Fair partitions of the plane into incongruent pentagons

TL;DR

This paper demonstrates that the entire Euclidean plane can be partitioned into convex pentagons that are all different from each other but share the same area and perimeter, addressing a longstanding geometric question.

Contribution

It introduces a novel method for partitioning the plane into incongruent convex pentagons with equal area and perimeter, solving a problem posed by R. Nandakumar.

Findings

Plane can be dissected into mutually incongruent convex pentagons

All pentagons have the same area and perimeter

Addresses a longstanding open problem in geometric dissections

Abstract

Motivated by a question of R.\ Nandakumar, we show that the Euclidean plane can be dissected into mutually incongruent convex pentagons of the same area and the same perimeter.