All Questions
Tagged with mg.metric-geometry discrete-geometry
671 questions
4
votes
2
answers
271
views
Centralizing four red vectors in six green sectors
Four red vectors are given, one per quadrant, $[0,90^\circ)$,
$[90^\circ,180^\circ)$, etc.
A rigid star of six green vectors separated by $60^\circ$
can be positioned at
$(\theta,
\theta+60^\circ,
\...
- 151k
6
votes
1
answer
450
views
Dissecting a tetrahedron into orthoschemes
Is there a way to dissect any tetrahedron into a finite number of orthoschemes?
I know that for a tetrahedron which only has acute angles, one can take the center of the inscribed circle and project ...
- 601
5
votes
1
answer
491
views
Isometric embedding of a positively curved polyhedral surface
Suppose you have a 2-dimensional polyhedral surface with specified lengths for the edges so that the vertices all have positive curvature. I believe this has a unique isometric embedding into 3-...
- 27.5k
5
votes
2
answers
523
views
Maximal area coverable by $k$ disjoint isosceles triangles contained in a triangle of area 1.
Given a triangle $\Delta$ of unit area, how much area can always be covered by $k$ isosceles triangles contained in $\Delta$ and intersecting at most at their boundaries?
The answer is easy for $k=1$....
- 17.9k
2
votes
2
answers
5k
views
Best fit for multiple shapes inside an area
Is there a forumla to come up with the best fit for multiple shapes inside a rectangular area, so that none of the shapes are overlapping?
- 121
20
votes
3
answers
3k
views
How many unit squares can you pack into a rectangle with nearly integer side lengths?
Earlier today, somebody asked what looks like a homework problem, but admits the following reading which I think is interesting:
Suppose $a_1,\dots, a_n$ are positive integers, and $\varepsilon$ is ...
94
votes
5
answers
9k
views
Is there a dense subset of the real plane with all pairwise distances rational?
I heard the following two questions recently from Carl Mummert, who encouraged me to spread them around. Part of his motivation for the questions was to give the subject of computable model theory ...
- 236k
2
votes
1
answer
247
views
Are combinatorial configurations whose Levi graphs may be represented as covering graphs over voltage graphs realizable with pseudolines?
This question is related to this previous question. Many combinatorial configurations have Levi graphs which may be represented as derived graphs obtained from voltage graphs over a cyclic group; in a ...
1
vote
1
answer
524
views
How to compute the number of regular spheres needed to fill a rectangular space
Computing the volume of a sphere is straightforward 4/3*pi*R^3
As is the volume of a rectangular space length*width*height (e.g. 10*10*6)
How might I go about determining how many spheres would fit ...
- 111
2
votes
4
answers
8k
views
Compute the Centroid of a 3D Planar Polygon
Given a list of 3D coordinates that define the surface( Point3D1, Point3D2, Point3D3, and so ...
- 381
28
votes
3
answers
2k
views
Is the ratio Perimeter/Area for a finite union of unit squares at most 4?
Update: As I have just learned, this is called Keleti's perimeter area conjecture.
Prove that if H is the union of a finite number of unit squares in the plane, then the ratio of the perimeter and ...
- 18.9k
19
votes
5
answers
21k
views
Dividing a square into 5 equal squares
Can you divide one square paper into five equal squares?
You have a scissor and glue. You can measure and cut and then attach as well. Only condition is You can't waste any paper.
- 383
5
votes
4
answers
1k
views
Coloring Points in the Plane
Suppose one wants to color the points in the plane so any two points at distance one apart are different colors. How many colors are needed?
I heard this problem when I was a kid. Back then the most ...
- 5,275
8
votes
6
answers
1k
views
Combinatorial distance ≡ Euclidean distance
Definition: A polytope has property X iff there is a function f:N+ → R+ such that for each pair of vertices vi, vj the following holds:
disteuclidean(vi, vj) = f(distcombinatorial(vi, vj))
with ...
- 9,720
3
votes
1
answer
439
views
Convex n-polytope general position vectors to general position vectors of tetrahedron
I asked this question in a comment to this question, but got no response. I thought that perhaps it needed more exposure, so I made it a question in itself.
Define a set of general position vectors $...
- 4,842
19
votes
2
answers
1k
views
Four Dimensional Origami Axioms
What are the axioms of four dimensional Origami.
If standard Origami is considered three dimensional, it has points, lines, surfaces and folds to create a three dimensional form from the folded ...
- 193
27
votes
6
answers
2k
views
When shorter means smaller?
Assume a convex figure $F\subset \mathbb R^2$ satisfies the following property: if $f:F\to \mathbb R^2$ is a distance-non-increasing map then its image $f(F)$ is congruent to a subset of $F$.
Is it ...
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
43
votes
12
answers
2k
views
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
views
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
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 ...
