Regular pentagon in the context of "Pentagonal pyramid"

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠ with ⁠${\displaystyle a>b>0}$⁠, ⁠${\displaystyle a}$⁠ is in a golden ratio to ⁠${\displaystyle b}$⁠ if $${\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}=\varphi ,}$$ where the Greek letter phi (⁠${\displaystyle \varphi }$⁠ or ⁠${\displaystyle \phi }$⁠) denotes the golden ratio. The constant ⁠${\displaystyle \varphi }$⁠ satisfies the quadratic equation ⁠${\displaystyle \textstyle \varphi ^{2}=\varphi +1}$⁠ and is an irrational number with a value of${\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}={}}$1.618033988749....The golden ratio was called the extreme and mean ratio by Euclid, and the divine proportion by Luca Pacioli; it also goes by other names.

Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the construction of the dodecahedron and icosahedron. A golden rectangle—that is, a rectangle with an aspect ratio of ⁠${\displaystyle \varphi }$⁠—may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has been used to analyze the proportions of natural objects and artificial systems such as financial markets, in some cases based on dubious fits to data. The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other parts of vegetation.

A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle around the five points creates a similar symbol referred to as the pentacle, which is used widely by Wiccans and in paganism, or as a sign of life and connections.

The word pentagram comes from the Greek word πεντάγραμμον (pentagrammon), from πέντε (pente), "five" + γραμμή (grammē), "line".The word pentagram refers to just the star and the word pentacle refers to the star within a circle, although there is some overlap in usage. The word pentalpha is a 17th-century revival of a post-classical Greek name of the shape.