The birth of
            E
            8
            out of the spinors of the icosahedron

E 8 is prominent in mathematics and theoretical physics, and is generally viewed as an exceptional symmetry in an eight-dimensional (8D) space very different from the space we inhabit; for instance, the Lie group E 8 features heavily in 10D superstring theory. Contrary to that point of view, here we show that the E 8 root system can in fact be constructed from the icosahedron alone and can thus be viewed purely in terms of 3D geometry. The 240 roots of E 8 arise in the 8D Clifford algebra of 3D space as a double cover of the 120 elements of the icosahedral group, generated by the root system H 3 . As a by-product, by restricting to even products of root vectors (spinors) in the 4D even subalgebra of the Clifford algebra, one can show that each 3D root system induces a root system in 4D, which turn out to also be exactly the exceptional 4D root systems. The spinorial point of view explains their existence as well as their unusual automorphism groups. This spinorial approach thus in fact allows one to construct all exceptional root systems within the geometry of three dimensions, which opens up a novel interpretation of these phenomena in terms of spinorial geometry.

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An exceptionally preserved conulariid from Ordovician erratics of Northern European Lowlands

PalZ ◽

10.1007/s12542-020-00534-7 ◽

2021 ◽

Author(s):

Consuelo Sendino ◽

Martin M. Bochmann

Keyword(s):

Internal Structure ◽

Late Pleistocene ◽

Point Of View ◽

Three Dimensions ◽

Northern European ◽

Current Location ◽

Transport Direction ◽

First Time ◽

Erratic Boulder

AbstractA conulariid preserved in three dimensions from Ordovician fluvioglacial erratics of the Northern European Lowlands (North German Plain) is described under open nomenclature. It is assigned to the genus Conularia with similarities to Baltoscandian conulariids. The lithology of the erratic boulder and fauna contained in it provide important information on the origin and transport direction of the sediment preserved in a kame from the Saalian glaciation. This paper deals with the site of origin of the boulder in Baltoscandia analysing the comprised palaeofauna, from a palaeostratigraphic and palaeogeographic point of view, from its deposition in Ordovician times until its arrival at its current location in the Late Pleistocene. It also reveals for the first time the internal structure of the conulariid aperture.

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Clifford Spinors and Root System Induction: $$H_4$$ and the Grand Antiprism

Advances in Applied Clifford Algebras ◽

10.1007/s00006-021-01139-2 ◽

2021 ◽

Vol 31 (3) ◽

Author(s):

Pierre-Philippe Dechant

Keyword(s):

Root System ◽

Root Systems ◽

Invariant Sets ◽

Simple Construction ◽

Reflection Groups ◽

Computational Work ◽

Algebraic Framework ◽

Underlying Geometry ◽

Coxeter Plane ◽

Insight Into

AbstractRecent work has shown that every 3D root system allows the construction of a corresponding 4D root system via an ‘induction theorem’. In this paper, we look at the icosahedral case of $$H_3\rightarrow H_4$$ H 3 → H 4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan–Dieudonné theorem, giving a simple construction of the $${\mathrm {Pin}}$$ Pin and $${\mathrm {Spin}}$$ Spin covers. Using this connection with $$H_3$$ H 3 via the induction theorem sheds light on geometric aspects of the $$H_4$$ H 4 root system (the 600-cell) as well as other related polytopes and their symmetries, such as the famous Grand Antiprism and the snub 24-cell. The uniform construction of root systems from 3D and the uniform procedure of splitting root systems with respect to subrootsystems into separate invariant sets allows further systematic insight into the underlying geometry. All calculations are performed in the even subalgebra of $${\mathrm {Cl}}(3)$$ Cl ( 3 ) , including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes, and are shared as supplementary computational work sheets. This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework.

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FEBID 3D-Nanoprinting at Low Substrate Temperatures: Pushing the Speed While Keeping the Quality

Nanomaterials ◽

10.3390/nano11061527 ◽

2021 ◽

Vol 11 (6) ◽

pp. 1527

Author(s):

Jakob Hinum-Wagner ◽

David Kuhness ◽

Gerald Kothleitner ◽

Robert Winkler ◽

Harald Plank

Keyword(s):

Volume Growth ◽

Point Of View ◽

High Fidelity ◽

Direct Write ◽

General Picture ◽

3D Space ◽

Design Flexibility ◽

Substrate Temperatures ◽

Very High ◽

Comprehensive Discussion

High-fidelity 3D printing of nanoscale objects is an increasing relevant but challenging task. Among the few fabrication techniques, focused electron beam induced deposition (FEBID) has demonstrated its high potential due to its direct-write character, nanoscale capabilities in 3D space and a very high design flexibility. A limitation, however, is the low fabrication speed, which often restricts 3D-FEBID for the fabrication of single objects. In this study, we approach that challenge by reducing the substrate temperatures with a homemade Peltier stage and investigate the effects on Pt based 3D deposits in a temperature range of 5–30 °C. The findings reveal a volume growth rate boost up to a factor of 5.6, while the shape fidelity in 3D space is maintained. From a materials point of view, the internal nanogranular composition is practically unaffected down to 10 °C, followed by a slight grain size increase for even lower temperatures. The study is complemented by a comprehensive discussion about the growth mechanism for a more general picture. The combined findings demonstrate that FEBID on low substrate temperatures is not only much faster, but practically free of drawbacks during high fidelity 3D nanofabrication.

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A biomimetic 3D airflow sensor made of an array of two piezoelectric metal-core fibers

Sensor Review ◽

10.1108/sr-12-2016-0278 ◽

2017 ◽

Vol 37 (3) ◽

pp. 312-321 ◽

Cited By ~ 1

Author(s):

Yixiang Bian ◽

Can He ◽

Kaixuan Sun ◽

Longchao Dai ◽

Hui Shen ◽

...

Keyword(s):

Design Methodology ◽

Spherical Surface ◽

Three Dimensional ◽

Three Dimensions ◽

Piezoceramic Cylinder ◽

Metal Core ◽

Content Type ◽

3D Space ◽

Highly Sensitive ◽

Airflow Sensor

Purpose The purpose of this paper is to design and fabricate a three-dimensional (3D) bionic airflow sensing array made of two multi-electrode piezoelectric metal-core fibers (MPMFs), inspired by the structure of a cricket’s highly sensitive airflow receptor (consisting of two cerci). Design/methodology/approach A metal core was positioned at the center of an MPMF and surrounded by a hollow piezoceramic cylinder. Four thin metal films were spray-coated symmetrically on the surface of the fiber that could be used as two pairs of sensor electrodes. Findings In 3D space, four output signals of the two MPMFs arrays can form three “8”-shaped spheres. Similarly, the sensing signals for the same airflow are located on a spherical surface. Originality/value Two MPMF arrays are sufficient to detect the speed and direction of airflow in all three dimensions.

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On Closed Subsets of Root Systems

Canadian Mathematical Bulletin ◽

10.4153/cmb-1994-050-4 ◽

1994 ◽

Vol 37 (3) ◽

pp. 338-345 ◽

Cited By ~ 6

Author(s):

D. Ž. Doković ◽

P. Check ◽

J.-Y. Hée

Keyword(s):

Weyl Group ◽

Root System ◽

Irreducible Component ◽

Product Space ◽

Root Systems ◽

Inner Product Space ◽

Inner Product ◽

Closed Sets ◽

Finite Dimensional ◽

Real Inner Product Space

AbstractLet R be a root system (in the sense of Bourbaki) in a finite dimensional real inner product space V. A subset P ⊂ R is closed if α, β ∊ P and α + β ∊ R imply that α + β ∊ P. In this paper we shall classify, up to conjugacy by the Weyl group W of R, all closed sets P ⊂ R such that R\P is also closed. We also show that if θ:R —> R′ is a bijection between two root systems such that both θ and θ-1 preserve closed sets, and if R has at most one irreducible component of type A1, then θ is an isomorphism of root systems.

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Extension and Longitudinal Growth During the Development of Red Pine Root Systems

Canadian Journal of Forest Research ◽

10.1139/x75-016 ◽

1975 ◽

Vol 5 (1) ◽

pp. 109-121 ◽

Cited By ~ 29

Author(s):

D. C. F. Fayle

Keyword(s):

Root System ◽

Root Systems ◽

Red Pine ◽

Longitudinal Growth ◽

Rooting Depth ◽

Soil Conditions ◽

Moisture Availability ◽

Below Ground

Extension of the root system and stem during the first 30 years of growth of plantation-grown red pine (Pinusresinosa Ait.) on four sites was deduced by root and stem analyses. Maximum rooting depth was reached in the first decade and maximum horizontal extension of roots was virtually complete between years 15 and 20. The main horizontal roots of red pine seldom exceed 11 m in length. Elongation of vertical and horizontal roots was examined in relation to moisture availability and some physical soil conditions. The changing relations within the tree in lineal dimensions and annual elongation of the roots and stem are illustrated. The development of intertree competition above and below ground is considered.

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LARGE QUANTUM GRAVITY EFFECTS: CYLINDRICAL WAVES IN FOUR DIMENSIONS

International Journal of Modern Physics D ◽

10.1142/s0218271800000633 ◽

2000 ◽

Vol 09 (06) ◽

pp. 669-686 ◽

Cited By ~ 17

Author(s):

MARÍA E. ANGULO ◽

GUILLERMO A. MENA MARUGÁN

Keyword(s):

Quantum Gravity ◽

Symmetry Axis ◽

Point Of View ◽

Three Dimensions ◽

Asymptotic Region ◽

Four Dimensions ◽

Cylindrical Waves ◽

Significant Difference ◽

Gravity Effects ◽

Linearly Polarized

Linearly polarized cylindrical waves in four-dimensional vacuum gravity are mathematically equivalent to rotationally symmetric gravity coupled to a Maxwell (or Klein–Gordon) field in three dimensions. The quantization of this latter system was performed by Ashtekar and Pierri in a recent work. Employing that quantization, we obtain here a complete quantum theory which describes the four-dimensional geometry of the Einstein–Rosen waves. In particular, we construct regularized operators to represent the metric. It is shown that the results achieved by Ashtekar about the existence of important quantum gravity effects in the Einstein–Maxwell system at large distances from the symmetry axis continue to be valid from a four-dimensional point of view. The only significant difference is that, in order to admit an approximate classical description in the asymptotic region, states that are coherent in the Maxwell field need not contain a large number of photons anymore. We also analyze the metric fluctuations on the symmetry axis and argue that they are generally relevant for all of the coherent states.

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Covariant Physics

10.1093/oso/9780198864899.001.0001 ◽

2021 ◽

Author(s):

Moataz H. Emam

Keyword(s):

Point Of View ◽

Theoretical Physics ◽

Newtonian Mechanics ◽

Theory Of Relativity ◽

College Physics ◽

Introductory Courses ◽

Tensor Calculus ◽

Research Papers ◽

Intended Reader ◽

Theory Of Fields

This book is an introduction to the modern methods of the general theory of relativity, tensor calculus, space time geometry, the classical theory of fields, and a variety of theoretical physics oriented topics rarely discussed at the level of the intended reader (mid-college physics major). It does so from the point of view of the so-called principle of covariance; a symmetry that underlies most of physics, including such familiar branches as Newtonian mechanics and electricity and magnetism. The book is written from a minimalist perspective, providing the reader with only the most basic of notions; just enough to be able to read, and hopefully comprehend, modern research papers on these subjects. In addition, it provides a (hopefully short) preparation for the student to be able to conduct research in a variety of topics in theoretical physics; with particular emphasis on physics in curved spacetime backgrounds. The hope is that students with a minimal mathematical background in calculus and only some introductory courses in physics may be able to study this book and benefit from it.

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Effects of Temperature on Root Morphology and Ectendomycorrhizal Development in Pinusresinosa Ait.

Canadian Journal of Forest Research ◽

10.1139/x75-024 ◽

1975 ◽

Vol 5 (2) ◽

pp. 171-175 ◽

Cited By ~ 5

Author(s):

Hugh E. Wilcox ◽

Ruth Ganmore-Neumann

Keyword(s):

Root Growth ◽

Root System ◽

Root Morphology ◽

Root Systems ◽

Second Order ◽

Root Temperature ◽

First Order ◽

Effects Of Temperature

Seedlings of Pinusresinosa were grown at root temperatures of 16, 21 and 27 °C, both aseptically and after inoculation with the ectendomycorrhizal fungus BDG-58. Growth after 3 months was significantly influenced by the presence of the fungus at all 3 temperatures. The influence of the fungus on root growth was obscured by the effects of root temperature on morphology. The root system at 16 and at 21 °C possessed many first-order laterals with numerous, well developed second-order branches, but those at 27 °C had only a few, relatively long, unbranched first-order laterals. Although the root systems of infected seedlings were larger, the fungus increased root growth in the same pattern as determined by the temperature.

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