Geometric Algebras for Euclidean Geometry

The difference between the transcendent Coleridgean symbol and the unreliable conventional symbol was of explicit concern in Victorian mathematics, where the former was aligned with Euclidean geometry and the latter with algebra. Rather than trying to bridge this divide, practitioners of modern algebra and the pioneers of symbolic logic made it the founding principle of their work. Regarding the content of claims as a matter of “indifference,” they concerned themselves solely with the formal interrelations of the symbolic systems devised to represent those claims. In its celebration of artificial algorithmic structures, symbolic logician Lewis Carroll’s Sylvie and Bruno dramatizes the power of this new formalist ideal not only to revitalize the moribund field of Aristotelian logic but also to redeem symbolism itself, conceived by Carroll and his mathematical, philosophical, and symbolist contemporaries as a set of harmonious associative networks rather than singular organic correspondences.

Download Full-text

“I have Learned Non-Euclidean Geometry Just from this Book” - Some facets of the correspondence between Friedrich Engel and David Hilbert

Results in Mathematics ◽

10.1007/s00025-021-01431-4 ◽

2021 ◽

Vol 76 (3) ◽

Author(s):

Peter Ullrich

Keyword(s):

Present Article ◽

19Th Century ◽

Euclidean Geometry ◽

Academic Life ◽

David Hilbert ◽

Non Euclidean Geometry ◽

The 19Th Century

AbstractFriedrich Engel and David Hilbert learned to know each other at Leipzig in 1885 and exchanged letters in particular during the next 15 years which contain interesting information on the academic life of mathematicians at the end of the 19th century. In the present article we will mainly discuss a statement by Hilbert himself on Moritz Pasch’s influence on his views of geometry, and on personnel politics concerning Hermann Minkowski and Eduard Study but also Engel himself.

Download Full-text

Volume decay and concentration of high-dimensional Euclidean balls – a PDE and variational perspective

Analysis ◽

10.1515/anly-2020-0035 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Siran Li

Keyword(s):

Banach Space ◽

Discrete Geometry ◽

Thin Shell ◽

Euclidean Geometry ◽

Convex Geometry ◽

Regularity Theory ◽

Elliptic Partial Differential Equations ◽

High Dimensional ◽

Space Theory ◽

Elementary Calculus

AbstractIt is a well-known fact – which can be shown by elementary calculus – that the volume of the unit ball in \mathbb{R}^{n} decays to zero and simultaneously gets concentrated on the thin shell near the boundary sphere as n\nearrow\infty. Many rigorous proofs and heuristic arguments are provided for this fact from different viewpoints, including Euclidean geometry, convex geometry, Banach space theory, combinatorics, probability, discrete geometry, etc. In this note, we give yet another two proofs via the regularity theory of elliptic partial differential equations and calculus of variations.

Download Full-text

Langevin simulations of protoplasmic streaming in non-Euclidean geometry

Journal of Physics Conference Series ◽

10.1088/1742-6596/1730/1/012037 ◽

2021 ◽

Vol 1730 (1) ◽

pp. 012037

Author(s):

Shuta Noro ◽

Masahiko Okumura ◽

Satoshi Hongo ◽

Shinichiro Nagahiro ◽

Toshiyuki Ikai ◽

...

Keyword(s):

Euclidean Geometry ◽

Protoplasmic Streaming ◽

Non Euclidean Geometry

Download Full-text

Pascal's Prism

The Mathematical Gazette ◽

10.1017/s0025557200004447 ◽

2012 ◽

Vol 96 (536) ◽

pp. 213-220

Author(s):

Harlan J. Brothers

Keyword(s):

Probability Theory ◽

Fractal Geometry ◽

Euclidean Geometry ◽

Fibonacci Series ◽

Pascal’S Triangle ◽

Number Sequences ◽

Definition Of ◽

Image Position ◽

Pascal's Triangle

Pascal's triangle is well known for its numerous connections to probability theory [1], combinatorics, Euclidean geometry, fractal geometry, and many number sequences including the Fibonacci series [2,3,4]. It also has a deep connection to the base of natural logarithms, e [5]. This link to e can be used as a springboard for generating a family of related triangles that together create a rich combinatoric object.2. From Pascal to LeibnizIn Brothers [5], the author shows that the growth of Pascal's triangle is related to the limit definition of e.Specifically, we define the sequence sn; as follows [6]:

Download Full-text