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Results tagged with mg.metric-geometry
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user 46140
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
3
votes
Accepted
Is there a method to make a rep-n rep-tile for any number n, using only triangles?
You can indeed do it for rectangles, by taking a ratio of sides equal to $\sqrt{n}$.
For triangles, M. Beeson (2012), Triangle Tiling I: the tile is similar
to ABC or has a right angle (pre-print) giv …
- 7,226
5
votes
Accepted
The exact number of points within a circle of radius r centered on a lattice point in a hexa...
By identifying the lattice points with numbers of the form $x - y\omega$, $\omega = e^{2\pi i / 3}$, $x, y \in \mathbb{Z}$, we find that we want to count Diophantine solutions to $x^2 + xy + y^2 \le r …
- 7,226
1
vote
Accepted
On triangulations and "coverage" of circumcircles
It's sufficient to prove $D(\triangle abc) \subseteq D(\triangle abd) \cup D(\triangle bcd)$ by symmetry under permutation of the labels $b,d$.
Divide the circumcircle of $abc$ into three arcs: $\frow …
- 7,226
3
votes
Partition of polygons into 'congruent sets of polygons'
By the Wallace-Bolyai-Gerwien theorem it suffices to cut the polygon into $n$ sets of equal area, which can certainly be done by continuity of the area on one side of a line as you move the line acros …
- 7,226
0
votes
Geodesics on the sphere
A couple of answers already propose using the triangle inequality, but I think they both apply it using a calculus argument. There's a simple geometric argument. Let $C$ be the shortest path from $A$ …
- 7,226
5
votes
Are there infinitely many "generalized triangle vertices"?
This is a report on an unsuccessful computational approach which is rather too long for a comment.
I work with complex numbers to represent the points in the obvious way.
It suffices to consider $\mu( …
- 7,226