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Results tagged with discrete-geometry
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user 2480
Finite or discrete collections of geometric objects. Packings, tilings, polyhedra, polytopes, intersection, arrangements, rigidity.
3
votes
How do you traverse a rectangular grid of points while turning as little as possible?
(I assume that the OP wants to minimize is the number of turns rather than the total amount of absolute turning angles.)
If we restrict to moving only parallel to the axes, here is an elementary proof …
- 5,103
10
votes
Are there Monohedra with odd numbers of faces?
This is too long for a comment.
The figure shows how to construct an $11$-face convex polyhedron whose faces all quadrilaterals, but not congruent. I think it's only interesting because it shows that …
- 5,103
8
votes
Fair cutting of the plane with lines
CLAIM. The only fair cutting with at least one quadrilateral is the square grid (2). It was already shown in the original answer (below) that there cannot be $n$-agons for $n\ge 5$.
LEMMA. No fair cut …
- 5,103
5
votes
Accepted
Convex lattice polygons with equal area and perimeter
This answers question 2 as well. But I think both questions are way more suitable for math.stackexchange.
I add another example because, unlike the first, it has the property that multiplied by $I^{n …
- 5,103
6
votes
Accepted
Tilting the $d$-cube to vertically separate its vertices
Robert Israel's answer is best possible.
As pointed out there, the problem is equivalent to finding a vector in $\mathbb{R}^d$ with minimal separation $1$ among sums of subsets of entries, and with th …
- 5,103
1
vote
Partitions of convex planar regions into zonogons
Any centrally symmetric convex polygon can be sudbivided into rhombuses, so assume only rhombuses are used. Now split them further (with lines parallel to the edges) so that no vertex of a rhombus fal …
- 5,103
14
votes
2
answers
539
views
Are all well behaved "mean" functions on $\mathbb{R}^+$ equivalent?
Given a set $S$, a function $M: S\times S \rightarrow S$ is a mean if it satisfies the properties:
$M(a,a)=a\qquad$ (identity)
$M(a,b)=M(b,a)\qquad$ (commutativity).
and possibly
$M(M(a,b),M(a,c …
- 5,103
13
votes
Accepted
infinite configuration of lines
I can sketch a proof based on assuming this "finite" result:
A). For any pentagonal star one of the 5 triangles will have area strictly
smaller than that of the central pentagon.
(I think a brute for …
- 5,103