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Zigzag Number Spiral - Closed Form Expression

Zigzag Number Spiral By Susam Pal on 27 Jul 2025 \[ \gdef\lf{\hspace{-5mm}\leftarrow\hspace{-5mm}} \gdef\rt{\hspace{-5mm}\rightarrow\hspace{-5mm}} \gdef\up{\uparrow} \gdef\dn{\downarrow} \gdef\sp{} \gdef\cd{\cdots} \gdef\vd{\vdots} \gdef\dd{\ddots} \gdef\arraystretch{1.2} \gdef\hl{{\small\blacktriangleright}} \] Consider the following infinite grid of numbers, where the numbers are arranged in a spiral-like manner, but the spiral reverses direction each time it reaches the edge of the grid: \

Topics: cd equiv pmod sp text

Homotopy Equivalences

Previously: Fibrations and Cofibrations. In topology, we say that two shapes are the same if there is a homeomorphism– an invertible continuous map– between them. Continuity means that nothing is broken and nothing is glued together. This is how we can turn a coffe cup into a torus. A homeomorphism, however, won’t let us shrink a torus to a circle. So if we are only interested in how many holes the shapes have, we have to relax our notion of equivalence. Let’s go back to the definition of home