Questions tagged [geometry]

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

10,041 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
48 votes
0 answers
587 views

What is the largest volume of a polyhedron whose skeleton has total length 1? Is it the regular triangular prism?

Say that the perimeter of a polyhedron is the sum of its edge lengths. What is the maximum volume of a polyhedron with a unit perimeter? A reasonable first guess would be the regular tetrahedron of ...
user avatar
44 votes
0 answers
2k views

What Rubik's Twist configuration has the lowest visible surface area?

The Rubik's Twist has been a fun time sink. From the wiki page, [It] is a toy with twenty-four wedges that are right isosceles triangular prisms. The wedges are connected by spring bolts, so that ...
user avatar
  • 491
26 votes
0 answers
756 views

Can the lion protect the sheep from three wolves?

Generally, in pursuit-evasion games, there's one prey and one or many pursuers. I'd like to know how extending the food chain would change the dynamics of such games. Specifically, let's consider a ...
user avatar
  • 1,667
26 votes
0 answers
421 views

How many "prime" rectangle tilings are there?

Given two tilings of a rectangle by other rectangles, say that they are equivalent if there is a bijection from the edges, vertices, and faces of the tilings which preserves inclusion. For instance, ...
user avatar
26 votes
0 answers
431 views

What are the known convex polyhedra with congruent faces?

A monohedral polyhedron is one whose faces are all congruent. Note that this is a weaker condition than being isohedral (face-transitive). We have a classification of all convex isohedral polyhedra, ...
user avatar
20 votes
1 answer
3k views

Triangle dissection, no shared edges

It's possible to divide a triangle into smaller triangles such that no edge lengths are shared. Alternately, no two internal triangles share two vertices. The top three are the known simplest ...
user avatar
  • 19.6k
17 votes
0 answers
282 views

If two convex polygons tile the plane, how many sides can one of them have?

The set of convex polygons which tile the plane is, as of $2017$, known: it consists of all triangles, all quadrilaterals, $15$ families of pentagons, and three families of hexagons. Euler's formula ...
user avatar
17 votes
0 answers
1k 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 ...
user avatar
  • 933
15 votes
0 answers
150 views

What is the covering density of a very thin annulus? Is it $\frac{\pi\sqrt{51\sqrt{17}-107}}{16}$?

Take some very small $\epsilon>0$, and consider the annulus/ring given by the set $\{(r,\theta)\ |\ 1-\epsilon\le r\le1\}\subset \mathbb{R}^2$. We wish to place translated copies of this annulus ...
user avatar
15 votes
0 answers
482 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 ...
user avatar
15 votes
1 answer
607 views

Volume of $n$-dimensional spherical orthant in upper diagonal halfspace

Consider an $n$-dimensional Euclidean Space. Consider orthants in that space. Each orthant occupies $\frac{1}{2^n}$ of the volume of an $n$-dimensional unit sphere. Let's call that a spherical ...
user avatar
  • 14.2k
15 votes
0 answers
1k 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,...
user avatar
  • 373
14 votes
0 answers
416 views

Minimizing the sum of cosines of non-obtuse angles formed by $n\geq4$ concurrent lines in $3$D space

Suppose I have two lines in $3$D space passing through the origin. The smallest angle formed between them would be between $0$ and $\pi/2$. Minimizing the cosine of this angle we'll get $\cos {(\pi/2)}...
user avatar
  • 328
14 votes
0 answers
313 views

Figures and Numbers: Relating properties of geometric shapes and their Fourier series

Consider two types of parametrized curves $\gamma:[0,2\pi]\rightarrow \mathbb{R}^2$ open curves $\gamma_\sim(t) = (t,a(t) + b(t))$ closed curves $\gamma_\bigcirc(t) = (a(t),b(t)) = a(t) + ib(t)$ ...
user avatar
14 votes
0 answers
1k 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:\...
user avatar
13 votes
1 answer
306 views

Fitting a tetrahedron through the smallest hole

I'm designing a child's toy consisting of a closed box with a hole on top; a unit tetrahedron must fit through this hole. What is the smallest possible area of the hole? Currently my hole is an ...
user avatar
  • 161
13 votes
0 answers
409 views

"Perfect" solutions to the kissing number problem besides in dimensions 1,2,8, and 24.

The kissing number problem asks how many n dimensional unit spheres can fit around a central one with no overlapping; a natural question is in what dimensions can this be done so that there is no ...
user avatar
13 votes
0 answers
536 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 ...
user avatar
12 votes
0 answers
422 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 ...
user avatar
11 votes
0 answers
107 views

Proving a manifold with certain homogeneous property is Einstein

Let $M$ be a Riemannian manifold such that for all pairs of points $(p, q), (r, s)$ satisfying $d(p, q) = d(r, s)$, there exists an isometry $f: M \to M$ which takes $p$ to $r$ and $q$ to $s$. Prove ...
user avatar
11 votes
0 answers
122 views

hypoid gears described mathematically

I bought a worm drive electric saw recently. The claim is that, rather than the worm drive I expected, the drive involves hypoid gears. From what I can see, hypoid is an abbreviation of hyperboloid. ...
user avatar
  • 131k
11 votes
0 answers
449 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). ...
user avatar
11 votes
0 answers
257 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\...
user avatar
  • 560
10 votes
0 answers
252 views

Is there an ellipse with rational major and minor axes which has a "simple" closed form perimeter?

Other than the trivial case of an ellipse with equivalent major and minor axes (a circle) or the degenerate case where one axis is $0$, is there any known ellipse that has rational length major and ...
user avatar
  • 694
10 votes
0 answers
168 views

Are there "close" solutions to Hilbert's third problem?

Hilbert's third problem (or a modern formulation thereof) asks whether two polyhedra $P,Q$ of equal volume are equidecomposable by cutting $P$ into finitely many polyhedral pieces and rearranging them ...
user avatar
10 votes
0 answers
120 views

Exemples of applications of "groupoidification" to linear algebra

I just read Baez's very nice blog notes about groupoidification, and around the beginning, he states : "From all this, you should begin to vaguely see that starting from any sort of incidence ...
user avatar
  • 41.6k
10 votes
0 answers
262 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 ...
user avatar
  • 19.6k
10 votes
0 answers
254 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 seemingly simple patterns. This got me wondering regarding optimal ...
user avatar
  • 31.8k
10 votes
0 answers
497 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 $n$ ...
user avatar
10 votes
1 answer
3k views

Vertices of a cyclic polygon have integer coordinates and sides. If odd $n$ divides the squares of the sides, it divides twice the area.

IMO 2016 Problem 3: Let $P = A_1 A_2 \cdots A_k$ be a convex polygon in the plane. The vertices $A_1, A_2, \ldots, A_k$ have integral coordinates and lie on a circle. Let $S$ be the area of $P$. An ...
user avatar
10 votes
0 answers
517 views

Decomposing geodesic tessellations over a sphere into parallelograms

I'm working with some icosahedron-based tessellations of triangles over the surface of a sphere. Class I and Class II tessellations have a nice property where, cutting along the edges of the ...
user avatar
  • 201
10 votes
0 answers
469 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 ?
user avatar
  • 966
10 votes
0 answers
466 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 $(x_0,x_1,x_2,x_3)\in\mathbb R^4$. $SU_2$...
user avatar
  • 9,338
10 votes
0 answers
237 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-...
user avatar
  • 19.6k
10 votes
2 answers
2k views

Relationship between two centers of circles in a Venn diagram

Let $S$ be a circle of 1 unit area. No part of circles $A$ and $B$ are outside the circle $S$. Let $n(S)=1$, $n(A)$, and $n(B)$ be the area of circle $S$, $A$, and $B$, respectively. For the given ...
user avatar
9 votes
0 answers
198 views

How these circles are congruent?

Here is a problem involving curvilinear incircles and mixtilinear incircles. Let a triangle$\triangle$$ABC$ have circumcircle $\gamma$.It's A-Excircle tangency point at side$BC$ is $D$ Let $\gamma_1$ ...
user avatar
9 votes
0 answers
163 views

Can every tree with total length $2$ be covered by a semi-disc of radius $1$?

Can every tree with total length $2$ be covered by a semi-disc of radius $1$? If the tree is actually a curve, or the convex hull of the tree is a triangle, I know this is correct after some attempts. ...
user avatar
9 votes
1 answer
146 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 ...
user avatar
  • 1,164
9 votes
1 answer
136 views

Do any of these concepts relating to similarity in general metric spaces have names?

Definition Let a metric space $(X,d)$ be given. A similitude is a function $f: X \to X$ such that, for all $x,y \in X$, $d(f(x),f(y)) = r \cdot d(x,y)$ for some positive real number $r>0$. ...
user avatar
  • 19.5k
9 votes
0 answers
390 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 ...
user avatar
9 votes
1 answer
240 views

Is a normal matrix satisfying $A^TA=...$ circulant?

Let $A=\{a_{ij}\}$ be a normal matrix such that $a_{ij}\geq 0$ with equality iff $i=j$. Suppose that $$ A^TA=\begin{pmatrix} a & b & \cdots & b\\ b & a & \ddots & \vdots\\ \...
user avatar
  • 28.6k
9 votes
1 answer
632 views

Could Euclid have bisected a line segment without his method of superposition?

In Book I Proposition 10 of the Elements, Euclid performs the bisection (i.e. finding a midpoint) of a line segment. In the course of doing so, he uses Book I Proposition 4, the Side-Angle-Side ...
user avatar
9 votes
0 answers
355 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 ...
user avatar
  • 39.4k
9 votes
1 answer
598 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 ...
user avatar
9 votes
1 answer
955 views

IMO 2016, Problem 3: Number Theory with the Area of a Polygon

IMO 2016 (Problem 3). Let $P=A_1A_2\cdots A_k$ be a convex polygon in the plane. The vertices $A_1, A_2, \cdots , A_k$ have integral coordinates and lie on a circle. Let $S$ be the area of $P$. An odd ...
user avatar
  • 3,347
8 votes
0 answers
58 views

What is the shortest way to enclose $n$ circles?

What is the shortest way to enclose $n$ circles of radius $1$? Is there a formula for the length of the shortest path? I have tried to find the shortest way, up to $n\leq10$. Below are the shortest ...
user avatar
  • 1,193
8 votes
0 answers
149 views

Rational point inside a rational polygon

I have the following conjectures. Conjecture 1: Hypotheses: Let $P = (v_1, v_2, …. v_n)$ be a (convex or concave) polygon drawn on a plane. The lengths of the edges $(v_1, v_2)$, $(v_2, v_3)$ ... $(...
user avatar
8 votes
1 answer
263 views

Given isosceles triangles $\triangle ABC$ and $\triangle DBF$ (all spherical chords), identify the chord $\overline {DF}$ so that $|AD| = |DF| = |FC|$

tl;dr: As shown in the image below, find the chord $\overline {DF}$ so that $|\overline {AD}| = |\overline {DF}|$, and have the answer be in the form of the ratio between $|\overline {AC}|$ and $|\...
user avatar
8 votes
1 answer
203 views

Can continued fraction of $\pi$ tile the plane?

By continued fraction, I mean a simple (canonical) continued fraction. By "tile the plane": I actually am interested in infinite sequences of tillable rectangles. Continued fraction of $e$ ...
user avatar
  • 12k
8 votes
2 answers
769 views

How small can the sum of the interior angles of a triangle in hyperbolic geometry get? Is there a lower bound?

I know that in Euclidean geometry, the sum of the interior angles of a triangle is exactly $\pi$. In hyperbolic geometry, I know that the sum of the interior angles of a triangle is $\leq \pi$, and ...
user avatar
  • 1,887

1
2 3 4 5
201