0
votes
1answer
48 views

How do 3 points define a plane?

I was solving a combinatorics problem which asked me to find the number of planes that can be constructed from a set of 25 points such that no 4 points in the set of 25 points are co-planar and then I ...
0
votes
0answers
50 views

How to establish this result using induction?

A point $(x,y)$ in the plane is called a lattice point if both coordinates $x$ and $y$ are integers. Let $P$ be a polygon whose vertices are lattice points. Then the area of $P$ is $I + \frac{1}{2}B ...
0
votes
1answer
16 views

Number of Lines formed by joining points formed by the intersection of lines in a plane

There are $\mathbf{n}$ lines in a plane no two of which are parallel. They intersect at $^{n}C_2$ distinct points in space. How many new lines are constructed by joining these points. The essential ...
0
votes
0answers
35 views

Average Degree of a Random Geometric Graph

A set of $N$ points are distributed randomly on a unit square with uniform distribution. Two points $\mathbf{p}_i$ and $\mathbf{p}_j$ are said to be connected if $\|\mathbf{p}_i - \mathbf{p}_j\| \leq ...
4
votes
1answer
124 views

Proof of “Japanese Theorem” — Triangulation of Cyclic Polygon

On Mathoverflow, I saw this great result on the "Japanese Theorem". “Japanese Theorem” on cyclic polygons: Higher-dimensional generalizations? Given triangulation of a cyclic polygon, the sum of ...
0
votes
1answer
65 views

What class am I most prepared for?

I've only taken up to calc 3, discrete, and linear algebra. Which course am I most prepared for? I'm going to be taking differential equations and advanced calc, but I want to take a 3rd class. I can ...
0
votes
4answers
127 views

Maximizing the Magnitude of the Resultant Vector

Given a set of $n$ two-dimensional unit vectors: $\left\{ \mathbf{v}_1, \dots, \mathbf{v}_n \right\}$, I want to find the coefficients $\left\{ \alpha_1, \dots, \alpha_n \right\}$, $0 \leq \alpha_i ...
4
votes
0answers
285 views

Counting simple quadrilaterals in a rectangular lattice.

I've been trying to make an algorithm to find the number of all possible simple quadrilaterals in a N*M lattice. I already have a brute force solution but since this is a Project Euler problem I ...
4
votes
1answer
87 views

Covering all the edges of a hypercube?

Consider an arbitrary $n$- dimensional hypercube: If we select $n - 1$ corners of that hypercube and highlight all $(n - 2)$ dimensional elements that originate from each of the corners is it ...
8
votes
1answer
167 views

combinatorial geometry: covering a square

I'm stuck with this problem. can anyone help me? A finite collection of squares has total area 4. show that they can be arranged to cover a square of side 1.
6
votes
2answers
248 views

What is a natural way to enumerate the symmetries of a cube?

I want to define a labeling, by small natural numbers, of the 48 symmetries of a cube — affine transformations which do not change the volume it occupies. What is a ...
2
votes
1answer
89 views

Arrangements of congruent rectangles

I have stumbled on an interesting problem. How many congruent rectangles on a plane can be arranged in such a way that they all touch each other but never overlap? I pondered this problem for a few ...
4
votes
3answers
433 views

Dissecting a square into congruent pieces that all touch the centre

Edited for clarity: I thought I had a complete set of solutions to this: Cut a square into identical pieces so that they all touch the center point. It became clear, after some discussions, ...
1
vote
0answers
102 views

Complexity of Counting the number of inducing $n$-gons

Definition: A $n$-gon is simple if it has no self intersection and is in general position if no pair of its edges are parallel. It is clear that by extending the edges of each simple $n$-gon in ...
3
votes
2answers
451 views

Computing the moments of a triangle

Assume you have a triangle in the plane defined by its three vertices at $(x_0,y_0)$, $(x_1,y_1)$, and $(x_2,y_2)$. Is there a general expression for the moments of the triangle, where the moments are ...