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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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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 ...
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527 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 ...
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288 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 ...
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617 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 ...
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971 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 ...
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198 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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249 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 ...
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371 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 ...
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974 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,...
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202 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\...
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159 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). ...
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148 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 ...
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267 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 $n$ ...
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396 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 ?
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169 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{...
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241 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 ...
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318 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 $(x_0,x_1,x_2,x_3)\in\mathbb R^4$. $SU_2$...
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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 ...
8
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141 views

Heesch numbers in 3D

At the Tiling Database there are 3, 20, 198, 1390 (A054361) non-tiling polyominoes of order 7 to 10. In 3D, solids with a particular Heesch number don't seem to be well known. Glenn Rhoads found ...
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116 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, ...
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172 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 ...
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186 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 ...
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255 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 ...
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220 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 ...
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649 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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48 views

Can a regular $n$-simplex have vertices in $\mathbb Z^n$ for $n > 1$?

Trivially, a regular $0$-simplex (point) and $1$-simplex (line segment) can have integer vertices in $0$ and $1$ dimensional Euclidean space respectively. On the other hand, a regular $2$-simplex (...
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136 views

Modular geometry: The parabolas of quadratic residues modulo $p$

[For using the available space better, I rotated the function graphs by 90 degrees.] For the quadratic function $f_1(x) = x^2$ (with $x \in \mathbb{R}$) there is only one parabola all integer points $...
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82 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 ...
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59 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 ...
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78 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 ...
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82 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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200 views

Axiom of Choice as similar to Parallel Postulate?

I'm looking for resources that have drawn a comparison between the Parallel Postulate and the Axiom of Choice. That is, if we treat ZFC as an analogue to Euclidean geometry, can we view the ...
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169 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 ...
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259 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 ...
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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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131 views

What makes a geometric construction more or less stable?

As anyone who's actually done geometric construction of n-gons knows, not all construction methods are made equal. Some are very stable (the shape you get is always close to ideal even if you're not ...
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118 views

Why Are Fresnel Functions Used For Splines?

Why are Fresnel functions still used in the research and implementation of clothoid splines? They cannot represent curves of constant curvature, which has led to a lot of research/implementation ...
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88 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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145 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 ...
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86 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 ...
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296 views

From Icosahedron to Pentagonal hexecontahedron (Floret Tessellation)

Inspired by this post: Floret Tessellation of a Sphere I tried to transform myself an icosahedron into its simplest Floret tessellation. But I am having trouble when applying the 'method' given in the ...
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245 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 ...
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444 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 ...
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396 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 ...
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335 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 ...
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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 ...
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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
783 views

The apex of parabolic motion forms an ellipse of constant ellipticity.

I am not sure how well-known this is idea is, but here is a .gif illustrating it: Basically, the set of highest points of parabolic motion at constant initial velocity forms an ellipse, with ...
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177 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-...
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76 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 ...