Constructions for large spatial point-line (nk) congurations

Gábor Gévay

Abstract


Highly symmetric figures, such as regular polytopes, can serve as a scaffolding on which spatial (nk) point-line configurations can be built. We give several constructions using this method in dimension 3 and 4. We also explore possible constructions of point-line configurations obtained as Cartesian products of smaller ones. Using suitable powers of well-chosen configurations, we obtain infinite series of (nk) configurations for which both n and k are arbitrarily large. We also combine the method of polytopal scaffolding and the method of powers to construct further examples. Finally, we formulate an incidence statement concerning a (1004) configuration in 3-space derived from the product of two complete pentalaterals; it is posed as a conjecture.

Keywords


spatial conguration, Platonic solid, regular 4-polytope, product of congurations, incidence statement

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DOI: https://doi.org/10.26493/1855-3974.270.daa

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications