to the closest dot from another colour; closest ...
14
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0answers
284 views

In what kind of space does this object live?

Let me quickly build up some background. One way to build a hypercube is to take cubes, and start gluing them together, face to face, such that each edge is shared by $3$ cubes. You complete the ...
13
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0answers
601 views

Nontrivial trivial integrals

Consider two propositions in geometry: Circumscribe a right circular cylinder about a sphere. The surface area of the cylinder between any two planes orthogonal to the cylinder's axis equals the ...
13
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0answers
938 views

Klein's Erlangen program taken seriously

Felix Klein suggested in his Erlangen program a way to classify geometries based on group theory. According to Wikipedia, we have the following definition: A Klein geometry is a pair (G, H) where G ...
12
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0answers
181 views

Points moving towards the nearest point, where will they meet?

Consider an arbitrary number of distinct points in some $n$ dimensional space whose locations are exactly known. Every point is moving towards its nearest point simultaneously, at constant speed. ...
12
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0answers
229 views

Fastest way to check existence of solution for a linear system of inequalities

What is the fastest way to check if there exists a solution to the inequality $A x \leq b$, with $A \in \mathbb R^{n \times m}$? I know this can be checked through the phase 1 of a linear programming ...
12
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0answers
232 views

Surface of the intersection of $n$ balls

Suppose there are $n$ balls (possibly, of different sizes) in $\mathbb R^3$ such that their intersection $\mathfrak C$ is non-empty and has a positive volume (i.e. is not a single point). Apparently, $...
12
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0answers
957 views

Gauss-Lucas Theorem (roots of derivatives)

Gauss-Lucas Theorem states: "Let f be a polynomial and $f'$ the derivative of $f$. Then the theorem states that the $n-1$ roots of $f'$ all lie within the convex hull of the $n$ roots $\alpha_1,\ldots,...
11
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0answers
198 views

Point set where each point has unity distance to all other points ($L_1$ metric)

I want to construct a point set where each point has the same (w.l.o.g., unit) distance to all other points in the $L_1$ metric. Example: The points $\left(\frac{1}{2},0\right)$, $\left(-\frac{1}{2},0\...
11
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0answers
367 views

How small parallelograms are we guaranteed to get, when we select the two sides from different plane lattices?

This a shortened version (motivation from telecommunications stripped away) of a question I asked in MO in late May (no answers). I am mostly checking, if somebody has seen this or a related question ...
10
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0answers
145 views

A conjecture about the connection between a Penrose tiling and the Fibonacci word fractal

Consider the Penrose tiling $P3$, inflated up to $6$ generations: We draw a line passing through the center of the tiling (red dot) and the outer vertex of the rightmost starting tile (black dot). ...
10
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0answers
113 views

Rep-tiles of order 4 and order 9

Rep-tiles are figures which are dissected by the same figures as itself. As you can see, the rep-tile of order 4 is also a rep-tile of order 9 in the above figures: Compare the L-shaped figure at ...
10
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0answers
255 views

Maximum number of regions of a sphere partitioned by $\binom{n}{3}$ planes from $n$ points

We can place $n\in\mathbb{N}$ points on the surface of a sphere in a configuration so as to maximize the answer. A plane is defined by $3$ points. We create all $\binom{n}{3}$ planes from the $n$ ...
10
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385 views

Singularities on a weighted projective curve

Let $C$ be the curve of degree $3$ defined over $\mathbb{C}$ given by $$x(y+z)=y^3-z^3$$ which lives in the weighed projective space $\mathbb{P}(x,y,z)=\mathbb{P}(2,1,1)$. Is the curve singular ?
9
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0answers
154 views

Strange point lies on common tangent of 9-point circle and incircle

Let $ABC$ be a triangle, with medial triangle $DEF$ and intouch triangle $PQR$. Let $J$ be the midpoint of $\overline{AD}$, and let $BJ$ meet $AP$ at $K$. Let $X$ be the point on ray $\overrightarrow{...
9
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0answers
303 views

Geometrical interpretation of a group action of $SU_2$ on $\mathbb S^3$

Background There're some nomenclatures from Michael Artin's Algebra to explain. 3-Sphere, or $\mathbb S^3$, is the locus of $x_0^2+x_1^2+x_2^2+x_3^2=1$, where $(x_0,x_1,x_2,x_3)\in\mathbb R^4$. $SU_2$...
8
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0answers
110 views

How can we formalize Jorge Luis Borges' Aleph?

Background. Jorge Luis Borges was a post-modern short-story writer of the 20th century, whose stories often invoke a healthy dose of surrealism. One of his works is called The Aleph. In this book, ...
8
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0answers
160 views

A variation of the square peg problem

This question spawned from this recent thread. The notorious square peg problem states that any continuous, simple and closed curve $\gamma$ in the plane contains the vertices of some square. It has ...
8
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0answers
177 views

Brownian Motion in Confined space, any results?

I am searching for work regarding Brownian motion in a confined space, like a sphere or a cylinder, where the wall will serve as reflection boundary. I am wondering if it is possible to derive results ...
8
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0answers
246 views

What type of aperiodic tiling is used by Turkish Airlines on their bathroom walls?

The walls and bulkheads of Turkish Airline flights are decorated with a pattern that appears to be some sort of aperiodic tiling. They are most prominent on the bulkheads of flights, and are also used ...
8
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212 views

Computing Hodge numbers of a complete intersection

The situation is this: I have a 5-dimensional irreducible projective variety $Y$ embedded in $\mathbb P^{13}$. This variety is singular, the singularities being a disjoint union of two curves. I have ...
8
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0answers
612 views

Properties of the projection onto a nonconvex set

Consider a set $\Omega\subseteq\mathbb{R}^n$ being "sufficiently regular", for example being the image of a $C^1$ mapping from $\mathbb{R}^p$ for some $p\ge1$. We may then consider the mapping $$ g:\...
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3k views

Geometry Proof: Convex Quadrilateral

A quadrilateral ABCD is formed from four distinct points (called the vertices), no three of which are collinear, and from the segments AB, CB, CD, and DA (called the sides), which have no ...
7
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0answers
77 views

Determining finitude or infinitude from a simple geometric construction

Playing with a pencil on a checkered sheet I encountered this construction: 1) take a point $A$ on the grid and a point $B$ that is distant from $A$ $n=2,3,4...$ horizontal steps and $1$ vertical ...
7
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0answers
58 views

Is this construction of the “edge polytope” known?

Given a convex polytope $P\subseteq\Bbb R^d$. I am going to construct a new polytope from its edges (I call it the edge polytope) with the following steps: Take the 1-skeleton of $P$. Extract the ...
7
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0answers
193 views

how to place a rope with a given length within an orthogonal triangle (see picture)

I would like to know what is the optimal way of placing the red rope of a given length $p$, where $\sqrt{2}<p<2$, in the orthogonal triangle ABCA, so that the green area is minimized (see ...
7
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0answers
160 views

Optimal Compass and Straightedge Constructions

I was recently looking over some Islamic geometry patterns, and was struck by the complexity of the constructions needed to create seeming simple patterns. This got me wondering regarding optimal ...
7
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0answers
246 views

If a surface $S$ admits two differentiable orthogonal families of geodesics $\Rightarrow$ The Gaussian curvature of $S$ is zero.

This question was previously posted in Orthogonal differentiable family of curves . But I'm facing unsolved issues. I'm still interested in solve the following exercise: QUESTION: We say that a ...
7
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0answers
165 views

Magic Cubic Curve Permutations

The permutation $(-2,9,-4,7,-6,5,-8,3,1)$ can be considered magical. With their negative values diametrical to $0$ at $(0,0)$, a placement of integers begins so that all zero-sum triples form straight ...
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0answers
232 views

Do side-rational triangles of the same area admit side-rational dissections?

Call a polygon side-rational if the lengths of all its sides are rational. Call a dissection of a polygon side-rational if all of the polygons within the dissection are side-rational. Then my ...
7
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0answers
87 views

Fast search of local positive quadruples on the sphere

Let $U = \{u_{1}, u_{2}, \ldots, u_{n}\} \subset \mathbb{R}^{3}$ be the finite set of points on the unit sphere in $\mathbb{R}^{3}$: $||u_{i}||_{2} = 1$ Definition: Quadruple of points $(u_{i}, u_{j},...
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0answers
143 views

Decorating eggs

I came across this question as I have been helping someone decorate styrofoam eggs by gluing wrapping paper onto it. The question is quite simple but I found myself stumped. Given an egg what is the ...
7
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0answers
85 views

Can ${n \choose 2}$ points be covered by lines determined by $n$ points?

Let $S=\{(a_1,b_1),\ldots,(a_{n \choose 2},b_{n \choose 2})\}$ be ${n \choose 2}$ points on the plane. Does there exist $n$ points, such that the lines determined by the $n$ points cover all the ...
7
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0answers
233 views

Characterization of Convex Polygons

John Lee's Axiomatic Geometry has an interesting characterization of convex and non-convex vertices for polygons. Let $P$ be a polygon. Consider a ray emanating from a vertex of $P$ which does not ...
7
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0answers
434 views

Family of geometric shapes closed under division

The family of rectangles has the following nice properties: Every rectangle $R$ can be divided to two disjoint parts, $R_1 \cup R_2 = R$, such that both $R_1$ and $R_2$ are rectangles (i.e. belong ...
7
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0answers
384 views

What's the average distance between two discs in the plane?

Consider two discs in the plane of radius $r$ and $s$, with centers separated by a distance $l$. If we choose a point uniformly at random from each disc, what is the expected distance between the two ...
7
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0answers
328 views

What’s the best way to cut an apple?

Take the apple in one hand, and the knife in the other. In the first cut, the apple is divided in two pieces: a small one that drops into the plate and a big one that is still hold with the hand. This ...
7
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0answers
192 views

Paper cylinder inside out

My question is related with paper folding: Given a cylinder of paper it is possible to turn it inside out using folding along lines. This is a Martin Gardner recreational puzzle. Secondly, Has ...
7
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0answers
494 views

Proof Strategy for a Dynamical System of Points on the Plane

I have a rather simple-looking system which exhibits a particular behaviour in simulation, and I would now like to attempt to prove this formally. The problem is, I don't really know where to start, ...
7
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0answers
175 views

Points at Integer Distances in 3-space

With the restriction no three points in a line, no four points on a circle, there is a 7 point configuration of points on the plane such that all pairs of points are at integer distances. 1 For 3-...
6
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0answers
75 views

Continuity of the Euler characteristic with respect to the Hausdorff metric

Hadwiger's theorem of integral geometry states that all continuous valuations which are invariant under rigid motions are expressible in terms of the intrinsic volumes. The continuity property means ...
6
votes
0answers
57 views

Construction involving regular polygons inside a circle

Let's make a construction involving regular polygons: ► First, we begin with a equilateral triangle, with side $\ell_3 = 1;$ ► After, we draw a square on the middle point each side of the initial ...
6
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0answers
702 views

Testing if two finite sets of points differ only by rotation (unordered, in polynomial time in size and dimension)?

Imagine we have two size $m$ sets (without order) of points $X=\{x^i\}_{i=1..m}, Y=\{y^i\}_{i=1..m} \subset \mathbb{R}^n$ and we want to answer the question if they differ only by rotation: if there ...
6
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0answers
50 views

Given number of regions, find maximum number of boundaries

Problem: Supoose we want to divide $\mathbb{R^2}$ into $N$ regions separated by straight lines. What is the maximum number of boundary lines we could have? I also pose the restriction that between any ...
6
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0answers
58 views

Constraints on conical coffee cup constructions of cardioids & catacaustics

The Mathologer video Times Tables, Mandelbrot and the Heart of Mathematics discusses several relationships. For the n=2 and 3 cases, the cardiod and catacaustic (or nephroid per @Rahul's comment) ...
6
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0answers
74 views

Can any curve in 3D space be described by an intersection of two surfaces?

Can any curve in 3D space be described by an intersection of two surfaces? If not, what assumptions I need to let it be true? If this is too general, what if I restrict the scenarios to twice ...
6
votes
0answers
115 views

Does inscribing a circle, then a triangle, then …, inside of an initial triangle telescope to some “center” of that triangle?

Start with a triangle, and construct its inscribed circle. Take the three points where the inscribed circle is tangent to the triangle, and construct a new triangle with those points as the vertices. ...
6
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0answers
72 views

Omar Khayyam and the tribonacci constant

While trying to find the tribonacci cousin of this post, I came across this nice short article A Geometric Problem of Omar Khayyam and its Cubic by Wolfdieter Lang. Given the figure, $\hskip1.7in$ ...
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0answers
78 views

There is some general meaning of angle in geometry?

The other day I discovered the concept of hyperbolic angle to denote angles in hyperbolic geometry, as the half of the area between the hyperbola defined by $x^2-y^2=1$ and the $x$-axis. To be honest ...