All Questions
4,826 questions
8
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8
answers
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Side-Angle-Side Congruence and the Parallel Postulate
Is there a link between the side-angle-side congruence of triangles and the parallel postulate? Specifically, does it follow from Euclid's first four axioms alone? In fact, does it even follow from ...
7
votes
2
answers
1k
views
What are the possible images of a square under an area-preserving map?
Let S be the open unit square in R^2: the set of points (x,y) with 0 < x < 1 and 0 < y < 1. Consider an area-preserving smooth map S --> R^2, that is, a map whose Jacobian has determinant ...
- 4,949
9
votes
3
answers
1k
views
Minimize Perimeter(S)/Area(S) for S inside the unit square.
This is a very silly question.
For all regions S contained inside the unit square, what is the infimum of the quantity Perimeter(S)/Area(S)? This ratio being considered is not scale invariant, so it ...
- 101
2
votes
4
answers
3k
views
Symmetrical Presentation of 4-Dimensional Rotation Matrix
This question is not urgent; just a matter of curiosity...
It is relatively easy to generate an arbitrary 3D or even 4D rotation matrix using conjugation (i.e. YXY−1) of orthogonal rotations. I ...
- 524
1
vote
3
answers
2k
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Various Cartan's Lemmata
I am a bit amazed by "Cartan's Lemma".. I have so far seen it in :
Algebraic Geometry sources:
Look at Proposition 2.9 of Freitag and Kiehl's Étale Cohomology where he used étale morphism to describe ...
- 2,275
13
votes
7
answers
2k
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Upper bound on the area of a midpoint pentagon?
Starting with a convex pentagon P, we define the "middle polygon" Q, whose vertices are the middle points of the sides of the initial pentagon. The ratio between the areas of this polygons seem to ...
- 566
3
votes
3
answers
311
views
Are there infinite sets of stellations of polyhedra?
Lists of stellations of polyhedrons are given particular rules like in the book The Fifty Nine Icosahedra which follows "Miller's Rules".
There seems to be no "correct" ruleset to use, so more ...
- 2,615
22
votes
4
answers
2k
views
Minimal surface in a ball
Assume a minimal surface $\Sigma$ has boundary on the unit sphere in the Euclidean space
and $r$ is the distance from $\Sigma$ to the center of the ball.
Is it true that
$$\mathop{\rm area} \Sigma\ge ...
5
votes
2
answers
447
views
Historical question re: ellipses obtained by certain geometrical constructions
I am a faculty member in the Forensic Science Program at PennState (UP). I am trying to obtain information of a historical nature concerning two closely related topics. I seek historical references ...
43
votes
12
answers
2k
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Can a discrete set of the plane of uniform density intersect all large triangles?
Let S be a discrete subset of the Euclidean plane such that the number of points in a large disc is approximately equal to the area of the disc. Does the complement of S necessarily contain triangles ...
- 17.9k
7
votes
3
answers
1k
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How can we count lines in an n-x-n rectangular array?
Is there a formula for the number of lines that contain exactly two points through an n x n rectangular array of points?
- 79
20
votes
4
answers
2k
views
Minkowski sum of small connected sets
Suppose that the convex hull of the Minkowski sum of several compact connected sets in $\mathbb R^d$ contains the unit ball centered at the origin and the diameter of each set is less than $\delta$. ...
- 61.9k
12
votes
3
answers
530
views
Making an l_2 distance out of l_1 distance
If we think of the l1 distance as a grid-distance between points, then we can think of l2 distance as what we get when we "shortcut" the grid by going "inside" a cell.
Making the grid finer doesn't ...
- 4,462
4
votes
1
answer
1k
views
How to find the Fermat Point using the construction of the tangent to ellipse?
Be done the triangle ABC, it is known the method to finding the point Q that minimises the sum QA+QB+QC among all points Q in the plane (The Fermat point).
I want a hint for solving this problem using ...
- 43
15
votes
3
answers
1k
views
Use of n-transitivity in finite group theory
Hello, apparently finite groups which are n-transitive with n>5 are only the permutation groups Sn or the alternating groups An+2, see e.g. page 226 this book by Isaacs http://books.google.fr/books?...
- 2,901
12
votes
3
answers
707
views
A "round" lattice with low kissing number?
Historically, the lattices with high density were studied intensively, e.g. E_8 lattice or Leech Lattice. However, there are situations that lattices with low kissing number are required. Specifically,...
- 139
3
votes
2
answers
384
views
High dimensional Steiner tree
Given n affinely independent points in n-1 dimensional Euclidean space, how is the minimum Steiner tree constructed? Or assuming that the topology of the Steiner tree is given, is there an easy way ...
- 31
7
votes
6
answers
2k
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How to partition R^3 into pairwise non-parallel lines?
Problem. How to partition R^3 into pairwise non-parallel lines?
A possible solution is to stack infinitely many ``concentric'' hyperboloids, by increasing radius and decreasing slope. And don't forget ...
- 1,110
23
votes
6
answers
2k
views
Is there a topological description of combinatorial Euler characteristic?
There are a collection of definitions of "combinatorial Euler characteristic", which is different from the "homotopy Euler characteristic". I will describe a few of them and give some references, and ...
- 54.6k
6
votes
3
answers
982
views
Boolean network as a gauge field
Consider a set of N binary-state nodes at "time" t, each of which is a (boolean) transition function of two nodes in the set, evaluated at time t-1. Thus there are N of these boolean functions of two ...
- 85
11
votes
4
answers
608
views
What is the right way to think about / represent general tilings?
For periodic/symmetric tilings, it seems somewhat "obvious" to me that it just comes down to working out the right group of symmetries for each of the relevant shapes/tiles, but its not clear to me if ...
44
votes
11
answers
26k
views
Algorithm for finding the volume of a convex polytope
It's easy to find the area of a convex polygon by division into triangles, but what is the optimal way of finding the volume of higher-dimensional convex bodies? I tried a few methods for dividing ...
- 441
22
votes
5
answers
3k
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How does one find the "loneliest person on the planet"?
I'm looking for the algorithm that efficiently locates the "loneliest person on the planet", where "loneliest" is defined as:
Maximum minimum distance to another person — that is, ...
- 331
5
votes
1
answer
586
views
a general theory of configurations?
Once I found by accident an article by MacPherson: "Classical projective geometry and modular varieties", in "Algebraic analysis, geometry, and number theory" (Baltimore, MD, 1988), whose introduction ...
- 10.8k
2
votes
4
answers
2k
views
Closest grid square to a point in spherical coordinates
I am programming an algorithm where I have broken up the surface of a sphere into grid points (for simplicity I have the grid lines are parallel and perpendicular to the meridians). Given a point A on ...
- 386
32
votes
4
answers
4k
views
Largest hyperbolic disk embeddable in Euclidean 3-space?
Hilbert proved that there's no complete regular ($C^k$ for sufficiently large $k$) isometric embedding of the hyperbolic plane into $\mathbb{R}^3$. On the other hand, the pseudosphere is locally ...
- 13.6k