Abstract
As starting point for patterns with seven-fold symmetry, we investigate the basic possibility to construct the regular heptagon by bicompasses and ruler. To cover the whole plane with elements of sevenfold symmetry is only possible by overlaps and (or) gaps between the building stones. Resecting small parts of overlaps and filling gaps between the heptagons, one may come to simple parqueting with only a few kinds of basic tiles related to sevenfold symmetry. This is appropriate for parqueting with a center of seven-fold symmetry that is illustrated by figures. Choosing from the basic patterns with sevenfold symmetry small parts as elementary stripes or elementary cells, one may form by their discrete translation in one or two different directions periodic bordures or tessellation of the whole plane but the sevenfold point-group symmetry of the whole plane is then lost and there remains only such symmetry in small neighborhoods around one or more centers. From periodic tiling, we make the transition to aperiodic tiling of the plane. This is analogous to Penrose tiling which is mostly demonstrated with basic elements of fivefold symmetry and we show that this is also possible with elements of sevenfold symmetry. The two possible regular star-heptagons and a semi-regular star-heptagon play here a basic role.
Highlights
Plane geometry and number theory are considered as the oldest disciplines of mathematics where the historical roots blur in ancient times
This is analogous to Penrose tiling which is mostly demonstrated with basic elements of fivefold symmetry and we show that this is possible with elements of sevenfold symmetry
If we look to pictures of aperiodic tessellations by Penrose tiles in the literature, in particular, in [26] [27] [28] and the first two articles in [30], we see that here basically elements of fivefold symmetry that means that regular pentagons and semi-regular star-pentagons play a main role
Summary
How to cite this paper: Wünsche, A. (2021) Delight and Frustration with Number “Seven” in Plane Geometry and the Regular Heptagon. How to cite this paper: Wünsche, A. (2021) Delight and Frustration with Number “Seven” in Plane Geometry and the Regular Heptagon. Received: November 9, 2020 Accepted: January 25, 2021 Published: January 28, 2021
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