# Intersection between hyperplane within a simplex

1. Aug 28, 2014

### Debashis

Hi,

If we contract a (n-1)d hyperplane with a n-simplex, then what is maximum number of intersection points with the egdes of the simplex and the hyperplane ?

For, if we draw a line within a 2-simplex (there are 3 edges), it will have a intersection of maximum two edges. For 3-simplex, any plane within the tetrahedron (where there are 6 egdes) can have maximum intersection with 4 edges. What is the value for n-simplex ?

2. Aug 28, 2014

### homeomorphic

The question is a bit unclear. A line can intersect the edges of a 2-simplex in infinitely many points if you just make the edge a subset of the line. Do you mean to ask what is the maximum number of edges that can be intersected, rather than the maximum number of points?

3. Aug 28, 2014

### Ben Niehoff

Think of this as a graph theory problem instead. An n-simplex is just a complete graph with n vertices; i.e., each vertex is connected to every other vertex. Then the question is, how should we partition a complete graph on n vertices into two subgraphs, such that the number of edges crossing between the two subgraphs is maximized.

A complete graph has $n!$ edges. If we take $k$ vertices and call them subgraph A, while the remaining $n-k$ vertices we call subgraph B. Now we want to know the number of edges that go between A and B. Since each vertex has an edge connecting to every other vertex, then the number of edges going between A and B must be given by

$$k (n-k)$$
To maximize this for a given $n$, we must clearly choose the $k$ that is closest to $n/2$. Therefore the maximum number of (isolated) points in which a hyperplane can intersect a simplex is given by

$$f(n) = \begin{cases} \frac{n^2}{4} & n \; \text{even} \\ \frac{n^2 - 1}{4} & n \; \text{odd} \end{cases}$$
which gives the expected result for $n=3$ (triangle) and $n=4$ (tetrahedron).

4. Aug 29, 2014

### Debashis

thank you very much for the answer. How can we represent that hyperplane in the respective graph of n-vertices ?

Last edited: Aug 29, 2014