All Questions
5 questions
27
votes
3
answers
13k
views
Which unfoldings of the hypercube tile 3-space: How to check for isometric space-fillers?
Recently Mark McClure constructed and displayed
the 261 unfoldings of the hypercube (tesseract)
in response to the question,
"3D models of the unfoldings of the hypercube?":
The first 9 unfoldings ...
- 151k
18
votes
1
answer
677
views
Higher dimensional generalization of: Any quadrilateral tiles the plane?
Any (non-self-intersecting) quadrilateral tiles the plane.
(MathWorld image.)
Q. What is the strongest known generalization of this statement to higher dimensions?
I.e., $\mathbb{R}^d$ ...
- 151k
24
votes
1
answer
1k
views
Which unfoldings of the $d$-dimensional hypercube tile $(d{-}1)$-space?
A six year old question,
Which unfoldings of the hypercube tile $3$-space?, has just been answered by
Moritz Firsching:
All $261$ unfoldings tile space!
So now we know:
For $d=2$, the unfolding of ...
- 151k
15
votes
1
answer
528
views
Dividing a polyhedron into two similar copies
The paper Dividing a polygon into two similar polygons proves that there are only three families of polygons that are irrep-2-tiles (can be subdivided into similar copies of the original).
Right ...
9
votes
1
answer
282
views
Thinnest covering of the plane by regular pentagons
Q. Is it known what is the thinnest covering of the infinite plane by regular pentagons?
By covering I mean every point of the plane is covered.
By thinnest I mean the proportion of the plane covered ...
- 151k