Omittable Planes

  • Branko Grünbaum
  • Jonathan Lenchner


In analogy to omittable lines in the plane, we initiate the study of omittable planes in $3$-space. Given a collection of $n$ planes in real projective $3$-space, a plane $\Pi$ is said to be omittable if $\Pi$ is free of ordinary lines of intersection – in other words, if all the lines of intersection of $\Pi$ with other planes from the collection come at the intersection of three or more planes. We provide two infinite families of planes yielding omittable planes in either a pencil or near-pencil, together with examples having between three and seven omittable planes, examples that we call "sporadic," which do not fit into either of the two infinite families.

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