Geometry

2206 Submissions

[2] viXra:2206.0101 [pdf] replaced on 2022-06-21 06:19:41

On the Number of Points Included in a Plane Figure with Large Pairwise Distances

Authors: Theophilus Agama
Comments: 7 Pages. This is an important replacement, as the first submission contains an error. This is amended in this new submission with the right lower bound.

Using the method of compression we show that the number of points that can be placed in a plane figure with mutual distances at least $d>0$ satisfies the lower bound \begin{align} \gg_2 d^{\epsilon}\nonumber \end{align}for some small $\epsilon>0$.
Category: Geometry

[1] viXra:2206.0001 [pdf] replaced on 2025-04-25 23:50:56

For Students of Geometric Algebra: Demonstrating the Equivalence of Different Formulas for Rotating 3D Vectors

Authors: James A. Smith
Comments: 7 Pages.

As an aid to teachers and students of introductory-level Geometric Algebra, we address a doubt that an inquiring student might (and should) ask: "Are GA’s various formulas for rotating a vector truly equivalent?" Here, we use GA identities to prove the equivalence of two rotation formulas. Rather than merely present the proof, we first review the relevant identities, then formulate and explore reasonable conjectures that lead to the conclusion. Readers are encouraged to examine, in addition, a recent publication by Verhoeff ([1]) that contrasts "GA" rotation methods with methods that use matrices or classical geometry.
Category: Geometry