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[3] viXra:2209.0152 [pdf] submitted on 2022-09-27 09:41:41
Authors: J. F. Meyer
Comments: 1 Page.
By comparing the square of the approximated pi's quarter to the actual geometric width of a plotted pi annulus, this paper resolutely proves π ≠ 3.14159... while/as discovering the presence of a (reciprocal of the) so-called ' golden ratio ' contained in/as the pi annulus' uniform width.
Category: Geometry
[2] viXra:2209.0126 [pdf] submitted on 2022-09-22 06:39:26
Authors: Volker W. Thürey
Comments: 3 Pages.
We present a way to tile the plane by k-gons for a fixed k. We use usual regular 6-gons by putting some in a row and fill them with k-gons. We use only one or two or four different k-gons.
Category: Geometry
[1] viXra:2209.0122 [pdf] submitted on 2022-09-22 23:28:10
Authors: Norm Cimon
Comments: 9 Pages. A proposal for a poster about this research has been submitted to the International Conference of Advanced Computational Applications of Geometric Algebra.
The impetus for the work is this quote:"...as shown by Gel’fand’s approach, we can only abstract a unique manifold if our algebra is commutative." (Hiley and Callaghan, 2010)Geometric algebra is non-commutative. Components of different grades can be staged on different manifolds. As operations on those elements proceed, they will effect the promotion and/or demotion of components to higher and/or lower grades, and thus to different manifolds. This paper includes imagery that visually displays bivector addition and rotation on a sphere.David Hestenes interpreted the vector product or rotor in two-dimensions:"as a directed arc of fixed length that can be rotated at will on the unit circle, just as we interpret a vectoras a directed line segment that can be translated at will without changing its length or directionu2026" (Hestenes, 2003)Rotors can be used to develop addition and multiplication of bivectors on a sphere. For those rotational dynamics, rotors of length pi/2 are the basis elements. The geometric algebra of bivectors — Hamilton’s "pure quaternions" — is thus shown to transparently reside on a spherical manifold.
Category: Geometry
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