Generalized Flatland

Abstract

With illustrations by the author A Hexagon* Based on the gospel of GENERALITY as proclaimed by the POLYGONS ~ ost of my readers will be familiar with the sad story of my grandfather, an honourable square and eminent mathematician of FLAT-LAND who was condemned to lifelong imprisonment for claiming to have been abducted to SPACELAND, a world somewhere "out there" that extends our two-dimensional FLATLAND by a third dimension. Of course nobody, not even I, his grandson (a hexagon), believed in his story until, on the eve of the new millennium, I myself was abducted to GENERALIZED FLATLAND. I discovered that this world extends our flat world and the worlds of graphs and projective planes in a completely natural manner. As our world is populated by polygons such as triangles, quadrangles/squares, pentagons, etc., this extension of our world contains generalized polygons, both us simple ones and much more complicated ones of breathtaldng abstract beauty. I also found that GENERALIZED FLATLAND coincides with the land of mathematical buildings of rank 2 as conceived by one of our foremost mathematicians J. Tits. This means that all non-trivial mathematical buildings are made up of natives of this mysterious land.Preface I will tell you my story and, as evidence of my claims, show you drawings of my abductors, the four smallest natives of proper GENERALIZED FLATLAND. These drawings are extensions of beautiful renderings of closely related highly homogeneous graphs such as the complete graph on four vertices, the Petersen graph, and the Coxeter graph (Fig. 1). In fact, closer inspection discloses that my abductors share many of the remarkable properties of these graphs and are even more symmetric than the graphs they extend. I hope that the overwhelming evidence I have compiled will convince even the most sceptical among you that there is really life "out there" beyond FLATLAND, and that we are able, and have an obligation, to claim our rightful place in full GENERALITY. A Painting in the SandIt was the last day of our 2000th year. I spent this all-important day at the site of some recently discovered ruins in the desert of OZ. After unearthing some mysterious mathematical writings and drawings in the ruins they were excavating at the time, the archaeologists in charge had in-*Dedicated to my dear grandfather Edwin E.

Three-qubit operators, the split Cayley hexagon of order two, and black holes

“…Having employed the basic relations of incidence (aka up-to-a-sign multiplications producing triples of pairwise commuting operators), we have thus managed to establish the correspondence between 35 points of the Hexagon and 35 symmetric combinations on the one hand, and between 21 H1-lines and 21 triples of symmetric combinations on the other hand. As a consistency check, one readily sees that the : A diagrammatic illustration of the structure of the split Cayley hexagon of order two (after [18]- [20]) and how this geometry grasps the core features of the algebra of the real three-qubit Pauli matrices by bijectively associating the latter with the points of the hexagon. The points themselves are represented by circles whose interior is colored in nine different ways reflecting the nine orbits of an automorphism of order seven; the lines are drawn as black, light and dark grey curves/joints.…”

“…One further sees that the 14 lines we have just described are of the point/line/flag type (e. g., the triple {XXI,ZZI,Y Y I}). These lines are called H1-lines [20], or lines of Coxeter type [27]. We have altogether 21 such lines (represented in Fig.…”

Проективные Кривые Над Кольцом, Включающие В Себя Два-Кубиты

“…[16]- [20]). Обобщенный четырехугольник, ассоциированный с нашими наблюдаемыми, имеет порядок два, т.е.…”

“…2 справа) (см. [19], [20]). Пять точек овоида отвечают пяти взаимно удаленным точкам на P 1 GF (4) и, соответственно, пяти (т.е.…”

“…Геометрическая гиперплос-кость H в конечной геометрии представляет собой набор точек такой, что каждая линия геометрии или содержит в точности одну точку из H, или полностью содер-жится в H [20], [22]. Нетрудно проверить, что для обобщенного четырехугольника порядка два гиперплоскость H может принадлежать одному из следующих трех классов [22]: 1) H ov -овоид (имеется шесть таких гиперплоскостей); 2) H cl (X) -множество точек, соединенных с данной точкой X, включая и саму эту точку (име-ется 15 таких гиперплоскостей); 3) H gr -решетка, определенная выше (имеется 10 таких гиперплоскостей).…”