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Benjamin Nasmith

Publications and source records attributed to Benjamin Nasmith.

5 recordsLinked to original sources
arXiv2023

Rational Angle Sets and Tight t-Designs

Given a finite subset of a sphere or projective space, known as a design, we can compute the strength and angle set of that design. When the strength and angle set meet certain bounds, the design is called tight. Hoggar sought to prove that, aside from certain known cases, the angle sets of tight projective designs must be rational. Lyubich found a counter-example and provided a repair for Hoggar's proof but excluded the exceptional octonion projective cases. This note extends Lyubich's repair of Hoggar's proof to the remaining projective cases and extends the proof to all spherical cases. It does so by using Jordan algebra primitive idempotents to treat all of the cases simultaneously. We thereby confirm that tight spherical and projective designs have rational angle sets except in specific cases.

math.CO↗
arXiv2022

Octonion Integers and Tight 5-Designs

The two strictly projective tight 5-designs are the lines spanned by the short vectors of the Leech lattice and a set of points in the octonion projective plane that define a generalized hexagon of order (2,8). A previous paper introduced a common construction that can generate these two tight 5-designs. This paper describes the same construction in terms of octonion arithmetic. An octonion integer construction of the Leech lattice is described using properties of the octonion integers taken modulo 2. The Leech lattice automorphism group is constructed from octonion reflections. The common construction and the Suzuki subgroup chain of Leech lattice automorphisms are described in terms of octonion integers.

math.CO↗
arXiv2020

Octonions and the two strictly projective tight 5-designs

In addition to the vertices of the regular hexagon and icosahedron, there are precisely two strictly projective tight 5-designs: one constructed from the short vectors of the Leech lattice and the other corresponding to a generalized hexagon structure in the octonion projective plane. This paper describes a new connection between these two strictly projective tight 5-designs -- a common construction using octonions. Certain octonion involutionary matrices act on a three-dimensional octonion vector space to produce the first 5-design and these same matrices act on the octonion projective plane to produce the second 5-design. This result uses the octonion construction of the Leech lattice due to Robert Wilson and provides a new link between the generalized hexagon Gh(2,8) and the Leech lattice.

math.CO↗
arXiv2013

Classical Electrodynamics and Absolute Simultaneity

Maxwell's equations and the Lorentz force density are expressed using an alternative simultaneity gauge. As a result, they describe electrodynamics for an observer travelling with a constant velocity through an isotropic medium. If desired, one may regard the isotropic medium as an absolute space. An observer at rest in the medium may be referred to as a preferred observer. The medium appears electrically polarized to the moving observer. Such an observer regards the speed of light as anisotropic by convention while sharing absolute simultaneity relations with the preferred observer.

physics.hist-ph↗