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Results tagged with mg.metric-geometry
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user 39495
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
15
votes
2
answers
1k
views
Hausdorff dimension of Apollonian circle packing, 1.305686729, 1.305688 or yet something else?
I am interested in the Hausdorff dimension of the Apollonian circle packing.
There seem to be two numerical calculations of the value:
1.305686729(10)
from P.B Thomas and D.Dhar, The Hausdorf[sic!] …
- 225
10
votes
Largest regular $k$-simplex inscribed in a $d$-cube, $k < d$
Allow me look at one aspect, or special case, of your question, namely "finding the largest regular 3-dimensional tetrahedron inscribed in a d-dimensional unit cube".
I. $4$-cube
I can find the follow …
- 12.6k
22
votes
1
answer
690
views
Rational inscribed realization of the regular dodecahedron
While it is clear that the regular dodecahedron $D$ cannot be realized with all integer coordinates, it is easy to find a polytope, which is combinatorially equivalent (face lattice isomorphic) to $D$ …
- 12.4k
7
votes
Maximizing the area of a region involving triangles
I find a larger area than @MattF in a non-symmetric solution. A numerical approximation to that solution is:
[[[-0.3204647107, -0.4111035999], [0.1858745389, -0.4555874153], [0.1345901718, 3.49883904 …
CommunityBot
- 1
3
votes
Maximizing the area of a region involving triangles
too long for a comment:
The exact numbers in @Matt's answers are all algebraic and their minimal polynomials are:
For the area: root of
$$81x^9 - 5184x^8 + 60012x^7 + 1080072x^6 - 30787658x^5 + 308408 …
- 7,226
3
votes
Definition of "regular" in Stringham's "Regular figures in n-dimensional space"
Too long for a comment:
Stringham gave a talk about the content of his thesis here in the Seminar of Felix Klein in Göttingen on Monday, 1880/11/29, you can look at the scans here:
Ueber reguläre Körp …
- 10.7k
40
votes
Accepted
Which unfoldings of the hypercube tile 3-space: How to check for isometric space-fillers?
Answer to Q1: All of the 261.
I looked at this question because of a video of Matt Parker and wrote an algorithm to find solutions. See here for an example of how a solution would look like. I dumped …
- 10.7k
51
votes
4
answers
7k
views
what-if.xkcd.com: stabbing (simply connected) regions on the 2-sphere with few geodesics
In the latest what-if Randall Munroe ask for the smallest number of geodesics that intersect all regions of a map. The following shows that five paths of satellites suffice to cover the 50 states of t …
- 18.6k
35
votes
Accepted
what-if.xkcd.com: stabbing (simply connected) regions on the 2-sphere with few geodesics
Looking at this old question again, I'm now fairly convinced that the easiest route of solving this problem is using similar ideas to the one suggested by David E Speyer in a comment, namely basically …
- 10.7k
23
votes
Accepted
Can every simple polytope be inscribed in a sphere?
Not all simple polytopes are incribable, e.g. the dual of the cyclic polytope $C_4(8)$ is simple and not inscribable, as shown recently in Combinatorial Inscribability Obstructions for Higher-Dimensio …
- 10.7k
3
votes
Accepted
Hausdorff dimension of Apollonian circle packing, 1.305686729, 1.305688 or yet something else?
It seems in the meantime, there are new results:
Bai, Zai-Qiao; Finch, Steven R. Precise calculation of Hausdorff dimension of Apollonian gasket. Fractals 26 (2018), no. 4, 9 pp.
claims a better appro …
- 10.7k
10
votes
A cube is placed inside another cube
The answer to your question is: no
Its also easy to give a concrete counter-example:
Take the cube with vertices
$$
\left(\frac{273}{340},\,\frac{79}{68},\,\frac{13}{20}\right) ,
\left(\frac{407 …
- 10.7k
15
votes
3D models of the unfoldings of the hypercube?
I used sage to make a 3d animation of all 261 unfoldings.
Here is a screenshot of the first few:
The file cube-unfoldings.txt contains all the unfoldings, each line contains a list of 8 points.
…
- 10.7k
56
votes
Does this geometry theorem have a name?
Even more is true for this theorem. Check out this drawing from Arseniy Akopyan wonderful book of Geometry in Figures (Second, extended edition, 2017). On page 65 we find Figure 4.7.29)
In the fore …
- 10.7k
24
votes
How to check if a box fits in a box?
A (trivial) necessary condition is that the diagonal of the inner one is not longer than the diagonal of the outer one.
So if $(a,b,c)$ is supposed to fit in $(x,y,z)$, then we should have
$$a^2+b^2+ …
- 10.7k