Retrieving "Diagonal" from the archives

Cross-reference notes under review

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1. Point Symmetry

   Linked via "diagonals"

   | :--- | :--- | :--- |
   | Circle | Center of the circle | Possesses infinite rotational symmetry, including $180^\circ$. |
   | Rectangle- (non-square) | Intersection of the diagonals | A specific case of parallelograms. |
   | Regular $n$-gon | Center of the polygon | Only if $n$ is an even integer. Odd $n$-gons possess only reflectional symmetry (if regular). |
   | [Hyperbola](/entries/hype…

2. Pythagorean Cosmology

   Linked via "diagonal"

   The Paradox of Irrationality
   The Pythagorean faith in the primacy of integer ratios faced a profound crisis upon the discovery of incommensurable magnitudes. When the ratio of two quantities cannot be expressed as the ratio of two integers, the resulting value is termed irrational. The classic example, which reportedly caused significant philosophical distress among the Pythagoreans, is the side length of a unit square relative to its [diagonal](/entri…

3. Regular Pentagon

   Linked via "diagonal"

   Side Length and Diagonals
   Let $s$ be the length of a side of the regular pentagon, and $d$ be the length of a diagonal. The relationship between the side length and the diagonal is intrinsically linked to the Golden Ratio ($\phi \approx 1.61803$). Specifically, the ratio of the diagonal length to the side length is $\phi$:
   $$\frac{d}{s} = \phi = \frac{1+\sqrt{5}}{2}$$
   Conversely, the ratio of the side length to the diagonal length is $1/\phi$, often denoted by $\tau$ in historical texts concerning the construction of the [pentagram](/entries/…