**5**

votes

**2**answers

505 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 ...

**30**

votes

**2**answers

1k views

### What is the oriented Fano plane?

One way to remember the multiplication table of the octonions is to use the following diagram (which I got from John Baez's online paper): if $(e_i,e_j,e_k)$ is one of the lines listed according to ...

**14**

votes

**3**answers

1k views

### What is the minimum N for which there exist N points in the plane that cannot be covered by any number of non-overlapping closed unit discs?

This problem was posed in March 2010 at G4G9 in a talk by the Japanese mathematician Hirokazu "Iwahiro" Iwasawa. He claims there is a simple proof that N > 10, though he did not share it with the ...

**2**

votes

**0**answers

185 views

### Is there any symmetric unshellable triangulation of a tetrahedron?

M. Rudin gave a triangulation of a tetrahedron with the property that after any small tetrahedron removed, the remaining part is not homeomorphic to a ball (An unshellable triangulation of a ...

**16**

votes

**3**answers

2k 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 $\varepsilon$ ...

**68**

votes

**5**answers

5k 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 ...

**47**

votes

**3**answers

4k views

### What is the status of the Gauss Circle Problem?

For $r > 0$, let $L(r) = \# \{ (x,y) \in \mathbb{Z}^2 \ | \ x^2 + y^2 \leq r^2\}$ be the number of lattice points lying on or inside the standard circle of radius $r$. It is easy to see that $L(r) ...

**13**

votes

**1**answer

1k views

### Is it possible to dissect a disk into congruent pieces, so that a neighborhood of the origin is contained within a single piece?

Problem: is it possible to dissect the interior of a circle into a finite number of congruent pieces (mirror images are fine) such that some neighbourhood of the origin is contained in just one of ...

**8**

votes

**3**answers

9k 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.

**5**

votes

**4**answers

996 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 ...

**15**

votes

**1**answer

752 views

### Are there analogues of Desargues and Pappus for block designs?

Finite projective planes are fascinating objects from many perspectives. In addition to the geometric view, they can be viewed as combinatorial block designs.
From the geometric perspective, there ...

**8**

votes

**6**answers

911 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 ...

**4**

votes

**4**answers

1k views

### Number of spanning trees in a grid

Given a $\sqrt{n}\times\sqrt{n}$ piece of the integer $\mathbb{Z}^2$ grid, define a graph by joining any two of these points at unit distance apart. How many spanning trees does this graph have ...

**3**

votes

**1**answer

335 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 ...

**19**

votes

**2**answers

887 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 ...

**25**

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 ...

**13**

votes

**2**answers

1k views

### The Sylvester Gallai Theorem and Sections of Varieties with “Simple Topology”.

The Sylvester-Gallai theorem asserts that for every collection of points in the plane, not all on a line, there is a line containing exactly two of the points.
One high dimensional extension ...

**40**

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 ...

**8**

votes

**4**answers

411 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 ...

**22**

votes

**8**answers

2k views

### Points and lines in the plane

Does a positive real number $k\geq1$ exist such that for every finite set $P$ of points in the plane (with the property that no three points of $P$ lie on a common line and $|P|\geq3$), one can choose ...