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Is it possible to construct a finite nontrivial arrangement of points, lines, and planes in 3-dimensional Euclidean space with the following properties?

  • every line is incident with four points and four planes
  • for every point incident to a plane (of which there is at least one), exactly four lines are both incident to that plane and that point
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  • $\begingroup$ You are asking for every line to be incident with $a$ points, every line to be incident with $b$ planes, and every point on a plane to be jointly incident with exactly $c$ lines, for $(a,b,c)=(4,4,4)$. Do you have solutions for $(a,b,c)=(4,4,3)$ or $(a,b,c)=(4,3,4)$ or $(a,b,c)=(3,4,4)$? $\endgroup$
    – user44143
    May 24, 2021 at 21:17
  • $\begingroup$ No, I’m only interested in cases where all three are equal and even. $\endgroup$ May 24, 2021 at 21:31
  • $\begingroup$ Do you care whether the arrangement is finite or infinite? $\endgroup$
    – M. Winter
    May 25, 2021 at 13:57
  • $\begingroup$ I want it to be finite. $\endgroup$ May 25, 2021 at 17:44
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    $\begingroup$ Does the existence of such a non-trivial arrangement imply a $(4,4)$-configuration in the Euclidean plane? I know Grünbaum-RIgby configuration with this property. $\endgroup$
    – Hao Chen
    May 26, 2021 at 5:34

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