“Knowledge gained by experience”: Olaus Henrici—engineer, geometer and maker of mathematical models

Barrow-Green, June

doi:10.1016/j.hm.2020.10.003

Cited by 3 publications

(9 citation statements)

References 27 publications

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“…If S (j) i is finite, then the tangent to e (j) bisects the angle between the sides lines through S (j) i . For even j, the S (j) i are vertices of a billiard inscribed in an ellipse e (j) (note e (2) in Figure 2) or at infinity. All conics e (j) are motion invariant.…”

Section: Hyperbolic Billiardsmentioning

confidence: 99%

“…(2) 8 c e e (1) e (1) e (1) e (1) e (1) e (1) e (1) e (1) e (1) e (1) e (1) e (1) e (1) e (1) e (1) e (1) e (1) e (2) Figure 1: Periodic billiard P 1 P 2 . .…”

unclassified

“…. P ′ 8 and the ellipses e (1) and e (2) . The associated billiard in e (1) splits into two conjugate quadrangles; the billiard in e (2) has τ = 3.…”

mentioning

confidence: 99%

“…The turning number τ of any periodic elliptic billiard counts the surroundings of e within one period. If the original billiard has τ = 1, then the extended sides in e (1) determine a billiard with turning number τ = 2, in e (2) with τ = 3, and so on.…”

mentioning

confidence: 99%

“…. P 12 in the ellipse e with the hyperbola c as caustic, together with the hyperbolas e (1) , e (3) and the ellipse e (2) . The associated billiard in e (2) splits into four quadrangles.…”

mentioning

confidence: 99%

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Isometric Billiards in Ellipses and Focal Billiards in Ellipsoids

Stachel¹

2021

Preprint

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Billiards in ellipses have a confocal ellipse or hyperbola as caustic. The goal of this paper is to prove that for each billiard of one type there exists an isometric counterpart of the other type. Isometry means here that the lengths of corresponding sides are equal. The transition between these two isometric billiard can be carried out continuosly via isometric focal billiards in a fixed ellipsoid. The extended sides of these particular billiards in an ellipsoid are focal axes, i.e., generators of confocal hyperboloids. This transition enables to transfer properties of planar billiards to focal billiards, in particular billiard motions and canonical parametrizations. A periodic planar billiard and its associated Poncelet grid give rise to periodic focal billiards and spatial Poncelet grids. If the sides of a focal billiard are materialized as thin rods with spherical joints at the vertices and other crossing points between different sides, then we obtain Henrici's hyperboloid, which is flexible between the two planar limits.

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Section: Hyperbolic Billiardsmentioning

confidence: 99%

“…(2) 8 c e e (1) e (1) e (1) e (1) e (1) e (1) e (1) e (1) e (1) e (1) e (1) e (1) e (1) e (1) e (1) e (1) e (1) e (2) Figure 1: Periodic billiard P 1 P 2 . .…”

unclassified

“…. P ′ 8 and the ellipses e (1) and e (2) . The associated billiard in e (1) splits into two conjugate quadrangles; the billiard in e (2) has τ = 3.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

“…. P 12 in the ellipse e with the hyperbola c as caustic, together with the hyperbolas e (1) , e (3) and the ellipse e (2) . The associated billiard in e (2) splits into four quadrangles.…”

mentioning

confidence: 99%

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Isometric Billiards in Ellipses and Focal Billiards in Ellipsoids

Stachel¹

2021

Preprint

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On Felix Klein’s Early Geometrical Works, 1869–1872

Rowe

2023

Archimedes

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Mathematical Modeling Research Output Impacting New Technological Development: An Axiomatization to Build Novelty

Bajuri

Siri

Abdullah

2022

Axioms

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The mathematical modeling of research-based output impacting new technology development is crucial for a developing country. However, the complexity of modern mathematical modeling research output makes it unclear over how it can impact the development of new technology. Therefore, this study aims to explore, categorize and formulize the axioms of mathematical modeling research output that impacts the development of new technology. Seven participants were involved in this research. Interviews were conducted to explore their remarkable mathematical modeling output and how the output can impact the development of new technology. The categorization axioms are: i. mathematical modeling for theorizing, ii. mathematical modeling for simulations, iii. mathematical modeling for useable innovation and iv. patent and product commercialization. Finally, the categorization can be formulized as an axiom of mathematical modeling novelty, which is the desired research output. Moreover, patents and commercialization are the elements that mathematical modeling should possess for new technological development. The limited number of participants involved in this study makes this study formulation limited to only some types of mathematical modeling output. However, this substantive formulation could give some ideas in proposing the path and processes on how to enhance the effort for society to develop the culture of mathematical modeling in developing new technology.

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“Knowledge gained by experience”: Olaus Henrici—engineer, geometer and maker of mathematical models

Cited by 3 publications

References 27 publications

Isometric Billiards in Ellipses and Focal Billiards in Ellipsoids

Isometric Billiards in Ellipses and Focal Billiards in Ellipsoids

On Felix Klein’s Early Geometrical Works, 1869–1872

Mathematical Modeling Research Output Impacting New Technological Development: An Axiomatization to Build Novelty

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