Octadécagone

In geometry, an octadecagon (or octakaidecagon) or 18-gon is an eighteen-sided polygon. A regular octadecagon has a Schläfli symbol {18} and can be constructed as a quasiregular truncated enneagon, t{9}, which alternates two types of edges. As 18 = 2 × 32, a regular octadecagon cannot be constructed using a compass and straightedge. However, it is constructible using neusis, or an angle trisection with a tomahawk. The following approximate construction is very similar to that of the enneagon, as an octadecagon can be constructed as a truncated enneagon. It is also feasible with exclusive use of compass and straightedge. The regular octadecagon has Dih18 symmetry, order 36. There are 5 subgroup dihedral symmetries: Dih9, (Dih6, Dih3), and (Dih2 Dih1), and 6 cyclic group symmetries: (Z18, Z9), (Z6, Z3), and (Z2, Z1). These 15 symmetries can be seen in 12 distinct symmetries on the octadecagon. John Conway labels these by a letter and group order. Full symmetry of the regular form is r36 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g18 subgroup has no degrees of freedom but can seen as directed edges. Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular octadecagon, m=9, and it can be divided into 36: 4 sets of 9 rhombs. This decomposition is based on a Petrie polygon projection of a 9-cube, with 36 of 4608 faces. The list enumerates the number of solutions as 112018190, including up to 18-fold rotations and chiral forms in reflection.

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2 31 polytope

DISPLAYTITLE:2 31 polytope In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group. Its Coxeter symbol is 231, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node branch. The rectified 231 is constructed by points at the mid-edges of the 231. These polytopes are part of a family of 127 (or 27−1) convex uniform polytopes in 7-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

3 21 polytope

DISPLAYTITLE:3 21 polytope In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 7-ic semi-regular figure. Its Coxeter symbol is 321, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 3-node sequences. The rectified 321 is constructed by points at the mid-edges of the 321. The birectified 321 is constructed by points at the triangle face centers of the 321.

Neusis

La neusis (du grec ancien νεῦσις venant de νεύειν neuein « pencher vers »; pluriel : νεύσεις neuseis) est une méthode de construction géométrique utilisée dans l'Antiquité par les mathématiciens grecs dans des cas où les constructions à la règle et au compas étaient impossibles. La construction par neusis consiste à placer un segment de longueur fixée a entre deux courbes données l et m, de telle sorte que la droite support du segment passe par un point fixé P.

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The Spindle Assembly Abnormal Protein 6 (SAS-6) forms dimers, which then self-assemble into rings that are critical for the nine-fold symmetry of the centriole organelle. It has recently been shown experimentally that the self-assembly of SAS-6 rings is st ...

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Integrative approach to analyze the proximal region of the Trichonympha agilis centriole

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The centrosome, the major microtubule organizing center in animal cells, is composed of two orthogonally arranged centrioles and surrounding pericentriolar material. Centrioles are evolutionary conserved organelles characterized by a nine-fold symmetry of ...

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SAS-6 engineering reveals interdependence between cartwheel and microtubules in determining centriole architecture

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