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### Centralizing four red vectors in six green sectors

Four red vectors are given, one per quadrant, $[0,90^\circ)$,
$[90^\circ,180^\circ)$, etc.
A rigid star of six green vectors separated by $60^\circ$
can be positioned at
$(\theta,
\theta+60^\circ,
...

**4**

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**2**answers

284 views

### Characterization of combinatorial manifolds in terms of links

I need to reference the following result. Do you know a good source?
The following conditions on an $n$-dimensional simplicial complex $S$ are equivalent:
a) $S$ is an $n$ manifold;
b) The link of ...

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**3**answers

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### Why is complex projective space triangulable?

In an exercise in his algebraic topology book, Munkres asserts that $\mathbf{C}P^n$ is triangulable (i.e., there is a simplicial complex $K$ and a homeomorphism $|K| \rightarrow \mathbf{C}P^n$). Can ...

**4**

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**1**answer

687 views

### The Join of Simplicial Sets ~Finale~

Background
Let $X$ and $S$ be simplicial sets, i.e. presheaves on $\Delta$, the so-called topologist's simplex category, which is the category of finite nonempty ordinals with morphisms given by ...

**2**

votes

**1**answer

186 views

### Are combinatorial configurations whose Levi graphs may be represented as covering graphs over voltage graphs realizable with pseudolines?

This question is related to this previous question. Many combinatorial configurations have Levi graphs which may be represented as derived graphs obtained from voltage graphs over a cyclic group; in a ...

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votes

**0**answers

254 views

### Drawing a combinatorial 3-configuration of points and lines with pseudolines

This question is related to the question of drawing a combinatorial 3-configuration of points and lines with straight lines. We only relax the condition and admit drawings with pseudolines. Let us ...

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votes

**2**answers

593 views

### Drawing 3-configurations of points and lines with straight lines

It is well-known that the black-and-white coloring of the Heawood graph on 14 vertices determines a combinatorial 3-configuration with 7 "points" and 7 "lines", known as Fano plane. Similarly, any ...

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**3**answers

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### Is the ratio Perimeter/Area for a finite union of unit squares at most 4?

Update: As I have just learned, this is called Keleti's perimeter area conjecture.
Prove that if H is the union of a finite number of unit squares in the plane, then the ratio of the perimeter and ...

**29**

votes

**4**answers

1k views

### Tiling A Rectangle With A Hint of Magic

Here's a a famous problem:
If a rectangle $R$ is tiled by rectangles $T_i$ each of which has at least one integer sidelength, then the tiled rectangle $R$ has at least one integer side length.
...

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**3**answers

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### Cutting a rectangle into an odd number of congruent pieces

We are interested in tiling a rectangle with copies of single tile (rotations and reflexions are allowed). This is very easy to do, by cutting the rectangle into smaller rectangles.
What happens when ...

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votes

**3**answers

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### Are there infinite sets of stellations of polyhedra?

Lists of stellations of polyhedrons are given particular rules like in the book The Fifty Nine Icosahedra which follows "Miller's Rules".
There seems to be no "correct" ruleset to use, so more ...