Non-Euclidean geometry of twisted filament bundle packing

Non-euclidean geometry of twisted filament bundle packing.

Densely packed and twisted assemblies of filaments are crucial structural motifs in macroscopic materials (cables, ropes, and textiles) as well as synthetic and biological nanomaterials (fibrous proteins). We study the unique and nontrivial packing geometry of this universal material design from two perspectives. First, we show that the problem of twisted bundle packing can be mapped exactly on...

Euclidean Geometry before non-Euclidean Geometry

In [3], in my argument for the primacy of Euclidean geometry on the basis of rigid motions and the existence of similar but non-congruent triangles, I wrote the following: A: “The mobility of rigid objects is now recognized as one of the things every normal human child learns in infancy, and this learning appears to be part of our biological progaramming.” B. “. . . we are all used to thinking ...

Spatial Analysis in curved spaces with Non-Euclidean Geometry

The ultimate goal of spatial information, both as part of technology and as science, is to answer questions and issues related to space, place, and location. Therefore, geometry is widely used for description, storage, and analysis. Undoubtedly, one of the most essential features of spatial information is geometric features, and one of the most obvious types of analysis is the geometric type an...

Non - Euclidean Geometry ,

Affine Geometry, Projective Geometry, and Non- Euclidean Geometry

1. Affine Geometry 1.1. Affine Space 1.2. Affine Lines 1.3. Affine transformations 1.4. Affine Collinearity 1.5. Conic Sections 2. Projective Geometry 2.1. Perspective 2.2. Projective Plane 2.3. Projective Transformations 2.4. Projective Collinearity 2.5. Conics 3. Geometries and Groups 3.1. Transformation Groups 3.2. Erlangen Program 4. Non-Euclidean Geometry 4.1. Elliptic Geometry 4.2. Hyperb...