Related papers: From the Icosahedron to E8
We study well-rounded ideal lattices from totally definite quaternion algebras. We prove existence and classification results, and illustrate our methods with examples.
It is shown that the $M$-algebra related with the $M$ theory comes in two variants. Besides the standard $M$ algebra based on the real structure, an alternative octonionic formulation can be consistently introduced. This second variant has…
The satisfactory development of Quaternionic Analysis has indicated new solutions for physical and mathematical problems. It is worth mentioning the fact that quaternions possess four dimensions, and in this way they may be considered as…
This work rests upon the certainty that only fields of real and complex numbers, quaternions and octonions have algebras of all four arithmetical operations. Also quaternions are good to represent 3-dimensional Euclid space and…
Following a previous proposition of quaternity spacetime for electronic orbitals in neon shell, this paper describes the geometrical course each electron takes as it oscillates harmonically within a certain quaternity space dimension and…
In this paper, we show how regular convex 4-polytopes - the analogues of the Platonic solids in four dimensions - can be constructed from three-dimensional considerations concerning the Platonic solids alone. Via the Cartan-Dieudonne…
We construct the fcc (face centered cubic), bcc (body centered cubic) and sc (simple cubic) lattices as the root and the weight lattices of the affine Coxeter groups W(D3) and W(B3)=Aut(D3). The rank-3 Coxeter-Weyl groups describing the…
There exist as many index-$k$ sublattices of the hexagonal lattice up to isometry as there exist lattice triangles with normalized volume $k$ up to unimodular equivalence, which can be explained using orbifolds. In dimension 3, it was noted…
Similar sublattices of the root lattice $A_4$ are possible, according to a result of Conway, Rains and Sloane, for each index that is the square of a non-zero integer of the form $m^2 + mn - n^2$. Here, we add a constructive approach, based…
In contrast to many known results concerning periodic tilings of the Euclidean plane with pentagons, here tilings with rotational symmetry are investigated. A certain class of convex pentagons is introduced. It can be shown that for any…
We discuss the local and global problems for the equivalence of geometric structures of an arbitrary order and, in later sections, attention is given to what really matters, namely the equivalence with respect to transformations belonging…
A quantum spin-1/2, and its associated su(2) algebra of Pauli spin matrices are familiarly linked to Clifford algebra and quaternions. Somewhat more loosely, we develop connections between the su(4) algebra of two spins and of its…
The idea of black-hole lattices as models for the large-scale structure of the universe has been under scrutiny for several decades, and some of the properties of these systems have been elucidated recently in the context of the problem of…
The Leech lattice, $\Lambda_{24}$, is represented on the space of octonionic 3-vectors. It is built from two octonionic representations of $E_{8}$, and is reached via $\Lambda_{16}$. It is invariant under the octonion index cycling and…
In this paper, constructions of regular pentagon and decagon, and the calculation of the main trigonometric ratios of the corresponding central angles are approached. In this way, for didactic purposes, it is intended to show the reader…
In this paper we provide some applications of the norm form in some quaternion division algebras over rational field and we give some properties of Fibonacci sequence and Fibonacci sequence in connection with quaternion elements. We define…
We give a simple and self contained introduction to quaternions and their practical usage in dynamics. The rigid body dynamics are presented in full details. In the appendix, some more exotic relations are given that allow to write more…
We introduce a generalization of the Euclidean algorithm for rings equipped with an involution, and completely enumerate all isomorphism classes of orders over definite, rational quaternion algebras equipped with an orthogonal involution…
This is part one of a series of four methodological papers on (bi)quaternions and their use in theoretical and mathematical physics: 1- Alphabetical bibliography, 2- Analytical bibliography, 3- Notations and terminology, and 4- Formulas and…
This article uses Clifford algebra of definite signature to derive octonions and the Lie exceptional algebra G2 from calibrations using Pin(7). This is simpler than the usual exterior algebra derivation and uncovers a subalgebra of Spin(7)…