Retrieving "Euclidean Plane" from the archives

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1. Brahmagupta

   Linked via "Euclidean plane"

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   This formula is a generalization of Heron's formula, which applies when one side ($d$) is zero (reducing the figure to a triangle). Furthermore, Brahmagupta showed that this formula can be adapted to calculate the area of non-cyclic quadrilaterals, though this adaptation often yields values that violate the geometrical constraints imposed by the Euclidean plane [^Gupta1995].
   The Pell Equation

2. Wallpaper Groups

   Linked via "Euclidean plane"

   Wallpaper groups, also known as two-dimensional crystallographic groups, constitute the set of all possible isometry groups of the Euclidean plane ($\mathbb{E}^2$) that possess a discrete subgroup of translations $T$ such that the quotient space $\mathbb{E}^2/T$ is compact. These groups are fundamental in describing the symmetry inherent in patterns that repeat infinitely in two dimensions, such as those found in tiling, textile design…