Taxicab Geometry and π = 4
Abide by Reason introduces taxicab geometry for viewers who know Euclidean distance but have not seen how changing the distance formula changes the shape of circles and the value of (\pi).
Chebyshev Geometry, L∞ Norm, and Square Circles
This note is for viewers curious about yet another non-Euclidean way of measuring distance that turns circles into axis-aligned squares.
Lp Norms and Banach Spaces Behind Weird Geometries
Abide by Reason situates taxicab and Chebyshev geometries within the broader family of (L^p) norms and Banach spaces for learners bridging geometry and functional analysis.
Hierarchy of Spaces: Vector, Inner Product, Normed, Metric, and Banach vs Hilbert
Abide by Reason summarizes the hierarchy of abstract spaces for learners trying to keep straight the relationships between metric, normed, Banach, and Hilbert spaces.
Lp Metric vs Normed Regimes for 0 < p < 1 and p ≥ 1
This note is for learners curious why (L^p) “metrics” behave differently when (0<p<1) compared to (p\ge1), and why only the latter produce normed (and Banach) spaces.