>>fortunately in sketch there wasn't a mathematician present... The solution *would* work (at least "on paper"), though probably not doable (in a practical sense) -- use more than three spatial dimensions, use non-Euclidean space, utilize relativistic effects, etc.
(I suppose we've all heard the joke in which various experts propose a solution to a problem [increase milk production], and it ends with the punchline where the mathematician starts his proposal with "Consider a spherical cow...")

Since invisible lines were explicitly permitted at the start of the clip: for seven red lines all perpendicular to each other I give you two perpendicular visible red lines and five invisible ones that I assure you are all perpendicular to each other in the invisible red dimensions they occupy. If there ever is a need to render the red lines in blue or any other colored ink, for a small additional fee I can ensure that the invisible perpendicularity is adjusted to suit. Surely the result will bring visible and invisible business flooding in the door.