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In Plane Geometry and the Regular Heptagon Advances in Pure Mathematics, 2021, 11, 63-100 https://www.scirp.org/journal/apm ISSN Online: 2160-0384 ISSN Print: 2160-0368 Delight and Frustration with Number “Seven” in Plane Geometry and the Regular Heptagon A. Wünsche Institut für Physik, Humboldt-Universität, Berlin, Germany (Formerly) How to cite this paper: Wünsche, A. (2021) Abstract Delight and Frustration with Number “Sev- en” in Plane Geometry and the Regular As starting point for patterns with seven-fold symmetry, we investigate the Heptagon. Advances in Pure Mathematics, basic possibility to construct the regular heptagon by bicompasses and ruler. 11, 63-100. To cover the whole plane with elements of sevenfold symmetry is only possi- https://doi.org/10.4236/apm.2021.111005 ble by overlaps and (or) gaps between the building stones. Resecting small Received: November 9, 2020 parts of overlaps and filling gaps between the heptagons, one may come to Accepted: January 25, 2021 simple parqueting with only a few kinds of basic tiles related to sevenfold Published: January 28, 2021 symmetry. This is appropriate for parqueting with a center of seven-fold symmetry that is illustrated by figures. Choosing from the basic patterns Copyright © 2021 by author(s) and Scientific Research Publishing Inc. with sevenfold symmetry small parts as elementary stripes or elementary This work is licensed under the Creative cells, one may form by their discrete translation in one or two different di- Commons Attribution International rections periodic bordures or tessellation of the whole plane but the seven- License (CC BY 4.0). fold point-group symmetry of the whole plane is then lost and there re- http://creativecommons.org/licenses/by/4.0/ mains only such symmetry in small neighborhoods around one or more Open Access 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. Keywords Bicompasses and Ruler Construction, Regular Heptagon, Regular and Semi-Regular Star-Heptagons, Point-Group Symmetry C7 and C7v, Parqueting, Tiling, Tessellation, Penrose Tiles, Symmetry and Antisymmetry, Magnetic and Non-Magnetic Classes, Time Inversion, Color Groups 1. Introduction Plane geometry and number theory are considered as the oldest disciplines of mathematics where the historical roots blur in ancient times. Most knowledge DOI: 10.4236/apm.2021.111005 Jan. 28, 2021 63 Advances in Pure Mathematics A. Wünsche from ancient mathematics is handed down to modern time by the 13 books of Euclid’s “Elements” (for example, Stillwell [1], Maor [2] from the many repre- sentations of history of mathematics). Geometric constructions with compass and ruler fascinated professional mathematicians and layman to every time. A particularly interesting special case is constructions of regular polygons (n-gons) and it is known since ancient time that beside the square and the regular triangle ( n = 3 ), the regular pentagon ( n = 5 ) and their combination to the regular pen- tadecagon ( n =⋅=3 5 15 ) are constructible in such way and, furthermore, all n-gons which arise from them by multiple bisections of the central inner angles of the basic isosceles triangles (i.e., nm=⋅32,52,352,mm ⋅ ⋅⋅ m( = 0,1,2,) . The young Gauss found in 1796 that in addition the n-gons are constructible by 2h compass and ruler if n is a prime Fermi number nF=≡+h 21 (plus possible bisections of the inner angles). From the many possible references we cite here Stillwell [1] because he mentions with citation on p. 512 of the original paper that Wantzel in 1837 finally proved that this is not only sufficient but that it is basically necessary for such constructibility that n is a product of different prime Fermi numbers and completes in this way the insight of Gauss that, apparently, is not very well known up to now (see also Maor and Jost [3], p. 76, for full name and life data of Wantzel). From other possible references, we cite here the very interesting work of Conway and Guy [4] and the nice booklet of Sutton [5]. For 22 h = 2 one finds the Fermi number F2 =2 += 1 17 that leads to the famous constructibility of the regular 17-gon, the very new case not known in ancient time. In addition to cases resting on the prime Fermi numbers, clearly, the square and all its cases obtained by bisection of the central inner angles are con- structible by compass and ruler (i.e., nm=⋅=2 2m , 0,1,2, ). Formally, the last corresponds to Fermi numbers Fh with h → −∞ (2-gon or di-gon) but then it is not clear whether or not are there geometric objects which in some sense correspond to the (irrational) Fermi numbers with finite negative integers −∞ <h < 0 . The n-gon with the lowest number n ≥ 3 which is not constructible by compass and ruler is the regular heptagon to number n = 7 . The prime number “Seven” as symmetry in nature and art is very seldom realized (see Section be- fore “Conclusion”) but it plays some role in mystics. We mention here only the very old myth that our world was created in seven days and everybody will find many other things even from daily life which more or less arbitrarily were re- lated by men with the number “Seven”. The regular heptagon can be constructed by a so-called “neusis” construction [4] [5] [6] which is not fully in the spirit of constructions by compass and ruler. It was shown in [7] [8] that a regular heptagon may be constructed by an in- strument which was called “rhombic bicompasses” and consists of two com- passes which are connected by movable arms of equal lengths (at least three arms altogether) and with fixed endpoints in their action. In this way the points on the two different circles which can be drawn by such an instrument are cor- related (see Section 2). It seems to us that such constructions are more in the DOI: 10.4236/apm.2021.111005 64 Advances in Pure Mathematics A. Wünsche spirit of constructions by compass and ruler than neusis constructions but one has to wait whether or not this will be accepted by the community of mathema- ticians. In combination with usual compasses the construction of much more regular n-gons becomes possible than without it. A technical realization of rhombic bicompasses, in particular with the possibility of variable arm lengths, would be interesting. The regular heptagon and also the two possible regular star-heptagons possess the point-group symmetry Cm7v 7 with 14 symmetry elements given in Schoenflies and in International notation (m stands here for 'mirror' symmetry) used by physicists. This group characterizes a relatively simple symmetry and may serve as starting point for some considerations to extensions. The pure ro- tation group C7 7 with 7 elements of rotations and the mirror group CCsv≡ 1 are the only possible nontrivial subgroups of C7v where C7v may be considered in its structure as built from the subgroup C7 of C14 where the coset (CC14− 7 ) to its subgroup C7 is multiplied by the “simplest element of anti- symmetry” that is the reflection at a line in the plane. This provides the oppor- tunity to consider groups with and without general anti-symmetries which be- came important in physics, for example, as magnetic groups and non-magnetic groups if this element of anti-symmetry is the inversion of time. However, the extensions of the point groups C7 or C7v to crystallographic groups is as it is well known not possible that restricts their possibilities. Furthermore, one may dis- cuss possible realizations of C7v and C7 for parqueting and tessellations of planes and this goes in direction of one- and two-dimensional reliefs and tiling and art-work and also to realizations in nature. The number of contributions to symmetry and to discrete groups and of their extension to anti-symmetries and color groups and to application in physics, in particular, in crystallography is really enormous and their notations are often very different in physics and mathematics. An early classical book about sym- metries in art and nature is that of H. Weyl [9]1 and later to parts the book of Steinhaus [10]. One of my earliest books about regular and semi-regular sym- metries was that of Lyusternik [11] with general propositions about convex fig- ures and with the regular and semi-regular Archimedean polyhedrons. The more extensive book of Fejes Tóth [12] contains also mathematical propositions to two-dimensional figures and three-dimensional objects and in addition good-quality reproductions to plane figures and red-green stereo-spectacles for spatially seeing three-dimensional geometrical objects. Next, we have to mention the Russian school (it seems that one may speak about such) with roots, in particular, from E. S. Fyodorov who in 1890 (and in- dependently Schoenflies in 1891) found the crystallographic spatial groups and later from Shubnikov who contributed to symmetries and anti-symmetries with applications in physics where they play an important rule (e.g., [13]). The most important of the many journal papers of Shubnikov are collected in the work [14] 1Translations of books from Western countries into Russian were welcome since up to the turn in GDR in 1990 it was impossible for me to buy them in original. DOI: 10.4236/apm.2021.111005 65 Advances in Pure Mathematics A. Wünsche which comprises as well physical as also mathematical contributions to symme- try groups with particularly beautiful illustrations to them and tabular material.
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