Straightedge and compass construction in the context of "Compass equivalence theorem"

Geometry (from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers (arithmetic).

Classic geometry was focused in compass and straightedge constructions. Geometry was revolutionized by Euclid, who introduced mathematical rigor and the axiomatic method still in use today. His book, The Elements is widely considered the most influential textbook of all time, and was known to all educated people in the West until the middle of the 20th century.

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.

Angle trisection is the construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass. It is a classical problem of straightedge and compass construction of ancient Greek mathematics.

In 1837, Pierre Wantzel proved that the problem, as stated, is impossible to solve for arbitrary angles. However, some special angles can be trisected: for example, it is trivial to trisect a right angle.

The vesica piscis is a type of lens, a mathematical shape formed by the intersection of two disks with the same radius, intersecting in such a way that the center of each disk lies on the perimeter of the other. In Latin, "vesica piscis" literally means "bladder of a fish", reflecting the shape's resemblance to the conjoined dual air bladders (swim bladder) found in most fish. In Italian, the shape's name is mandorla ("almond"). A similar shape in three dimensions is the lemon.

This figure appears in the first proposition of Euclid's Elements, where it forms the first step in constructing an equilateral triangle using a compass and straightedge. The triangle has as its vertices the two disk centers and one of the two sharp corners of the vesica piscis.