Question 8
If $P$, $Q$ are points in space the point $[PQ]$ is the point on the line $PQ$ on the opposite side of $Q$ to $P$ and the same distance from $Q$.
$K_0$ is a set of points in space. Given $K_n$ we derive $K_{n+1}$ by adjoining all the points $[PQ]$ with $P$ and $Q$ in $K_n$.
(1) $K_0$ contains just two points $A$ and $B$, a distance $1$ apart, what is the smallest $n$ for which $K_n$ contains a point whose distance from $A$ is at least $1000$?
(2) $K_0$ consists of three points, each pair a distance $1$ apart, find the area of the smallest convex polygon containing $K_n$.
(3) $K_0$ consists of four points, forming a regular tetrahedron with volume $1$. Let $H_n$ be the smallest convex polyhedron containing $K_n$. How many faces does $H_1$ have? What is the volume of $H_n$?
$K_0$ is a set of points in space. Given $K_n$ we derive $K_{n+1}$ by adjoining all the points $[PQ]$ with $P$ and $Q$ in $K_n$.
(1) $K_0$ contains just two points $A$ and $B$, a distance $1$ apart, what is the smallest $n$ for which $K_n$ contains a point whose distance from $A$ is at least $1000$?
(2) $K_0$ consists of three points, each pair a distance $1$ apart, find the area of the smallest convex polygon containing $K_n$.
(3) $K_0$ consists of four points, forming a regular tetrahedron with volume $1$. Let $H_n$ be the smallest convex polyhedron containing $K_n$. How many faces does $H_1$ have? What is the volume of $H_n$?