# Maximum of Three Mutually Perpendicular Lines in Ordinary Space

## Theorem

In ordinary space, there can be no more than $3$ straight lines which are pairwise perpendicular.

Thus, in a configuration of $4$ straight lines in space, at least one pair will not be perpendicular to each other.

## Proof

By assumption and popular belief, ordinary space (on the measurable local level) is an instance of a $3$-dimensional Euclidean space.

Hence the results of Euclid's *The Elements* can be applied.

Let there be $4$ straight lines $a$, $b$, $c$ and $d$ in space.

Let $a$ be perpendicular to each of $b$, $c$ and $d$.

From Three Intersecting Lines Perpendicular to Another Line are in One Plane, $b$, $c$ and $d$ are all in the same plane.

Let $b$ be perpendicular to $c$ and $d$.

Either $c$ and $d$ are the same line or not.

If $c = d$ then $c$ is not perpendicular to $d$.

Otherwise, from Equal Corresponding Angles implies Parallel Lines, $c$ is parallel to $d$.

In either case $c$ is not perpendicular to $d$.

Hence the result.

$\blacksquare$

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $3$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $3$